As anticipated, parameter $a_1$ results from imposing the convergence speed towards the sliding-mode regime: $a_1 = -1/T_{sm}$, with $T_{sm}$ being the desired time constant. The value of $a_2$ results from imposing the optimal operating point (OOP) as "target" steady state, corresponding to $\lambda_{opt}$. Thus, by constraining the sliding-mode dynamic (3.21) to have the OOP as equilibrium point, $a_2$ is obtained as $a_2 = -a_1 \cdot \Omega_{h_{opt}}/\Gamma_{G_{opt}}$. A dynamic modification of $a_2$:

$$a_2 = -a_1 \cdot \frac{\Omega_{h_{opt}}}{\Gamma_{G_{opt}} \cdot \left( 1 + k \cdot \frac{\Omega_h - \Omega_{h_{opt}}}{\Omega_{h_{opt}}} \right)}, \quad (3.24)$$

with $k \ge 0$, can be adopted in order to reduce the operating point variations around the OOP.

The argument behind (3.24) relies upon some practical reasons, as follows. Because $(\Omega_{h_{opt}}, \Gamma_{G_{opt}})$ pair depends on the wind speed, the sliding surface (3.22) is variable with the wind speed, and so does parameter $a_2$. Thus, parameter $a_2$ can take in practice sufficiently large values such that the system leaves the normal operating regime. The larger the value of $k$ is, the more quickly the optimal steady state is reached and therefore the ORC tracking has a better quality – real-time simulation results presented further confirm these expectations. The newly added degree of freedom, $k$, serves in this way to adjust the ORC tracking accuracy. However, too aggressive ORC tracking may induce significant control input variations, affecting the reliability. The conclusion is that parameter $k$ can be used to play a role in the design of a desired energy-reliability trade-off. Accordingly, as $k$ increases, the slope of sliding surface’s image in the rotational speed – power plane also increases, corresponding to a tighter ORC tracking. On the contrary, a more reduced value of $k$ is suitable to diminish mechanical fatigue.

The on-off component of the sliding-mode control law, $u_N$, results from the quite habitual choice of the Lyapunov (energy) function as being the square of the obtained sliding surface, $\sigma$ (3.22):

$$u_N = -\alpha \cdot \operatorname{sgn}_h(\sigma), \quad (3.25)$$

where $\operatorname{sgn}_h(\cdot)$ is a hysteretic sign function of width $h$ and $\alpha$ is the amplitude. Finally, the total sliding-mode control law is the sum of the equivalent component (3.23) and the on-off component (3.25): $u = u_{eq} + u_N$.

A low-power (6-kW) rigid-drive-train horizontal-axis-turbine-based WECS was chosen to implement the above described sliding-mode control approach – results shown next are resumed from [65]. Before presenting some real-time simulation results – illustrative from the viewpoint of time evolution of variables of interest – some MATLAB®/SimulinK®results are first presented, which give an intuitive visual representation of the achieved dynamic optimization in the ORC plane. Thus,

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /><figcaption>Figure 3.17: WECS optimized by sliding-mode control MATLAB®/SimulinK®results [65]: response to step changes in the wind speed for $k = 0$: a) state space evolution; b) image of the wind-speed-dependent sliding surface in the ORC plane.</figcaption></figure>

Fig. 3.17 presents simulation results concerning the closed-loop evolution in response to step changes in the wind speed for trade-off parameter $k = 0$ in (3.24). Fig. 3.17a) shows the operating point excursion in the state space, $(\Omega_h, \Gamma_G)$, on a trajectory alternating sliding and non-sliding portions between two optimal steady-state points corresponding to the wind speed of 7 m/s and 8 m/s, respectively. This evolution shows that, once the step in the wind speed takes place, the operating point is attracted to the new sliding surface and then evolves in sliding mode to the new optimal point.
Fig. 3.17b) shows the relative position of the ORC vs. the sliding surface's image in the ORC plane for different wind speeds, as well as the role of $a_2$ in tuning the quality of ORC tracking: **the larger $a_2$ is, the closer to the ORC the operating point evolves.**

Validation results obtained on a **physical simulator** designed according to HILS methodology (see Section 2.3 in Chapter 2) are presented in the next two figures. For sake of practical compliance with the real-time tests, the initial sliding-mode control law had to suffer some modifications, which are briefly explained below.

* The original form (3.23) of the equivalent control input, $u_{eq}$, was computed with a modified version of parameter $a_2$, in order to compensate for losses; thus, $a_2 = -a_1 \cdot \eta_{em} \cdot \Omega_{h_{opt}} / \Gamma_{G_{opt}}$, where $\eta_{em}$ is an estimation of the WECS electromechanical efficiency, having a sigmoid variation with the rotational speed, as established by field measurements.

* The sliding-mode control law uncertainty due to its variability with the constructive parameters – e.g., $A(\lambda, v)$ in (3.23) – can be reduced if increasing the value of parameter $\alpha$ of its on-off component, $u_N$, with consequent increase of the sliding surface's attractiveness, so of the robustness, but meanwhile of the chattering, potentially responsible for unacceptable generator torque/current variations. A practical manner to alleviate this drawback is to use a continuous hysteretic sigmoid (e.g., hyperbolic tangent) function instead of a pure sign (relay) function in (3.25), without significantly affecting the control law robustness.

A set of ControlDesk® captures in Fig. 3.18 illustrate the evolution of the main variables when an energy-reliability optimization by variable speed is implemented in the form of the proposed sliding-mode control.

The trade-off coefficient was set to $k = 12$ and the $u_N$ term is implemented as a sigmoid hysteresis of amplitude $\alpha = 0.3$. The test signal displayed in Fig. 3.18a) is a 2-min pseudorandom wind speed sequence with medium turbulence, computed according to the von Karman spectrum in the IEC standard. The tracking precision of the optimal conversion regime can be estimated from the time evolutions of the tip speed, $\lambda$ (Fig. 3.18b)) and of the power coefficient, $C_p$ (Fig. 3.18c)). Fig. 3.18d) shows a non-uniform effectiveness of the control law along

<figure><img src="image_40.png" /><figcaption>Figure 3.18: WECS optimized by sliding-mode control real-time simulation results [65]: overall behaviour of interest variables.</figcaption></figure>

the concerned operating range, in the $(\lambda, \Gamma_G)$ plane. The ORC tracking precision in the rotational speed – power plane can be viewed in Fig. 3.18e). The evolution of the active power fed into the electrical grid, $P_{grid}$, is plotted in Fig. 3.18f), thus giving indications about the final energy conversion result.

Finally, Fig. 3.19 shows the influence of the trade-off parameter, $k$, on accomplishing the control goal.
<figure><img src="image_41.png" /><figcaption>Figure 3.19: WECS optimized by sliding-mode control real-time simulation results [65]: ORC-tracking-control-effort trade-off as k parameter increases (up to bottom).</figcaption></figure>

Indeed, the ORC tracking precision increases as k increases (Figs. 3.19d)-f)), but with the price of increasing the mechanical stress, as control input (electromagnetic torque) variations also increase (Figs. 3.19a)-c)). Therefore, the possibility of adjusting the newly added degree of freedom, k, confers flexibility to the sliding-mode controlled WECS, so that the wind energy conversion efficiency be significantly increased when the particular conditions of the site allow it (i.e., when the mechanical stress induced by the turbulences is not important).

### 3.2.3 Wind turbulence employed as searching signal in MPPT for WECS

This subsection summarizes the main ideas behind using the turbulence component of the wind speed as searching signal in an extremum-seeking-control (ESC)-based MPPT for a generic class of variable-speed WECS, irrespective of their aerodynamic, transmission and generation configurations [70].

Without loss of generality, the case of a squirrel-cage-induction-generator (SCIG)-based WECS (see Fig. 3.8a)) subjected to MPPT control is taken as an example. The approached system presents a higher level control loop superposed on the generator torque control: the high-speed shaft (HSS) rotational speed control, thus allowing the tip speed ratio, $\lambda$, direct control, according to the ESC principle presented in Fig. 3.5.

It is assumed that the wind speed signal is described by a fixed-point spectrum, with a seasonal component, $v_s$, and a turbulence component, $\Delta v = v - v_s \equiv v_t$, whose intensity varies in the usual range, i.e., [0.12;0.18] [44]. It is also assumed that measures of the wind speed, rotational speed and electrical power are available for feedback. Constructive parameters of the whole mechanical-to-electrical conversion chain, as well as its conversion efficiency, $\eta$, are supposed known, whereas the aerodynamic efficiency curve, $C_p(\lambda)$, is totally unknown. Under this set of assumptions, the considered model of the variable-speed WECS can be sketched as in Fig. 3.20a). The wind turbulences, together with rotational speed variations around a steady-state value, induce some $\lambda$ variations around its average value and consequently some nonharmonic, but bounded-spectrum, power coefficient variations, $\Delta C_p$, whose presence is suggested in Fig. 3.20a).

$\lambda(t)$ can be computed by using measures of the wind speed and rotational speed, as to its definition relation (3.1). Under the previously stated assumptions, the instantaneous value of the power coefficient, $C_p(t)$, can be estimated from the measured electrical power, $P(t)$ – related to the wind turbine power, $P_{wt}$, by means of conversion efficiency $\eta$: $P_{wt} = P(t)/\eta$ – and that of the wind speed, $v(t)$, by using $P_{wt}$ definition relation (3.3):

$$C_p(t) = \frac{P(t)}{0.5 \cdot \eta \cdot \rho \cdot \pi R^2 \cdot v(t)^3} \qquad (3.26)$$

The main idea of this approach relies upon how the available feedback information is further used. If supposing a Fourier decomposition of $\lambda(t)$ and $C_p(t)$ signals, then each $\lambda(t)$ harmonic component will generate a response
which is a part of $C_p(t)$. Next, by composing/averaging the effects of the respective components of the two Fourier decompositions, the operating point (OP) position on the $C_p$ curve can be obtained by *demodulation*.

<figure><img src="image_42.png" /><figcaption>Figure 3.20: Illustration of main ideas and concepts of WECS MPPT control using wind turbulence as searching signal [29], [70]: a) principle of WECS modelling and control; b) feedback processing to deduce the operating point position; c) block diagram of the control structure: feedback information processing and generation of the rotational speed reference.</figcaption></figure>

In particular, demodulation is here implemented by using the *Discrete Fourier Transform* (DFT) on an appropriately chosen time window, in order to extract the phase of each harmonic component of $\lambda(t)$ and $C_p(t)$ and then compute the phase lag between them. An average of these phase lags, further denoted by $\theta(t)$, contains the OP average position information. Values of $\theta(t)$ will get closer to 0 if the OP is on the left (rising) slope of the $C_p(\lambda)$ curve or closer to $\pi$ if the OP is on the right (falling) one. Hence, Zone 1) from Fig. 3.20b) will be characterized by $\theta < \pi/2$ and Zone 2) by $\theta \ge \pi/2$.

The control law main idea is **to move slowly** (as compared to the turbine's dynamics) **the average OP to the top of the aerodynamic efficiency curve**, $C_p$, **by regulating the average value of the tip speed**, $\lambda$, computed on a relatively large time window, of width $T_C$. Thus, the control is implemented using an **integrator** fed by the OP position information, which gives the **search direction** and whose **gain**, denoted by $k$, represents the **search step**. The control structure, presented in Fig. 3.20c), has two main parts, namely an information processing block and a rotational speed reference generator.

The role of the **information processing block** is to output the $C_p(t)$ and $\lambda(t)$ normalized signals – obtained from the electrical power, rotational speed and wind speed measurements – to further feed a FFT algorithm for obtaining their phase spectrum and the corresponding average OP position signal, $\theta(t)$. For information consistency, a sufficiently large window (compared to the turbulence dynamics) must be chosen. The measure information is updated at each sampling period, denoted by $T_S$, whose choice depends mainly on the turbine dynamics. A time window of width $T_C = 2^n \cdot T_S$ can then be a reasonable choice for FFT computation, where $n$ is a positive integer resulting from a trade-off between suitably tracking the wind speed's variations and complying with the system inertia. It is obvious that a given system will not be ever made to track an unlimitedly turbulent wind.

As the OP position changes smoothly around the optimal one (OOP), the role of the **rotational speed reference generator** – at the right of block diagram in Fig. 3.20c) – starts with a preliminary processing of $\theta(t)$ around the OOP. Note first that, when the OP is placed on the positive slope of the $C_p$ curve, the integrator will process a positive value, thus determining the tip speed reference to increase. In contrast, when the OP is on the negative slope of the $C_p$ curve, the tip speed reference must diminish. This can be achieved by integrating a nonlinear function of average phase information, $\operatorname{sgn}(\theta - \pi/2)$. As the slope of $C_p$ is continuous, instead of discontinuous feedback information in the form of the sign function a continuous, e.g., a hyperbolic tangent, approximation of the sign function can be used in practice. The result of this operation, applied to the integrator, produces a tip speed reference, renewed every $T_C$ seconds after the computation of the phase shift, $\theta$. The control structure finally outputs the rotational speed reference, computed based upon the tip speed reference.

The integrator gain, $k$, is a key parameter, whose tuning relies upon both the chosen $T_C$ and the desired search
speed. Some large deviations of the instantaneous tip speed, $\lambda$, around its optimal value, $\lambda_{opt}$, are possible, whereas its average value should remain close to $\lambda_{opt}$. Thus, a good performance of this control structure can be concluded upon a sufficiently small standard deviation of $\lambda$ around its average value. The value of $k$ can thus be chosen such that the variable-speed turbine compensates $\lambda$ standard deviation in the computing time $T_C$.

Some numerical simulation results are shown below, which synthesize the main validation aspects detailed in [70] for a low-power (6 kW) variable-speed fixed-pitch rigid-drive-train WECS, having $\lambda_{opt} \approx 7.2$ and $C_p(\lambda_{opt}) = 0.475$.

Figure 3.21: Performance of the ESC-based MPPT control with wind turbulence as searching signal [70]: a) the 50-min wind speed sequence used as excitation signal; b) achieving energy optimization by tracking the ORC, with integrator gain $k = 0.02$. A wind sequence of about 8 m/s of average speed, with a medium turbulence intensity of $I = 0.15$ (Fig. 3.21a)), obtained using the von Karman spectrum in the IEC standard, was chosen to represent a sufficiently rich and challenging excitation signal. Fig. 3.21b) shows the operating point's distribution around ORC (in the speed-power plane), showing a better ORC tracking accuracy of the wind-turbulence-based MPPT control law as compared with the classical MPPT version (see [70] for details).

Figure 3.22: Influence of the integration gain, $k$, on the quality of ESC-based MPPT with wind turbulence as searching signal [70]: a) tip speed's density of probability; b) $C_p$ evolution in relation to $C_p(\lambda_{opt}) = 0.475$. Different values of the control parameter, $k$, in the range from 0.005 to 0.5, were tested in simulation for closed-loop performance. Fig. 3.22a) shows how the tip speed's density of probability, $p(\lambda)$, is changing. For small $k$ – corresponding to slow searching – smaller $\lambda$ standard deviations are obtained, but the average value differs from the optimal one, which denotes poor energy efficiency. For large $k$ – i.e., large searching speed – the standard deviation becomes large despite the mean of the tip speed remaining close to the optimal value, thus also indicating poor efficiency. Fig. 3.22b) suggests that, once the system is in steady-state regime, a small value of $k$ can produce better performance. These remarks suggest the existence of a domain of $k$ values for which optimality close tracking can be successfully designed: a good tracking of the maximum power along with a sufficiently small variance.

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>
## 3.3 Control of micro-hydro energy conversion systems (miHECS)

Out of the renewable-energy conversion systems (RECS) with demonstrated potential as sustainable alternative to the classical electrical power generation technologies, the *new hydraulic capturing devices* aiming at harvesting the energy of *marine currents and river flows* represent a distinct class having received significant attention in the energy transition context [71]–[73].

Like in the case of all kind of RECS, and especially of *wind turbines*, the power generation by the *water turbines* should take place with minimal environmental impact, involving the use of limited-span civil engineering structures for conditioning the incoming flows. The water-turbine generated power can be controlled by using pitchable blades or by implementing the concept of *variable-speed operation of the electrical generator*, thanks to the associated power electronic interface. As stated in the RECS-control introductory Section 2.1 in Chapter 2 of this memoir, generator control is important in both *off-grid* mode (stand-alone microgrid) [74] and also in *grid-connected* applications, for providing ancillary services, such as improving power system stability or damping inter-area oscillations [75], [76].

Water-turbine-based generation systems installed in rivers, generally named as **microhydro energy conversion systems (miHECS)**, use various application-dependent capturing devices, of either classical design – such as (semi) Kaplan or Francis water turbines – or more modern trends, such as axial-flow, vertical-axis or cross-flow water turbines (CFWT) [77]–[79]. The free-water-flow operation invariantly requires use of general and already well-established RECS technology concepts, such as generator vector control, variable-speed control, MPPT, etc. [72], [80], [81].

The contributions synthesized in this section were obtained for miHECS being based on an original concept of **cross-flow water turbine (CFWT)**, the *Achard turbine*, developed and patented at the Laboratoire des Écoulements Géophysiques et Industriels (LEGI) in Grenoble, France [82].

CFWTs represent variations of Darrieus turbines [83], with several advantages vs. the axial-flow ones (smaller cavitation [84], simpler mechanical structure, more flexible power generation system), as well as several drawbacks (pulsating turbine torque, lower starting torque and lower conversion efficiency). In this context, control contributions briefed here were focused on fully using the advantages of CFWTs, while minimizing their drawbacks.

Fig. 3.23a) presents the three-bladed vertical-axis Achard turbine resulted as an evolution of Darrieus and Gorlov turbines: it has flying-wing-shaped blades fixed on the rotation axis by means of profiled central arms [82], [86]. This basic prime mover was further used within more complex harvesting structures in order to achieve better performance and to overcome the drawbacks inherent to vertical axis turbines; thus, a complex power take-off device was conceived through successive steps, as follows.

First, *four three-bladed CFWTs* were pilled-up into the same shaft to yield a **CFWT tower**. In order to smooth the mechanical torque, each turbine was installed with angular position shifted with $\pi/2$ with respect to the previous one, thus achieving a spatial filtering of the output power, using the same idea as in [87], [88]. In this way the CFWT tower power coefficient is almost constant irrespective of the tower's position with respect to the water flow direction.

A CFWT tower directly drives an electrical generator, situated in its lower part, as visible in Fig. 3.23b); the same figure shows that **two CFWT towers rotating in opposite senses** are further placed side by side for purpose of structural loads alleviation in certain operating regimes [86]. Finally, the two CFWT towers are encompassed by fairings to increase the water upstream-*vs*.-downstream differential pressure and, consequently, the captured power under certain water conditions.

<figure><img src="image_47.png" /><figcaption>Figure 3.23: Considered CFWT-based hydrokinetic harvesting device: a) basic concept of three-bladed Achard CFWT with straight blades; (b) power take-off structure in the form of two adjacent CFWT towers, each with four pilled-up turbines [85].</figcaption></figure>
It is the **CFWT tower** that plays the role of **prime mover** for the class of miHECS considered here. Contributions summarized next were obtained during my 2008–2009 post-doctoral internship at G2ELab, within the collaboration with Ph.D. students Ana-Maria ANDREICA-VALLET and Matthieu HAUCK and post-doctoral fellow Iulian MUNTEANU, working under the supervision of Professors Daniel ROYE and Seddik BACHA. It is worthy to note that validation of the different proposed control strategies spanned all the phases, from preliminary checking by MATLAB®/Simulink® numerical simulation, passing through HILS validation on a test bench at G2ELab – with the help of R&D engineer Axel RUMEAU – and finishing with prototype implementation on a pilote site near Grenoble, within the collaboration framework between the two Grenoble research laboratories, LEGI and G2ELab. All these contributions regard electrical generator control exploiting the variable-speed operation capability – they are as follows.

* An original method of direct active power control of a CFWT-tower-based miHECS in the stall region – characterized by unstable dynamic behaviour – referring to grid-connected (otherwise said, *PQ*) operation, was proposed, investigating the possibility of operating the miHECS as a dispatchable power generation source despite absence of a storage unit. Main results, synthesized in Subsection 3.3.1, were published in:

M. Hauck, I. Munteanu, A.I. Bratcu, S. Bacha, D. Roye (2011). Operation of grid-connected cross-flow water turbines in the stall region by direct power control. *IEEE Transactions on Industrial Electronics*, 58(4), 1132–1140.

* As a miHECS – like any other RECS employed within a microgrid (recall the general topology in Fig. 2.1 in Section 2.1 and the control-oriented RECS characterization in the same section) – may be required to switch between the grid-forming (*stand-alone*) and the grid-following (*grid-connected*) modes, systematic design of some proper management of "smooth" switching between the two modes – *Vf* and *PQ* – was approached. Subsection 3.3.2 briefs these results, resumed from:

M. Andreica Vallet, S. Bacha, I. Munteanu, A.I. Bratcu, D. Roye (2011). Management and control of operating regimes of cross-flow water turbines. *IEEE Transactions on Industrial Electronics*, 58(5), 1866–1876.

* *Synchronization of two CFWT towers* for some global purpose, e.g., alleviating mechanical efforts inside of a configuration like the one in Fig. 3.23b), was extensively investigated, the towers being also designed as *twin towers* in this case. Results concerning this topic – grouped together in Subsection 3.3.3 – made the object of several publications, starting by a patent explaining the main idea of synchronizing two rotating devices, as being *inspired and adapted from communications technology*:

A.M. Andreica, S. Bacha, A.I. Bratcu, J. Guiraud, I. Munteanu, D. Roye (Grenoble Institute of Technology, Electricité de France) (2009). *Hydraulic turbomachine for recovering and converting kinetic energy of water currents into electricity, has regulation units electrically regulating power from generator to ensure that shaft rotates at same speed relative to another shaft*. Patent WO2010020735-A1, FR2935159-A1,

a journal article reporting on some HILS validation results:

M. Vallet, I. Munteanu, A.I. Bratcu, S. Bacha, D. Roye (2012). Synchronized control of cross-flow-water-turbine-based twin towers. *Renewable Energy*, 48, 382–391

and the journal article:

M. Hauck, A. Rumeau, I. Munteanu, A.I. Bratcu, S. Bacha, D. Roye (2018). Identification and control of a river-current-turbine generator – application to a full-scale prototype. *IEEE Transactions on Sustainable Energy*, 9(3), 1365–1374,

which presents results of *on-site control validation* – also including other control methods, like the MPPT – performed on a *prototype of CFWT-tower-based miHECS* installed on a headrace canal feeding a hydroelectric plant near Grenoble in France.
### 3.3.1 Direct active power control of a cross-flow-water-turbine (CFWT)-based miHECS

Despite its obvious interest, RECS power control is a topic that is seldom dealt with because the primary resource intermittence prevent such systems from being fully dispatchable – that is, able to guarantee a certain imposed level of the provided power when grid connected – unless complemented by storage units.

Control methods ensuring that a low-impedance grid or a load is fed at constant power are reported in the case of hybrid photovoltaic/wind RECS, possibly with energy storage devices [89], [90].

In general, power control of a generation unit concerns its grid-connected – otherwise said, grid-following or PQ – operation mode. To fix ideas, a grid-connected CFWT-tower-based miHECS equipped with a permanent-magnet synchronous generator (PMSG) and a back-to-back AC-DC-AC power-electronic converter interfacing the grid is here considered (Fig. 3.24 [91]). For this topology, control roles are as follows: the grid-side converter ensures the transfer of the entire available DC-link power to the AC grid by regulating the DC-link voltage, $V_{DC}$, while the machine-side converter is used to control PMSG power by means of its three-phase currents.

Modeling of water turbines is similar to the one employed for wind turbines. Thus, the hydrodynamic model of a CFWT tower is based on its power coefficient characteristic, $C_p(\lambda)$ in Fig. 3.25a), having the same allure as the one of a single Achard turbine [92].

The extractible power, $P_T$, depends on the water flow speed cubed and on the turbine parameters in a manner similar to wind turbines (see also (3.3) in Section 3.1):

$$P_T = 0.5 \cdot \rho \cdot S \cdot w^3 \cdot C_p(\lambda), \quad (3.27)$$

where $\rho$ and $w$ are this time the water density and speed, respectively, $S$ is the surface swept by the turbine blades – $S = 2R_T H_T$, with $R_T$ and $H_T$ being the turbine radius and height, respectively – and the tip speed ratio, $\lambda$, preserves the same definition as in the case of wind turbines: $\lambda = R_T \cdot \Omega_T / w$, with $\Omega_T$ being the tower’s rotational speed.

Fig. 3.25b) shows a tower’s mechanical torque, $T_T$, and power, $P_T$, curves for a constant water flow speed. It is assumed that all four turbines composing the tower have the same mechanical torque characteristic, so the tower output torque increases proportionally with the number of turbines in the tower. Also, the water flow speed variations are considered very slow in relation to the system dynamics; however, flow turbulences (low-amplitude high-frequency variations) are likely to be present.

For considerations of lower mechanical and hydrodynamic stresses, the system is likely to be operated at low rotational speeds, in the so-called hydrodynamic stall region, i.e., in the $[0, \Omega_{Pmax}]$ range, identifiable in Fig. 3.25b), where $\Omega_{Pmax}$ corresponds to the maximum power available for a certain value of water flow speed.

<figure><img src="image_48.png" /><figcaption>Figure 3.24: Topology of a grid-connected PMSG-based miHECS with Achard CFWT tower [91].</figcaption></figure>

<figure><img src="image_49.png" /><figcaption>Figure 3.25: Mechanical characteristics of a CFWT tower: a) power coefficient vs. tip speed ratio; b) torque and power curves vs. rotational speed for a given water flow speed [91].</figcaption></figure>
<figure><img src="image_50.png" /><figcaption>Figure 3.26: CFWT tower torque-vs.-rotational-speed curves for various water speed values [91].</figcaption></figure>

Fig. 3.26 depicts the family of torque-vs.-rotational-speed curves when the water flow speed varies. $T_T (\Omega_T)$ can be linearized for a given water flow speed value, around a certain operating point, leading to a C-slope linear approximation. C is strongly variant, depending on the water flow speed, being either positive or negative, according to the operating point relative position on the mechanical characteristic with respect to the maximum torque point. Hence, the stall region is characterized by positive values of slope C, as in this case operating points are likely to be placed on the left, ascending branch of $T_T$ curve (see also Fig. 3.25b)).

As the PMSG is directly driven by the CFWT tower, a first-order linear dynamic of rotational speed, $\Omega_T$, results from its swing equation involving the linearized version of mechanical torque, $T_T$, and PMSG electromagnetic torque, $T_G$, in the form:

$$ \Omega_T(s) = -\frac{1}{Js-C} \cdot T_G(s), \qquad (3.28) $$

with J being the inertia of the tower-PMSG coupling. System (3.28) is **unstable** over most of the operating range of interest (**stall region**), as C is positive. Whereas employing a rotational speed regulatory control is the usual way to solve this problem, our contribution consists in **only stabilizing the rotational speed using generator current as the control input**. The argument behind this idea is simple, as follows. The classical dq-frame modelling of the ensemble PMSG-machine-side-converter [67], [68] allows the PMSG torque control by means of the quadrature current, $i_q$, the direct-current component, $i_d$, being maintained at zero; thus, $T_G = -K_g \cdot i_q$, where $K_g$ is the torque constant. The current loop dynamic being the fastest within the system, the current value may safely be taken as equal with its reference, $i_{q-ref} = i_q$. Hence, a first-order transfer from current $i_q$ to $\Omega_T$ results:

$$ \Omega_T(s) = \frac{K_g}{Js + D} \cdot i_q(s), \qquad (3.29) $$

where $D < 0$ is mainly given by the slope C of the $T_T (\Omega_T)$ curve, but may also contain a friction coefficient.

Following the above arguments, the reference of the $i_q$ PMSG current is obtained as the algebraic sum of a **stabilization** component, noted by $i_{qs}$, and a **regulation** component, noted by $i_{qc}$:

$$ i_{q-ref} = -i_{qs} + i_{qc}, \qquad (3.30) $$

where the two components, although they have current dimensions, do not represent variables with a physical meaning, but their sum plays the role of reference of the $i_q$ control loop.

Whereas $i_{qs}$ results from suitably selecting a stabilizing feedback gain, computation of $i_{qc}$ is based on having experimentally proved the remarkable property of $P_{DC}$ ($i_{qc}$) static curve of being **unimodal**, i.e., having the same aspect as the power-vs.-rotational-speed curve (Fig. 3.25b)). This remark allows envisaging that power control – including the MPPT – can be achieved directly in relation to this current input. Thus, an **upper-level, slower DC-link power control loop** can be designed to yield $i_{qc}$ as the output of a PI controller aiming at regulating $P_{DC}$ at an imposed level, $P_{DC-ref}$. Consequently to having identified a first-order open-loop transfer function $i_{qc} \to P_{DC}$, penalized by a non-minimum-phase behaviour due to presence of an "unstable" zero, PI controller design results from imposing some desired closed-loop second-order dynamics, where the non-minimum-phase behaviour is still present, but it is strongly attenuated. Open-loop identification and closed-loop design, as well as the global miHECS direct power control block diagram, are detailed in [91].

**HILS validation** results presented next are also resumed from [91]; they illustrate direct power control of a CFWT-tower-based miHECS of 1825 W rated power at 2.3 m/s of water flow speed.
<figure><img src="image_51.png" /><figcaption>Figure 3.27: CFWT-tower miHECS direct power control closed-loop performance (HILS oscilloscope captures) [91]: a) Ch 1 and Ch3 – DC-link power measure, $P_{DC}$, and reference, $P_{DC,ref}$, respectively (250 W/V); Ch 4 – PMSG rotational speed, $\Omega_T$ (50 rad/s/V); b) Ch 1 – regulation current, $i_{qc}$ (50 A/V); Ch 2 and Ch3 – generator current measure, $i_q$, and reference, $i_{q,ref}$, respectively (5 A/V); Ch 4 – stabilization current, $i_{qs}$ (50 A/V).</figcaption></figure>

Fig. 3.27a) shows the performance of the direct power regulation loop when the power demand steps up from 150 to 700 W and backwards, for a water flow speed of 2 m/s. The about-12-s power transient still keeps a non-minimum-phase behaviour, although much reduced. Note that the DC-link power controller is here a pure integral gain; a complete PI controller offering a second degree of freedom can be used if a faster response is aimed at. The rotational speed remains at low values on the left side of the power characteristic (it has the same sense of variation as the captured power).

Fig. 3.27b) shows the variations of the control variables, i.e., the stabilization and regulation currents, $i_{qs}$ and $i_{qc}$, respectively, their sum, which is the $i_q$ reference, as well as the measured value of $i_q$. The stabilization current, $i_{qs}$, evolves synchronously with the rotational speed, its absolute value is much higher than the one of $i_{qc}$, so the resulting quadrature current, $i_q$, is always negative (which is coherent with PMSG working as a generator).

To conclude, one of the significant merits of contribution in [91] concerns the broader impact it can have for a larger class of rotating-prime-mover-based RECS. Indeed, experimental validation of the considered miHECS configuration on a HIL simulator allowed an analogy with the operation of WECS, namely by identifying the same operating regions, i.e., partial load, full load and stop at the cut-out water flow speed (see Fig. 3.2 in Section 3.1). This allows the statement of basic control goals for CFWT-based systems in a similar manner as for WECS. Note that water turbines operation is more restrictive than wind turbines operation – as the formers have no pitchable blades and they cannot be put out of the water flow – and even more, one might not expect that the water flow speed could decrease to zero. Therefore, all basic control goals – i.e., power control, rotational speed limitation, MPPT, start and stop – must be entirely achieved by means of the electrical generator control. The direct power control framework in [91] opens the way towards implementing all

miHECS basic control goals in a simple manner, involving minimal tuning effort, irrespective of the water flow speed value and without measuring it.

### 3.3.2 Supervision of smooth switching between operating modes of miHECS

In this subsection, focus is on the management of the switching between the two main operating modes of a CFWT-tower-based miHECS: the grid-connected (grid-following or PQ) mode – when the system is required to provide imposed levels of active and reactive power to a "strong" grid – and islanded (stand-alone, grid-forming or Vf) mode, when the system must supply an isolated-load-demanded power and ensure the desired output voltage amplitude and frequency. The same PMSG-based generation configuration is considered. A synthetic control-engineering practical viewpoint of how a miHECS should be controlled across the main operating regimes in order to preserve its continuity of service is aimed at here.

The CFWT miHECS architecture in Fig. 3.28 shows potential for a large application field, its modularity
allowing adaptation to either river or tidal flows and both PQ and Vf operation.

<figure><img src="image_52.png" /><figcaption>Figure 3.28: CFWT-tower-based miHECS and its control structures for both PQ and Vf modes, emphasizing supervisor's role in switching between the two [93].</figcaption></figure>

The prime mover is rigidly coupled to the PMSG. A back-to-back three-phase PWM power electronic converter implements the interface between the generator and the grid/load. Free-fluid-flow conditions are assumed, the tower being driven by a river flow. When operating in stand-alone (Vf) mode, the generator feeds a mainly resistive load through an LC filter. The Vf operation may not require storage devices in a first place if, in a preliminary design phase, load supply is guaranteed for the lowest water speed value. Indeed, the storageless operation is possible as discontinuities of the water resource are less probable than, for example, those of the wind, and its relatively slow variations can be reliably predicted.

The control roles assigned to each of the two power electronic converters in Fig. 3.28 are specific to each of the operating modes. The control structure including blocks drawn with a dashed line concerns the PQ mode, while the blocks drawn with a solid line represent controllers enabled within Vf mode – details concerning control goals and design can be found in [94] and [95], respectively.

Thus, in PQ mode, the grid-side converter regulates the DC-link voltage, whereas the generator-side converter controls the turbine's rotational speed (e.g., MPPT in its well-known hill-climbing version explained in Fig. 3.4 and Table 3.1). The rotational speed control is preferred to the torque control as it can ensure stable operation of CFWTs at low speeds (see, e.g., details in previous Subsection 3.3.1 and [91]), as well as overspeed regime being avoided [84]. In Vf mode, the load-side converter ensures the load voltage and frequency regulation, while the generator-side converter keeps the DC-link voltage at the required value. Hence, in this mode, the turbine's rotational speed is no longer feedback-controlled.

<figure><img src="image_53.png" /><figcaption>Figure 3.29: Switching between the miHECS operating modes [93]: a) diagram of transition between operating modes; b) contactor signals; c) regulations enabled; d) time evolution of the smooth switching weights.</figcaption></figure>
The **supervisor** visible in Fig. 3.28 implements an upper-level control, its role being to ensure the smooth switching between the *PQ* and *Vf* control strategies when required, consequently switching the system output between the grid and the load and providing the controlled variables' setpoints to the low-level controllers. Binary-state contactors $K_1$ and $K_2$ achieve the connection/disconnection of the grid/load. Signals $S_1$ and $S_2$ enable the low-level controllers corresponding to the *PQ* and *Vf* operating modes, respectively.

Next, the **soft switching procedure** implemented in the supervisor is briefly explained. Transition is smoothed out by allowing certain switching time intervals, instead of instantaneously switching. A quite common control engineering practice to do this is **to weight references sent to the low-level controllers** by some signals $w_1$ and $w_2$, which are set to vary in opposite senses during a conveniently chosen time interval $\tau_3$ such that $w_1+w_2 = 1$.

Fig. 3.29a) shows the **transition diagram** between the two operating modes, where the intermediary states correspond to the necessary delays introduced for sake of smoothness. Thus, concerning the $PQ \rightarrow Vf$ transition, supplementary delays $\tau_1$ and $\tau_2$ allow overlapping of the electrical loads and hence limitation of the output current decreasing and of the DC-link voltage excessively increasing. The drawback of this choice is a nonnegligible delay when performing this transition (several hundreds of milliseconds).

In Fig. 3.29a), six **triggering events** were represented, whose meanings are as follows.

* Events $e_1$ and $e_2$ correspond to asynchronous decisions of the upper-level user/operator of passing from *PQ* mode to *Vf* mode and *vice versa*, respectively.

* Unlikely, events $e_3$, $e_4$, $e_5$ and $e_6$ are synchronous with the time.

Event $e_3$ marks the end of time interval $\tau_1$ passed since entering in state $IS_1$; the same holds for event $e_4$, time interval $\tau_2$ and state $IS_2$.

Events $e_5$ and $e_6$ correspond to the end of transients of signals $w_1$ and $w_2$: the former for $w_1$ decreasing and $w_2$ increasing and the latter for $w_1$ increasing and $w_2$ decreasing.

The **time switching sequences** from *PQ* mode to *Vf* mode and backwards can be viewed in Figs. 3.29b)–d). Time intervals that the system has to spend within each of the intermediary states $IS_1$–$IS_4$ may result from formulating an optimal control problem under state constraints according to, for example, the formalism developed in [96], which would require in this case both *system stability being preserved and DC-link voltage and output current magnitude constraints being respected during switching*. However, adopting **some covering values** for $\tau_1$–$\tau_4$, instead of optimal switching moments – according to some rather **qualitative** guidelines than to precisely quantitative values – can be a sufficiently convenient practical choice in this case.

* Thus, time intervals $\tau_1$ and $\tau_2$ can be chosen such that to allow canceling the load transients when coupling the load into the system ($K_1 = 1, K_2 = 1$, state $IS_1$) and respectively decoupling the system from the grid ($K_1 = 0, K_2 = 1$, state $IS_2$). In this case, $\tau_1 = \tau_2 = 0.1$ s is considered long enough to ensure load transient cancelling.

* As regards the time intervals $\tau_3$ and $\tau_4$ during which the control laws are switched, their selection is made in relation to the slowest closed-loop dynamic of the electrical variables, i.e., that of the DC-link voltage, $U_{DC}$. Thus, $\tau_3 = \tau_4 = 0.5$ s is sufficiently long for obtaining quite smooth transients on $U_{DC}$ along with reasonable control effort, that is, maintaining the generator's $q$ current, $i_{qG}$, and the inverter's current, within reasonable limits.

This subsection ends with some **HILS validation** results resumed from [93], which illustrate effective smooth $PQ \rightarrow Vf \rightarrow PQ$ switchings (Fig. 3.30) and their coherence with the different above-selected switching time intervals. The HILS test bench is based on a **CFWT tower of 2.3 kW rated power at water speed of 2.3 m/s**.

In Fig. 3.30a), a global view of the *a*-phase grid-side variables and the DC-link voltage evolutions are presented, with grid voltage and current zeroing when in *Vf* mode and their quite soft transients at commutation instants. Zoomed images of the same variables for both ways of switching are shown in Figs. 3.30b) and c), while emphasizing the switching timing.
<figure><img src="image_54.png" /><figcaption>Figure 3.30: Variable evolutions during $PQ \rightarrow Vf$ and $Vf \rightarrow PQ$ switchings (HILS oscilloscope captures) [93]: a)–c) Ch1 – grid voltage, $V_{G,a}$ (25 V/V); Ch2 – inverter duty cycle, $u_{L,a}$ (0.1/V); Ch3 – grid current $i_{L,a}$ (5 A/V); Ch4 – DC-link voltage, $U_{DC}$ (75 V/V); b) and c) are left and right zoom outs of a), respectively; d) – f) Ch1 – load voltage, $V_{ch,a}$ (25 V/V); Ch2 – quadrature PMSG current, $i_{qG}$ (5 A/V); Ch3 – load current $i_{ch,a}$ (5 A/V); Ch4 – PMSG rotational speed, $\Omega_G$ (50 rad/s/V); e) and f) are left and right zoom outs of d), respectively.</figcaption></figure>

Evolutions of both PMSG-side (mechanical) and load-side (electrical) variables in response to mode switching are exhibited in Fig. 3.30d). As expected, the system behaves differently when switching as $PQ \rightarrow Vf$ than $Vf \rightarrow PQ$, due to its genuine nonlinearity. Internal variables – such as q-component of generator's current, $i_{qG}$, rotational speed, load current and voltage – remain within admissible limits during switching. Zoomed images are shown in Figs. 3.30e) and f), where the transient $PQ \rightarrow Vf$ appears to be longer than expected.

### 3.3.3 Master-slave synchronized operation of miHECS for structural loads alleviation

In this subsection the miHECS under study is based on *two twin Achard CFWT towers* as prime movers, operating in a river stream and subject to *an imposed restriction between their angular positions* in order to achieve improved hydrodynamic performances, e.g., structural loads alleviation. The need for synchronization can be effectively achieved by a variable-speed-based control approach inspired from *phase-locked loop (PLL) synchronization techniques* used in electronics and communications [97], that is, without using any power coupling (mechanical or electrical) between the two towers.

The main idea, detailed in our patent [87], relies upon considering the CFWT tower as a system outputting a rotational position. Being speed-controlled, each tower can consequently be modelled as a frequency-controlled oscillator with a non-negligible dynamic, thus justifying application of PLL principle for synchronization purpose. Oscillators’ synchronization is a topic of general interest, with extensive applications in all fields of electrical engineering [98]–[100].

The considered CFWT-twin-tower miHECS architecture is depicted in Fig. 3.31a), where each tower is rigidly coupled to its associated PMSG and individually interfaced to the power grid by means of a back-to-back three-phase converter; hence, the system operates in PQ mode.
Ensuring that the two juxtaposed towers turn at the same speed, but in opposite senses, leads to practically nullifying the global lift force action and increasing the hydrodynamic conversion efficiency [82], [84]. The control goal is thus stated as *keeping equal the two rotational speeds, meanwhile ensuring the opposite symmetry of the blades*. Fig. 3.31b) shows a schematic upper view of the twin towers, emphasizing their relative angular position lag, noted by $\alpha$.

As the water speed varies quite slowly, the time response performance is sacrificed for sake of reliability wherever possible, priority being given to guaranteeing the system's continuity of service based upon initial minimal control tuning effort. Within an exclusively-information-coupled *master-slave* configuration, the control problem is *to control the slave tower to track the master tower's rotational speed, while simultaneously regulating their relative angular position lag at a desired constant value*.

Modelling and control design are based upon the master-slave cascade control block diagram in Fig. 3.32a), where the master tower is controlled in MPPT – e.g., the conventional *hill-climbing* version recalled in Section 3.1 – both towers being equipped with rotational speed control loops and incremental optical encoders to convey the feedback signals. All "1" indices refer to the master and the "2" indices refer to the slave. The *phase detector* senses the occasionally occurring master-slave angular position difference, which is fed to the loop filter. This latter provides the rotational speed difference, $\Delta\Omega$, necessary to adjust the slave rotational speed reference such that to zeroing the difference.

<figure><img src="image_55.png" /><figcaption>Figure 3.31: CFWT twin towers operation [88]: a) simplified architecture; b) upper view emphasizing the position lag, $\alpha$.</figcaption></figure>

<figure><img src="image_56.png" /><figcaption>Figure 3.32: Modelling and control structure for purpose of CFWT towers synchronization [88]: a) master and slave towers modelled as two coupled cascade-control systems, emphasizing their respective PMSGs' controls; b) linearized model of the slave tower position control loop.</figcaption></figure>

As the PMSG is torque controlled, the tower-PMSG coupling linearized dynamic is described by a first-order transfer function of form $\frac{1}{Js-C}$ (see also (3.28) in previous Subsection 3.3.1), where $J$ is the tower-PMSG coupling
inertia and $C = \partial T_T / \partial \Omega$ is the slope of the tower-torque-speed $T_T(\Omega)$ curve for a given value of water speed (see Fig. 3.26 in the current subsection). A rotational speed PI controller can be designed such that to obtain good performance on both the right side of $T_T(\Omega)$ (stable open-loop behaviour, $C < 0$) and its left side (unstable open-loop behaviour, $C > 0$), with the particular demand to have a gain large enough to stabilize the behaviour in this latter case. For example, imposing a second-order closed-loop dynamic with about hundred-of-millisecond time constant is a reasonable choice.

In order to obtain reasonable control effort, the master's rotational speed reference, $\Omega_1^*$, is low-pass filtered by $\frac{1}{T_T s + 1}$, as indicated in Fig. 3.32a). For reasons of obtaining the same dynamic for both master and slave, $\Omega_2^*$ must also be filtered by the same filter, where $T_T$ can be selected at about seconds, i.e., large enough in relation to the rotational speed closed-loop dynamic. This choice allows the first-order dynamic given by $T_T$ to prevail in the dynamic of $\Omega_2^* \to \Omega_2$ transfer. Such choice also corresponds to some particular PLLs design of first achieving the superposition of frequencies and then that of phase angles [101].

The synchronization loop controller is tuned such that a desired closed-loop behaviour from the master's angular position, $\theta_1 + \alpha$ – where $\alpha$ is the imposed angular position lag between the twin towers – to the slave's angular position, $\theta_2$ to be obtained. Fig. 3.32b) contains a block diagram of this transfer.

Some differences due to the nature of the input can be remarked in relation to an analog PLL, namely concerning the information containing the phase angle to be tracked. In the classical linear approach, the sinusoidal waveform containing the phase information is the loop reference and the phase of a local oscillator is adjusted by the output of the loop filter (phase controller) in order to achieve the synchronization with this reference waveform [97]. Here the phase information has the genuine form of a sawtooth signal, as provided by the master encoder. The slave encoder generates a similar signal representing the output of the local oscillator.

The two sawtooth signals can easily be transformed into harmonic signals (i.e., by applying sinus function), that carry the same phase information. These signals are further applied to a phase detector, just as in the PLL case. The phase detector formula used here is:

$$ \varepsilon_{\theta} = \sin(\theta_{1} - \alpha - \theta_{2}) = \sin(\theta_{1} - \alpha) \cdot \cos(\theta_{2}) - \sin(\theta_{2}) \cdot \cos(\theta_{1} - \alpha), \quad (3.31) $$

thus allowing to $\varepsilon_{\theta}$ to be approximated by the sinus argument, $\theta_1 - \alpha - \theta_2$, for sufficiently small values of this latter, hence, to further using a linear PLL design approach. Therefore, a PID controller can be designed to compensate the main dynamic $T_T$ and to ensure tracking of angle lag reference variations, according to imposing the slowest closed-loop dynamic in the system, as placed on the upper-most control level – design details, as well as steady-state and stability analysis of the synchronization closed loop can be found in [88].

Next two figures present some HILS experimental results, resumed from [88], which validate the reference-tracking and disturbance-rejection behaviour, respectively. The CFWT tower characteristics are the same as in the test bench used in previous Subsection 3.3.2 – dedicated to management of a miHECS operating modes – that is, each CFWT tower has 2.3 kW of rated power at water speed of 2.3 m/s. While the master's dynamic behaviour is numerically simulated, the focus is on illustrating the slave's physically-simulated behaviour, while also emphasizing the generation system's electromechanical variables.

Fig. 3.33a) shows the variables' evolutions when a gradient-limited angle lag variation (from 0 to $\pi/2$ rad and backwards) is imposed between the two CFWT towers, for a short time horizon (one minute), the water flow being constant at 2.3 m/s during this time. The master is operated in MPPT; as the slave is synchronized in position, it will find itself at about the same optimal point, as indicated by the power coefficient evolutions, $C_{p1}$ and $C_{p2}$. The associated control effort is visible on the PMSG currents, $i_{qG1}$ and $i_{qG2}$, and rotational speed evolutions, $\Omega_1$ and $\Omega_2$, as the slave is either accelerated or decelerated to follow the master angular position; short periods of synchronization loss may thus occur, as Figs. 3.33 a) and b) show. Such intervals are much shorter compared to the slow variations of the water flow speed; therefore, the resulting energy loss is negligible. The tracking dynamic performance can be assessed by analyzing both the evolution of angle lag response, $\Delta\theta = \theta_1 - \theta_2$, and that of the angle lag error, $\varepsilon_{\theta} = \Delta\theta - \alpha$.
Fig. 3.33b) displays a six-second zoomed image of the same curves as in Fig. 3.33a). With about-1.5-s transients, the synchronization process is sufficiently fast in order to prevent from long de-synchronization time lags, which are not desired because of hydrodynamic reasons. Oscillations visible on $i_{qG2}$ are due to some unmodelled dynamics and parameter variations, but they are not critical.

<figure><img src="image_57.png" /><figcaption>Figure 3.33: HILS validation of miHECS PLL-like synchronized operation – reference tracking performance (ControlDesk®captures) [88]: a) variables evolution when the imposed angle lag, $\alpha$, steps like 0 $\to \pi/2 \to 0$ rad; b) zooms of the same variables.</figcaption></figure>

<figure><img src="image_58.png" /><figcaption>Figure 3.34: HILS validation of miHECS PLL-like synchronized operation – disturbance rejection performance (ControlDesk®captures) [88]: a) variables evolution under water flow speed filtered step; b) zooms of the same variables.</figcaption></figure>

Figs. 3.34a) and b) show the evolution of the system main variables in response to a filtered-step variation of the water flow speed acting as a disturbance, with, this time, the angle lag $\alpha$ being constant at $\pi/2$ rad. Thus, the water flow speed varies from 2.3 m/s to 2.0 m/s and backwards, while the master generator operates in MPPT. The synchronization controller succeeds in cancelling the angle lag error in about 12 s.

The speed correction imposed to the slave, $\Delta\Omega$, evolves as a glitch, which is either positive or negative. Thus, when the water speed decreases, the slave is advanced in relation to the master, because the *physically*-simulated slave's rotational speed decreases *more slowly* than the *numerically*-simulated master's speed, due to some unmodelled dynamics, e.g., inherent supplementary frictions. Hence, $\Delta\Omega$ must be negative in order to re-cancel the angle lag error. When the water speed increases, the inverse situation happens.

Successful HILS validation of the synchronization control was continued with a *pilote-site validation* on a *prototype of two-twin-CFWT-tower-based miHECS* installed on a headrace canal feeding a hydroelectric plant near Grenoble in France. Indeed, due to its simplicity and good robustness – as revealed by both formal
analysis and experiments [88] – the above-briefed synchronization method proved to be easily and effectively implementable into practice, with results that confirm theoretical predictions. Some of these results, resumed from [85], are presented next.

The generation system has 16 kW of total output rated power at 2.8 m/s of water flow velocity, being characterized by maximum power coefficient $C_{p,max} = 0.95$ and optimal tip speed ratio $\lambda_{opt} = 2.7$. The PMSG rated rotational speed is 330 rpm.

Fig. 3.35 shows how the position error – above defined as $\Delta\theta = \theta_1 - \theta_2$ – evolves when the angle offset reference, $\alpha$, is set to zero and master tower’s rotational speed increases as a slowly-variable ramp. Evolutions of the two rotational speeds – master’s and slave’s – are practically superposed, which indicates effective synchronization.

<figure><img src="image_59.png" /><figcaption>Figure 3.35: Validation of PLL-like master-slave synchronization on a miHECS pilote site [85]: rotational speeds and position error between the two twin towers.</figcaption></figure>

<figure><img src="image_60.png" /><figcaption>Figure 3.36: Validation of PLL-like master-slave synchronization on a miHECS pilote site [85]: time evolutions of main variables when the two twin towers are synchronized and the master tower operates in MPPT.</figcaption></figure>

Fig. 3.36 synthesizes the results obtained in a more complex scenario, that is, when the two towers are synchronized and the master tower is operated at MPPT. When in MPPT mode and under variable water flow speed, the speed reference must be continuously updated, which renders the synchronization more challenging than in other cases. Moreover, a continuous triangle variation of the angle offset reference, $\alpha$, is also applied to challenge even more the synchronization loop with MPPT. Rotational speed variations show how the system tracks the water velocity changes. Evolutions of the power provided by each of the two twin CFWT towers are also represented in this figure. The position error is effectively corrected to zero despite the continuous variation of $\alpha$, thus proving a perfect coherence with HILS validation results reported in [88].

As a conclusion to this subsection, some remarks can be made towards generalizing the main merits of the PLL-like synchronization control of rotating prime movers. Indeed, this method is easily applicable for other types of RECS, e.g., for WECS using a similar structure, i.e., employing two vertical-axis adjacent wind turbines. The main idea can also be extended to several synchronized turbines composing a plant. A pertinent example is the synchronization of multiple horizontal-axis wind turbines in the same wind farm, in order to alleviate the power pulsations due to the tower shadow effect.

## 3.4 Power optimization control approaches for photovoltaic (PV) systems

Both stand-alone and grid-connected photovoltaic (PV) installations, as well as hybridizations of PV with some other renewable energy sources and/or storage devices, make nowadays part of a familiar landscape [102], [103]. Rendering the PV systems more adequate to the wide use in terms of power, efficiency, grid compliancy and communication capacity for those grid-connected ones, reliability and service time, safety and security, etc., continues to be a topic of great interest. In a larger framework, sizing and operation design of PV systems are usually the result of a co-design optimization procedure guided by various and often contradictory criteria, oriented towards
finding the best trade-off within a specific application [104]–[106].

Control approaches are necessary for optimizing the PV systems dynamic performance, in terms of *reactivity to the variability of* the primary energy source – i.e., the *solar irradiance* – and *robustness* to as various as possible kinds of disturbances. A general remark is that, being based upon a very fast physical phenomenon (the *photoelectric effect*) and not supposing any moving mechanical part, PV systems have *the smallest inertia among all the RECS* and this comes with the undesirable side effect of a globally reduced robustness.

As in the case of other RECS operating in grid-connected mode, maximization of the captured power is the main goal. Simple, efficient, and minimal-knowledge-demanding MPPT methods aiming at maximizing the extracted energy irrespective of the irradiance conditions were investigated in the literature [107], [108]. In this case, it is about tracking the *Optimal Regimes Characteristic* (ORC) like the one presented in Fig. 3.3 in the introductory Section 3.1 of the current chapter.

Among the various MPPT methods specific to PV systems, the ones relying upon injecting high-frequency small-amplitude (usually harmonic) perturbations in the system in order to detect the sign of the power gradient are mostly known for PV systems as the *perturb-and-observe (P&O) class of methods* [109], [110]. Their basic idea is the same as the one used in *Extremum Seeking Control* (ESC), which has been detailed in Section 3.1. It is worthy to note that ESC can also be applied by using *intrinsic disturbances*, already present in the system, therefore without a need for inducing exogenous excitation signals – such an approach for WECS made the object of one of our contributions briefed in Subsection 3.2.3 in this memoir and its application also proved pertinent in the case of PV systems, as further detailed in Subsection 3.4.3 in the current section.

Whereas MPPT is practically solved in the case of PV systems having unimodal power-vs.-voltage, $P_{PV}$ ($V_{PV}$), characteristic, it may not be the case of *large spatially distributed systems*, with high probability of undertaking partial shading, so *strongly imbalanced levels of irradiance*. Indeed, specific configurations of PV modules may have global *multiple-maxima power characteristics* – see, for example, the allure of curves in Fig. 3.37, obtained for the same PV array, but under different irradiance conditions [111]. Adaptations of the general MPPT methods for tracking multiple peaks under rapidly changing irradiance conditions were proposed [112], [113], which also motivated our contribution [114] in this regard.

Having been initiated during my 2008–2009 post-doctoral internship at G2ELab, work on the PV systems control topic also continued during my first years at GIPSA-lab (2012–2014).

*Synthetically speaking, our contributions propose some original improvements and adaptations to basic MPPT for grid-*,

*connected PV systems.*

Thus, contributions summarized next in the two first subsections were developed within the collaboration with Ph.D. student Damien PICAULT, post-doctoral fellow Iulian MUNTEANU and Associate Professor Bertrand RAISON, under the supervision of Professor Seddik BACHA. A short description follows.

* Application of Extremum-Seeking-Control (ESC)-based MPPT for grid-connected PV systems is systematically considered *under conditions of strongly variable irradiance*, while emphasizing possible existence of *multiple (local) power maxima* under *partially-shading* conditions, depending on the PV panels' connection. Main results were published in:

A.I. Bratcu, I. Munteanu, S. Bacha, B. Raison (2008). Maximum power point tracking of grid-connected photovoltaic arrays by using extremum seeking control. *Control Engineering and Applied Informatics* – Romanian Society of Control Engineering and Technical Informatics (SRAIT), 10(4), 3–12,

which is among the first publications dealing with systematic application of ESC techniques for achieving the MPPT for PV systems, and makes the object of Subsection 3.4.1.

<figure><img src="image_61.png" /><figcaption>Figure 3.37: Example of a PV array whose $P_{PV}$ ($V_{PV}$) curve exhibits: a) a single maximum; b) multiple maxima under certain irradiance conditions [111].</figcaption></figure>
* Approaching the *captured power maximization of cascaded* – or *series*, instead of the more habitual parallel – *DC-DC converter architectures of PV modules* led to an *MPPT-degraded control strategy*, where simultaneous MPPT of all PV modules needs to be given up whenever safety operating constraints require it. Results concerning this topic – presented next in Subsection 3.4.2 – made the object of an initial conference paper:

A.I. Bratcu, I. Munteanu, S. Bacha, D. Picault, B. Raison (2009). Power optimization strategy for cascaded DC-DC converter architectures of photovoltaic modules. In: *Procs. of the IEEE International Conference on Industrial Technology – ICIT 2009*, Churchill, VIC, Australia, February 2009, pp. 1–8,
further extended as a journal publication:

A.I. Bratcu, I. Munteanu, S. Bacha, D. Picault, B. Raison (2011). Cascaded DC-DC converter photovoltaic systems: power optimization issues. *IEEE Transactions on Industrial Electronics*, 58(2), 403–411,
which has received *the largest number of citations among all my journal publications* (e.g., 225 citations in Web of Science since its publication, in 2011).

The final Subsection 3.4.3 concerns formalization of the ESC-based MPPT and control design guidelines for a class of grid-connected PV systems for which *the searching signal is already present in the form of the sinusoidal ripple of some variables*. The related main results made the object of a collaboration with post-doctoral fellow Iulian MUNTEANU during 2012–2014 at GIPSA-lab and were published in the journal article:

* I. Munteanu, A.I. Bratcu (2015). MPPT for grid-connected photovoltaic systems using ripple-based extremum seeking control: analysis and control design issues. *Solar Energy*, 111, 30–42.

### 3.4.1 Extremum-seeking control (ESC)-based MPPT for photovoltaic (PV) systems

Without loss of generality, a two-conversion-stage topology of a single-phase grid-connected PV energy conversion system is considered in this subsection (Fig. 3.38), which is composed of an array of PV modules whose captured energy is sent to the utility grid after being converted into electrical energy by means of a two-stage power electronic system: a DC-DC converter and a DC-AC converter (inverter) connected through a DC-link. The capturing device consists in a PV array composed of *n* series connections connected in parallel, each of which is composed of *m* PV modules. It is supposed that, in general, each module experiences its own irradiance conditions, $I_{r_{ij}}, i = 1, 2, ...n, j = 1, 2, ...m$. The PV array output current is $I_{PVa}$ and its output voltage is $V_{PVa}$.

Fig. 3.38 also contains the *global control block diagram*, organized on *two levels*. The desired operating regime is decided in the upper level, whereas the low-level control effectively implements the decision taken at the upper level. In the case of grid-connected systems, imposing a certain operating point is equivalent with imposing a certain level of the power provided – in particular, this can be the maximum available in the irradiance. In this latter case, the upper-level control structure from Fig. 1 becomes an optimization control loop, required to track the maximum power operating point – which is in general not precisely known and varies with the irradiance level – instead of tracking an imposed power setpoint.

The *ESC implementation block diagram* of this *upper-level control* is detailed in the lower part of Fig. 3.38, a PV-adapted version of the general ESC principle illustrated in Fig. 3.5 in Section 3.1. The primary resource, *i.e.*, the solar irradiance, is supposed to be strongly time varying, as well as variable from a module to another. Due to taking into account this double variability, PV-power-*vs.*-voltage curves of the *global PV array* can either be *unimodal* – thus replicating the allure of the power-voltage curve of a *single PV module* – or they can be multimodal. Here the results of applying this method to unimodal PV power curve are only resumed, whereas details of multiple-maxima power curve case can be found in [111].

The role of the DC-DC converter is to boost the voltage level in the DC-link; it is controlled to track either a current or a voltage reference, $I_{PVa}^*$ or $V_{PVa}^*$, corresponding to the operating point imposed by the upper decision level. The inverter output current, $i_{grid}$, is controlled such that the DC-link voltage to be maintained at an imposed value, $V_{DC}^*$.
The ESC block diagram in the lower part of Fig. 3.38 shows that the MPPT is achieved by controlling the DC-DC converter input voltage. As recalled in Section 3.1 of this chapter, the controller performs a modulation/de-modulation operation, outputting a harmonic component called the *probing – or searching – signal*. The controller habitually contains a so-called washout filter, a demodulator, a low-pass filter and an integrator for obtaining the average component of the control input, as well as a summation with the probing signal block.

<figure><img src="image_62.png" /><figcaption>Figure 3.38: Grid-connected PV energy conversion system and the associated two-level control structure, where the **upper-level control** is particularized as an **ESC-based MPPT** with the associated DC-DC converter voltage as control input [111].</figcaption></figure>

To further explain operation of ESC-based MPPT, some modelling details are necessary. A grid-connected PV generator having an unimodal power characteristic is modelled by a generic power function, $P_{PV}(V_{PV})$, having a unique maximum at $V_{PVopt}$, whose argument has two components: an average one, $\overline{V_{PV}}$, and a **harmonic probing one, artificially induced**, of frequency $\omega_s$ and amplitude $a$ (Fig. 3). This function is approximated by its two-term Taylor’s series around its maximum, $P_{PV}(V_{PVopt})$. The searching error, defined as $\overline{V_{PV}} = V_{PV} - \overline{V_{PV}}$, has a negative gradient:

$$ \dot{\overline{V_{PV}}} = 0.25 \cdot k \cdot a^2 \cdot P''_{PV}(V_{PVopt}) \cdot \overline{V_{PV}}, \qquad (3.32) $$

as constants $k$ and $a$ are positive and $P''_{PV}(V_{PVopt})$ is negative, thus achieving a convergent searching process. The closed-loop stability is ensured by a sufficiently large excitation frequency, $\omega_s$, which the washout filter parameter, $h$, depends on [13] – see also (3.4) in Section 3.1. Input of the integrator in Fig. 3.38, $d\overline{V_{PV}}/dt$, will toggle its sign following the excursion of the operating point from a side to the other of the maximum, with a convergence speed depending proportionally on $k$, $a$ and $1/\omega_s$.
The low-level control is another cascade control structure itself, namely having an external control loop for controlling the PV array voltage, $V_{PVa}$, and an internal loop dedicated to chopper current, $I_{ch}$, control, as suggested in the upper part of Fig. 3.38. Control tuning details can be found in [111].

Some representative numerical simulation results are next extracted from [111] to illustrate performance of the ESC-based MPPT. They were obtained on a PV array with $n = 10$ modules in parallel ($m = 1$), 1100 W of maximum global power, global voltage of 48 V. A 0.1-V-amplitude sinus MPPT probing signal of frequency $\omega_s = 2\pi \cdot 100$ rad/s was used.

Two aspects were studied about the effectiveness of the ESC-based MPPT: the first one concerns the dynamic performance of tracking unimodal power-vs.-voltage curves depending on ESC controller integral gain, *k* (see the lower part of Fig. 3.38), while the second aspect investigates the capability of tracking multimodal power curves. In the latter case, as the ESC capability of finding the global extremum is not guaranteed theoretically, simulations showed that performance of tracking the global maximum is affected by the initial state [111]. Next, illustration of the first aspect is only resumed.

If all the modules of the PV array are submitted to the same irradiance level, then global unimodal power curves are obtained. The evolutions exhibited in Fig. 3.39 correspond respectively to two irradiance scenarios: first, the case a), when the common level of irradiance is affected by step changes (from 900 W/m² to 600 W/m², and then back to 900 W/m²); then b), c) and d), which present the dynamic behaviour under strongly (stochastically) variable irradiance conditions.

<figure><img src="image_63.png" /></figure>

Figure 3.39: ESC performance in tracking the maximum of unimodal PV power curves under strongly variable irradiance – MATLAB®/Simulink® results [111]: a) power responses at step changes of irradiance versus maximum power for different values of the ESC controller gain, *k*; b) stochastic profile of irradiance and corresponding variation of the maximum power; c) excursion of the operating point in the PV-power-voltage plane; d) grid current under the irradiance profile from b). The irradiance dynamic model used for obtaining curves in Fig. 3.39b), c) and d) is set in analogy with the dynamic modelling of another irregular renewable energy source, the wind speed – the wind turbulence modelling has been recalled throughout Section 3.2, dedicated to wind energy conversion systems (WECS) optimal control. Thus, a two-spectral-component model – containing an average component and a "turbulence" component – is used here to represent the irradiance dynamic behaviour.

The system is set to start from the imposed voltage of $V_{PVa}=43$ V at 900 W/m² irradiance, which does not correspond to the maximum power point. Thus, the first power transient in Fig. 3.39a) shows the MPPT performance in response to the first irradiance step. Three power time evolutions, corresponding to three different values of the ESC controller parameter, *k*, can be seen. The larger the value of *k* is, the better the tracking quality, with an effect of performance saturation and even over-tuning for $k_3 = 50$, as it corresponds to a longer and more turbulent transient.
Intuitively, as the light is however more regular than the wind, the frequency of variations would be smaller than in the case of wind. Also, the turbulence component of the irradiance would depend inversely of its average – which would express the assumption that the irradiance turbulence is more significant on cloudy days – unlike the wind, whose turbulence level depends directly on the average wind speed (see (3.13) in Subsection 3.2.1 earlier in this chapter and also Chapter 3 of our Springer monograph [29]). The following irradiance model was used in simulations:

$$ Irr(t) = \underbrace{\overline{Irr}}_{\text{average irradiance}} + \underbrace{\left(\frac{Irr_{max}}{Irr}\right)^3 \cdot f_t(t)}_{\substack{\text{"turbulence" irradiance} \\ \text{irradiance}}} \quad (3.33) $$

where $f_t(t)$ is a band-limited white noise passed through a suitably chosen low-pass filter with a gain proportional with the desired turbulence variance. The irradiance stochastic signal, shown in Fig. 3.39b) with dashed line, exhibits quite fast variations compared to the usual natural behaviour, but this contributes at challenging even more the robustness of the ESC-based MPPT. The same figure also contains the corresponding variation of the maximum power (with solid line), showing a direct and almost instantaneous relation between the irradiance and the maximum power. The ESC controller integral gain is $k = 30$, with $h = 50$ as washout filter parameter.

Figs. 3.39c) and d) present the ORC tracking performance in the PV-power-voltage plane and the time evolution of the grid-injected current, $I_{grid}$, respectively. As expected, the PV array voltage around the maximum power point is almost constant. $I_{grid}$ amplitude depends proportionally of the irradiance level.

### 3.4.2 MPPT-degraded operation of some particular grid-connected PV topologies

This subsection is dedicated to the use of ESC-based MPPT in a more extended framework, which focuses on *implementing MPPT along with complying with some critical global operating constraints*. Indeed, for some particular PV configurations, it may happen that power optimization to be occasionnaly given up for sake of security, which most of the time means nonviolation of admissible variation limits of some variables. In such cases, MPPT should be integrated within a *larger supervising structure*, able to detect and manage the different operating modes.

One such particular topology is considered next (Fig. 3.40), where n PV generators, each coupled to its DC-DC boost converter, are **cascaded – instead of paralleled**, which is more commonly used – on the same DC bus and interfaced to the grid by means of an inverter. The problem is stated as to find a strategy of operating this architecture at MPPT, given that its different PV generators may undertake supplementary constraints – expressed mainly as **output-voltage-limitation (OVL) requirements** – when exposed to strongly variable irradiance conditions.

The architecture in Fig. 3.40 features a high-voltage PV string connected to a single inverter, having the advantages of a "converter-per-panel" approach, obviously cheaper, but also more efficient than individual grid-connected inverters [116]. Following the result that the buck and boost converters are the most efficient topologies for a given cost [116], boost DC-DC converters were here employed.

Thus, for a given DC-bus voltage, the individual boost converters can be operated at a relatively small step-up ratio (usually three to four), further resulting in good efficiency along with low cost. On the other hand, the series

<figure><img src="image_64.png" /><figcaption>Figure 3.40: Block diagram of the considered cascaded PV architecture [115].</figcaption></figure>
connections has somehow *limited MPPT performance, because the global power is limited by the power given by the worst irradiated panel*. Thus, the total power is susceptible to diminish in the case of, e.g., partial shading on some panels. In this context, our paper [115] provides a numerical-simulation-based comparison between the cascaded case and a parallel case of the same power.

In Fig. 3.40 each converter module *i* is based on the PV generator *PV*<sub>*i*</sub>, consisting of either a single panel or a connection of panels, supposed to receive all the same irradiance, such that the power characteristic remains unimodal. Such a PV generator can independently control and thus optimize the power flow from its irradiance *Irr*<sub>*i*</sub> by means of the duty cycle *u*<sub>chi</sub> ∈ {0, 1} of its associated chopper *i*. In grid applications, each chopper performs MPPT for its PV generator, while the grid inverter regulates the DC-bus voltage. At steady state, the same current *I*<sub>DC</sub> passes through all the choppers, with the sum of their output voltages being the DC-bus voltage Σ<sub>*i*=1</sub><sup>*n*</sup> *V*<sub>oi</sub> = *V*<sub>DC</sub>. The chopper output voltages are:

$$V_{oi} = V_{DC}^{*} \cdot \underbrace{\frac{I_{PVi}^{*} \cdot V_{PVi}^{*}}{\sum_{i=1}^{n} I_{PVi}^{*} \cdot V_{PVi}^{*}}}_{w_i}, \quad i = 1, 2, \dots, n, \qquad (3.34)$$

where $V_{DC}^*$ is the desired (reference) DC-bus voltage and ($I_{PVi}^*, V_{PVi}^*$), $i = 1, 2, \dots, n$, are the n operating points. According to (3.34), the DC-bus voltage distribution on the converter modules depends on the weights, $w_i$, $i = 1, 2, \dots, n$, of the individual PV power levels in the global provided power.

As long as the irradiance levels are almost the same for all PV generators in the grid-connected cascaded PV topology in Fig. 3.40, one can impose practically the same operating point – in particular, the maximum power one – ensuring that the DC-bus voltage be almost equally supported by the n DC–DC converters and the total power be maximized. Indeed, numerical simulations show that simultaneous MPPT operation of all PV generators is effective only if these ones are almost equally illuminated [115]. Otherwise, if the irradiance levels are sensibly different for some time, then the weights $w_i$ become **unbalanced** in (3.34), and some DC–DC converters undergo **over-voltages** (possible violation of voltage-limitation-type constraints). Hence, in order to cope with the most unfavourable case, the global power optimization requires a **supervisor that detects a sufficient degree of degrading the initial optimal strategy such that to allow meeting the constraints**.

Implementation of a supervisor for the above-explained purpose supposes that each PV generator to be equipped with two control laws instead of a single one: the **MPPT** (the main one) and the over-voltage limitation (**OVL**, the secondary one). The supervisor looks at variation trends in a relevant time window, therefore its algorithm runs every $T_{sv}$ s, where $T_{sv}$ is suitably chosen depending on the fastest dynamic of the supervised system.

Assuming that all the generators are in MPPT, the supervisor must be able to detect OVL violations due to unbalance between the power weights of generators and, thus, between their output voltages $V_{oi}$, $i = 1, 2, \dots, n$, according to (3.34). The admissible threshold can reasonably be set at $1.2 \cdot V_r$, where $V_r$ is a chopper's *rated* output voltage, whose value results in the initial sizing of the PV system, namely by imposing that the ratio between the DC-bus voltage *ideal* value, $V_{DC}^{*ideal}$, and $V_r$ to approach the number of PV generators. Therefore, the following holds:

<figure><img src="image_65.png" /><figcaption>Figure 3.41: Supervisor implementing the global power optimization strategy: block diagram and connections with the choppers' and inverter's control blocks [115].</figcaption></figure>

$$V_r = \frac{V_{DC}^{*ideal}}{n} \qquad (3.35)$$
When at least one such limitation is detected, then a new possible value of the DC-bus voltage reference is first sought for, in a way such that to reestablish the balance between voltages, meanwhile laying in between ±20% around $V_{DC}^{*ideal}$. If such a value could not be found out, then the supervisor decides switching from MPPT to OVL for the generators having surpassed the admissible threshold voltage, while the other generators remain in MPPT. If the power weight of a generator operating in OVL decreases, this means that the global power has increased. This can only be due to the increasing of the power provided by the generators still operating in MPPT, which indicates that the irradiance balance is going to be reestablished and the MPPT is again possible. The control of the concerned generator can then be switched back to MPPT based on estimating the gradient of the power weight.

The *supervisor block diagram* is shown in Fig. 3.41. The formal steps of the supervising algorithm can be found in [115]. Below a description and justifying details of its operation are briefed.

Conveniently designed low-pass filters are used to reflect the trend on the desired time window. Based upon computing the power weights $w_i$, $i = 1, 2, ...n$, and their gradients, the supervisor outputs $n$ binary decisions referring to the operating modes of the $n$ PV generators: output $i$ is 1 for MPPT and 0 for OVL. Switching between the two controls supposes only changing the voltage reference. These outputs feed block 1 in Fig. 3.41, which generically denotes the $n$ chopper control structures; these latter further provide the duty cycles $u_{chi}$, $i = 1, 2, ...n$, to the choppers.

If the voltage limitation is violated for at least one PV generator, this means that the power available from another generator (or maybe from several others) has decreased. The supervisor should in this case compute a new, reduced, value of the DC-link voltage reference, by using $V_r$, the chopper *rated* output voltage, given by (3.35):

$$V_{DC}^{*new} = \frac{1.2 \cdot V_r}{\max_{i=1,2,...n}\{w_i\}}, \qquad (3.36)$$

which would allow for all the generators to remain in MPPT, meanwhile meeting all the new constraints $V_{o1}^{new} \le 1.2 \cdot V_r$, $i = 1, 2, ...n$, if the new DC-link voltage (3.36) is declared suitable for the inverter control loop. This takes place if $V_{DC}^{*new} \ge 0.8 \cdot V_{DC}^{*ideal}$ – in this case, it can be provided as a new reference to the inverter control block (block 2 in Fig. 3.41).

A binary variable noted "$V_{DC}^{*new}$ found" is also computed, which, together with $w_i$ and $dw_i/dt$, $i = 1, 2, ...n$, is used within a finite-state automaton to provide the operating modes of the PV generators. If $V_{DC}^{*new}$ found = 0, the supervisor degrades the MPPT of the generators having surpassed the voltage limit by switching their control to OVL at $1.2 \cdot V_r$. This suboptimal regime corresponds to the less restrictive degradation as it still ensures the constraints to be met at the limit.

A supervised three-PV-generator cascaded DC-DC converter system having $V_{DC}^{*ideal} = 450$ V was simulated in MATLAB®/Simulink® under the irradiance scenario shown in Fig. 3.42a), with all the PV generators initially operating in ESC-based MPPT – according to control design and implementation details presented in previous Subsection 3.4.1 – under equal irradiance levels at $Irr_1 = Irr_2 = Irr_3 = 900$ W/m². Step variations of irradiance, seldom encountered in nature, were used, such as to be most challenging for the power optimization strategy. Supervisor is set to take decisions each $T_{sv}=0.1$ ms.

The considered scenario was built such as to illustrate supervisor's decisions in both the case where a new DC-link voltage reference is possible to be found out, and in the contrary case, as detailed by the results resumed below from [115].

As $n = 3$ for this system, then $V_r = V_{DC}^{*ideal}/n = 150$ V. Fig. 3.42b) indicates that $V_{o2}$ and $V_{o3}$ reach the maximum threshold of $1.2V_r = 180$ V in response at $Irr_1$ having decreased from 900 to 400 W/m². According to (3.36), a new acceptable value of the DC-bus voltage reference results (Fig. 3.42c)):

$$V_{DC}^{*new} = \frac{1.2 \cdot 150}{\frac{900}{2200}} = 440 \text{ V} > 360 \text{ V} = 0.8 \cdot V_{DC}^{*ideal}$$

Note that (3.36) provides a *theoretical* level of the new DC-bus voltage, as the measured PV power levels and, therefore, their weights $w_i$ are supposed to be in steady state. The abrupt irradiance variations determine that
the optimal power levels change, thus requiring from MPPTs to readjust; moreover, the PV power signals will always exhibit ripples when in MPPT – as it has been shown, for example, in simulation results in Fig. 3.39 in Subsection 3.4.1. It is very probable that this time cannot be waited, so the supervisor must decide the change of $V_{DC}^*$ faster than the MPPTs reach their new steady states. Thus, variables appearing in (3.36) have some transient values; therefore, actual computation will provide a slightly different value of $V_{DC}^{*new}$ compared to the one achieved with steady-state values – as suggested by the zoom in Fig. 3.42c) – which must be low-pass filtered before being imposed to the inverter.

<figure><img src="image_66.png" /><figcaption>Figure 3.42: Performance of the supervised cascaded PV system under irradiance step variations [115]: a) irradiance scenario; b) output voltages of the three choppers; c) variations of the DC-bus voltage and grid current; d)–f) time evolutions of the power provided by the three PV generators.</figcaption></figure>

Between $t_2$ and $t_3$, all the PV generators operate in MPPT (Figs. 3.42d)–f)). At time $t_3$, $I_{rr1}$ comes back to its initial level and the generators remain in MPPT until $t_4$, when both $I_{rr1}$ and $I_{rr2}$ decrease abruptly and significantly, from 900 to 400 and to 500 W/m², respectively. Shortly after, only $V_{o3}$ reaches the threshold of 180 V, but this time a new DC-bus voltage reference cannot be found any longer. Consequently, the control of the third PV generator is switched from MPPT to OVL (Figs. 3.42b) and f)). The other two generators continue to operate in MPPT, obviously at reduced power levels (Figs. 3.42d) and e)).

Note that this decision is, as the matter of fact, different from the theoretical one the supervisor should have taken. Indeed, the new DC-bus voltage value computed according to (3.36) would be $1.2 \cdot 150/(900/1800) = 360$ V if using the steady-state values of weights $w_i$. On the other hand, 360 V is exactly the lower admissible threshold for $V_{DC}$, i.e., $360$ V = $0.8 \cdot V_{DC}^{*ideal}$.

Therefore, the supervisor should have theoretically allowed the MPPT operation of all PV generators. However, the values of $w_i$ effectively used by the supervisor are smaller than the steady-state ones (Figs. 3.42d) and e)), so finally $V_{DC}^{*new}$ effectively computed is smaller than 360 V. Thus, the supervisor's actual decision in this particular case is to degrade the MPPT on PV generator 3, thus making sure that voltage constraints are met. Note that slower-than-step variations of irradiance and smaller transients are more probable in nature; in this case, the supervisor is more likely to decide keeping the MPPT operation by reducing the DC-bus voltage reference.

To wrap-up the above-briefed results towards attempting some generalization, the designed supervisor exemplifies a practical way of dealing with different PV-topology-dependent constraints that can act in contradiction with MPPT operation, as long as the harvested power maximization remains the main purpose.
### 3.4.3 Intrinsic sinusoidal ripple used as searching signal in ESC-based MPPT for grid-connected PV systems

Taking inspiration on wind-turbulence-based ESC for achieving MPPT of wind energy conversion systems (WECS) – see the approach in [70], described earlier in Subsection 3.2.3 of this chapter – this subsection considers the *ESC-based MPPT for a class of grid-connected PV systems without intermediary DC-DC converter stage*, that is, directly connected to the grid inverter. Thus, different from the classical ESC implementation, the perturbation signal is *not artificially generated* by the MPPT algorithm; instead, the method briefed here uses the *DC-link voltage ripple*, which "naturally" results from the grid inverter operation [117], [118], *as already present, intrinsic searching (dithering) signal*.

This technique that allows for simplifying the control structure to a certain extent is also known as **ripple correlation control** [119], [120], having been subsequently linked to ESC [118], [121]. Our analysis in [122] emphasized the significant variation of system behaviour with the irradiance and proposed an adaptive control approach for fast-changing irradiance as a possible solution. Here the main ideas are presented.

The PV generation system considered here is detailed in Fig. 3.43 and consists in a string of series-connected PV modules, which is in turn connected directly on the inverter DC-link and further to the "strong" grid [123]. Such a topology avoids use of intermediate DC-DC boost converters, with obvious advantages like reduced power losses, simplified overall hardware, instrumentation and control and enhanced reliability.

A single-phase voltage-source inverter is employed in order to transfer the generated DC electrical energy to the grid; to this end, the inverter is classically operated as AC current source, supposing DC-link voltage regulation at a certain value irrespective of the current injected into DC-link by the PV string. Fig. 3.43 shows the considered PV generator structure, where the outer voltage controller imposes the necessary magnitude of AC current, $i_{AC}^*$, in order to establish the DC-link voltage balance – see also the two-loop control structure presented earlier in

Fig. 3.38 in Subsection 3.4.1 [111]. At its turn, the upper-level MPPT control loop must provide to the voltage controller that DC-link voltage reference, $v_{DC}^*$, corresponding to maintaining the PV string operating point at the MPP, irrespective of irradiance and parameter variations (e.g., panel temperature).

For the PV topology in Fig. 3.43, implementing the MPPT by means of a modified ESC structure which exploits an *already existing high-frequency sinusoidal ripple component in voltage $v_{DC}$* proved to be a profitable idea. Thus, the MPPT algorithm computes the DC-link voltage reference *by measuring the effect of this high-frequency perturbation on the string output power signal, $p_{PV}$*. This controlled behaviour supposes relatively **larger flexibility in DC-link voltage variation**, which hence must be allowed as early as in the initial setup design. Thus, domain of DC-link voltage variations should include the range of MPP PV voltage variations. However, MPPT at low irradiance levels may not be possible, because the inverter cannot operate below a certain $v_{DC}$ value, but power loss is not a critical issue in this case.

The MPPT control structure using ripple-based ESC is given in Fig. 3.44, where acronym CCP stands for current-controlled plant and acronym VVP denotes voltage variation plant. Although in a different form in relation to the habitual one – recalled in Fig. 3.5 in Section 3.1 earlier in this chapter – the ESC structure preserves the **washout** (perturbation effect separation) **filter** and the **demodulator**. As the perturbation is intrinsic to the

<figure><img src="image_67.png" /><figcaption>Figure 3.43: Considered PV topology as a string direct connection to AC single-phase grid, where the two-level control system implements classical grid-connected inverter operation [122].</figcaption></figure>
system itself, it must also be extracted by means of a separation filter. Choice of a band-pass filter (BPF in Fig. 3.44), instead of a high-pass filter used in classical ESC, is here motivated to improve closed-loop behaviour under fast variation of irradiance conditions, as justified further.

<figure><img src="image_68.png" /><figcaption>Figure 3.44: PV MPPT by ripple-based ESC: control block diagram [122].</figcaption></figure>

In Fig. 3.44 $H_{0DC}$ represents the second-order closed-loop DC-link voltage transfer function, which results by adequately designing a PI controller for the open-loop plant described by the transfer from $i_d$ – the d component of inverter current, $i_{AC}$, in the dq stationary frame – to the DC-link voltage, $v_{DC}$; modelling details and choice of closed-loop performance for this low-level control can be found in [122].

Operations performed sequentially within the MPPT algorithm are delimited by dashed line at the right of Fig. 3.44. The MPPT algorithm uses measured signals $v_{DC}$ and $p_{PV}$ as inputs and outputs the DC voltage reference, $v_{DC}^*$. At each time step, this algorithm performs the following operations: band-pass filtering of signals $v_{DC}$ and $p_{PV}$, multiplication of these filtered signals, multiplication of the result by the positive gain $K_{ES}$ and finally integration of the result.

Next are resumed the main arguments supporting the ripple-based ESC MPPT convergence analysis.

The PV generator power characteristic, $p_{PV}$ ($v_{PV}$), is supposed unimodal, having its maximum at an a priori unknown value, noted as $v_{mpp}$. As also suggested in Fig. 3.44, notation $p_{mpp} = p_{PV}$ ($v_{mpp}$) is adopted. Note that irradiance variations are almost instantaneously experienced by both $v_{mpp}$ and $p_{mpp}$. The argument of this function has two main components: an average one, $\overline{v_{PV}} \equiv \overline{v_{DC}}$, and the a-magnitude $2\omega$-sinusoidal perturbation, with $\omega$ being the grid pulsation:

$$ \tilde{v}_{DC} = a \cdot \sin(2\omega t), \qquad (3.37) $$

which plays here the role of probing signal. The three-term truncation of Taylor's series of the power function around its maximum, $p_{mpp}$:

$$ p_{PV}(v_{DC}) \approx p_{mpp} + p'_{PV}(v_{mpp}) \cdot (v_{DC} - v_{mpp}) + \frac{1}{2} p''_{PV}(v_{mpp}) \cdot (v_{DC} - v_{mpp})^2, \qquad (3.38) $$

makes appear the first-order and the second-order derivative of PV power curve with respect to voltage, $p'_{PV}$ and $p''_{PV}$, respectively. Both $f_1 = p'_{PV}(v_{mpp})$ and $f_2 = p''_{PV}(v_{mpp})$ vary with the irradiance.

According to (3.38), system dynamics while tracking the MPP is different depending on whether the operating point is near to the MPP or it is far from it. The two regions can respectively be identified on the P-V curve in Fig. 3.45, along with a parabola useful for estimating the second-order derivative at MPP, which is a value characterizing the behaviour "close to MPP".

Notation $\varepsilon = \tilde{v}_{DC} \cdot \tilde{p}_{PV}$ is already implicitly introduced in Fig. 3.44, whereas notation:

$$ \varepsilon_v = \overline{v_{DC}} - v_{mpp} \qquad (3.39) $$

<figure><img src="image_69.png" /><figcaption>Figure 3.45: Typical PV-power-voltage curve for constant irradiance, where the two regions – "far from MPP" and "close to MPP", respectively – are identified for convergence analysis purpose [122].</figcaption></figure>
is introduced to denote **the voltage searching error**. As indicated in Fig. 3.44, reference $v_{DC}^*$ applied to the PV voltage control loop is finally computed by integrating signal $\varepsilon$ with gain $K_{ES}$. Thus, **convergence analysis** results obtained in [122] can be synthesized as follows.

* When in operation "close to MPP":

$$v_{DC}^* = \frac{1}{2} a^2 f_2 K_{ES} \int \varepsilon_v dt \qquad (3.40)$$

The integrator input, $\varepsilon_v$, toggles its sign as the operating point moves from a side to the other of the $P_{PV}$ curve's maximum. As constant $K_{ES}$ is positive and value $f_2$ is negative in this region (see Fig. 3.45), (3.40) defines a process convergent to $v_{mpp}$ – as the voltage searching error, $\varepsilon_v$, diminishes – whose convergence speed depends proportionally on $K_{ES}$ and $a^2$ [13].

Choice of the washout filter's parameters depends on the excitation frequency, $2\omega$, but here this frequency is fixed, as imposed by the grid, so it is not a tuning parameter. Therefore, the other parameters make the object of a suitable design. Oscillations magnitude, $a$, depends on the irradiance value corroborated with the operating point position on the PV power characteristic, thus yielding variability of dynamic response. Rendering the value of $K_{ES}$ adaptive with $a$ can be a pertinent solution for obtaining more uniform dynamic behaviour.

* When in operation "far from MPP":

$$v_{DC}^* = \frac{1}{2} a^2 f_1 K_{ES} \int dt, \qquad (3.41)$$

which also defines a process convergent to abscissa of maximum, as constant $K_{ES}$ is positive and value $f_1$ is either positive or negative, depending on whether the current voltage is smaller or larger than that of maximum, respectively (see Fig. 3.45). It results that convergence speed does not depend on error $\varepsilon_v$ in this case, indicating an approximately linear decreasing of maximum searching speed on a constant-irradiance $P-V$ curve.

Coming now back to use of a band-pass filter (BPF) rather than a high-pass one (HPF) as washout filter (see ESC ripple-based MPPT control diagram in Fig. 3.44), justification of such a choice appears pertinent especially **when irradiance variations are strong enough to produce some side effects on maximum searching dynamic performance**. To this end, Taylor-series-truncation modelling of signal $p_{PV}$ in the region "close to MPP" is resumed:

$$p_{PV} \approx p_{mpp} + \frac{1}{4} f_2 a^2 + f_2 \varepsilon_v a \sin(2\omega t) - \frac{1}{4} f_2 a^2 \cos(4\omega t) \qquad (3.42)$$

Because irradiance variations are practically instantaneously repercuted in PV power variations, variables such as $p_{mpp}$, $f_2$ and voltage oscillations magnitude $a$ (which depends on PV power) also vary strongly. Term $p_{mpp} + f_2 a^2/4$ is not constant any longer, but represents a signal of the same bandwidth $f_E$ as the irradiance, whose spectrum is presented in Fig. 3.46a). This figure also presents spectrum of term $f_2 \varepsilon_v a \sin(2\omega t)$, which is approximately $\pm f_E$ centred around frequency $2\omega$, taking account of variations of $a$ and $f_2$. Higher-frequency components are not presented here.

Supposing that an HPF is used to select the $2\omega$-sinusoidal component, as represented with dashed-dotted line in Fig. 3.46a), HPF folding frequency must ideally be chosen larger than $f_E$ and smaller than $2\omega - f_E$. As grid frequency $\omega$ cannot be modified, such choice may prove restrictive, even more as bandwidth $f_E$ is larger, i.e., it represents fast irradiance variations. Therefore, one cannot guarantee that the two spectra are not superposed. It is thus very probable that, whatever choice of HPF frequency is, a part of irradiance spectrum be further replicated by modulation in high frequency, around $2\omega$ (triangle-shaped spectrum with dashed line in Fig. 3.46b) [124]. It may also be the case of higher-frequency components (at $4\omega$ in (3.42)) to be replicated by demodulation around frequency $2\omega$ (trapezoidal spectrum with dashed line in Fig. 3.46b)).
It results that possible alterations of expected dynamic performance to be experienced because of intrusion of undesired and uncontrollable dynamics like those of irradiance, suggesting that HPF to be replaced by a more precise filtering like, e.g., band-pass filtering – also suggested by dashed-dotted line in Fig. 3.46a) – for sake of preserving accuracy of imposed performance. Moreover, use of HPF as washout filter possibly shortens convergence time to MPP. Trading off advantages and drawbacks of different types of washout filters may further make the object of a more-in-depth analysis.

As a conclusion of the above analysis, abrupt, large-spectrum irradiance variations are likely to be not accurately tracked and may potentially induce instability. These results are coherent with the theoretical analysis of ESC performance and limitations in [12], while their more intuitive presentation form contribute at formulating clearer guidelines for control design. On the other hand, assumption of stepwise irradiance variations corresponds to the most unfavourable case, quite improbable to occur in the nature.

Concerning control design of this specific MPPT method for PV systems, it is based upon average, low-frequency, modelling of ESC around the MPP, leading to a parameter-varying third-order closed-loop transfer function, whose parameters depend on both gain $K_{ES}$, as a tuning parameter, and on irradiance level. Root locus method can be a solution for imposing suitable closed-loop dynamic performance, along with an adaptive formula of $K_{ES}$.

Thus, based on relatively easy measuring voltage oscillation magnitude, $a$, and supposing that desired dynamic performance is calibrated at the rated operating point – usually, the one under reference irradiance of 1000 W/m² – with $K_{ES\_rated}$, then for a given operating point different from rated the gain is rendered adaptive as:

<figure><img src="image_70.png" /><figcaption>Figure 3.46: Symbolic representation of filtering and modulation process that precedes integration in ESC ripple-based MPPT control diagram in Fig. 3.44, to justify choice of a band-pass filter as wash-out filter [122], where spectra of interest are: a) before filtering; b) after modulation.</figcaption></figure>

$$K_{ES} = \frac{a_{rated}^2}{a} \cdot K_{ES\_rated}, \qquad (3.43)$$

where $a_{rated}$ and $a$ are voltage oscillation magnitudes at rated and at the current operating point, respectively. The continuous gain scheduling in (3.43) only compensates a part of plant variation by reasonably exploiting the limited knowledge about the system. Closed-loop performance is consequently expected to alter along the operating domain, but less than without any adaptation.

This subsection will end with some numerical simulation results reported in [122], which were performed for validation purpose on a PV system of class described in Fig. 3.43, having 2020 W of rated power and MPP voltage $V_{mpp} = 462$ V, both at reference irradiance $E = 1000$ W/m². For this system, $a_{rated} = 11$ V and $K_{ES\_rated}=2.4$. Strongly-variable irradiance scenarios were used, as considered relevant for challenging the adaptive-gain MPPT performance.

Irradiance scenario in Fig. 3.47 consists in a ramp-wise positive and then negative variation occurring when system operated initially at 1000 W/m² of irradiance, its reference value. PV power evolution copies practically instantaneously irradiance variation (Fig. 3.47a)). Fig. 3.47b) shows how voltage oscillation magnitude, $a$, varies, as well as evolution of gain $K_{ES}$, rendered adaptive according to (3.43). Fig. 3.47c) shows comparatively evolution of PV voltage with and without adaptation, indicating a response time significantly improved in the case of an adaptive gain. Fast tracking is necessary to avoid that error between $v_{mpp}$ and $\overline{v_{DC}}$ to become larger than oscillation magnitude, $a$; otherwise, tracking would not be any longer effective.

Last simulation scenario aims at illustrating system dynamic behaviour under strongly variable irradiance
conditions, close to those encountered in the nature. To this end, the same **stochastic** two-component dynamic model of *irradiance* as in [115] – explicited in (3.33) in Subsection 3.4.1, earlier in this section – was used, to reflect realistic irradiance variation. Irradiance variation according to this model is shown in Fig. 3.48a); the corresponding PV maximum power variation is shown in Fig. 3.48b).

<figure><img src="image_71.png" /><figcaption>Figure 3.47: Closed-loop performance of ESC ripple-based MPPT in response to *irradiance ramp variations* of –1000 W/m²/s starting from 1000 W/m² during 0.5 s and backwards – time evolution of interest variables [122]: a) PV power; b) voltage oscillation magnitude and adaptive ESC gain; c) PV voltage evolution comparatively with and without $K_{ES}$ adaptation.</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>Figure 3.48: ESC ripple-based MPPT performance in response to **stochastic variation of irradiance** [122]: a) irradiance variation; b) corresponding variation of maximum PV power; c) PV voltage tracking performance; d) variation of ESC perturbation signal magnitude, *a*, and limited variation of adaptive gain $K_{ES}$; e) grid current variation; f) PV voltage evolution.</figcaption></figure>

Fig. 3.48c) presents performance of tracking voltage at maximum power, $v_{mpp}$, by the average value of PV voltage, $\bar{v}_{DC}$, emphasizing significant alteration of the voltage tracking performance when irradiance goes abruptly too low (smaller than 300 W/m²). This happens because perturbation signal's magnitude *a* also decreases and, consequently, according to adaptation formula (3.43), value of ESC gain, $K_{ES}$, increases rapidly with potentially negative effects on the global stability. Gain limitation was here the preferred solution for preserving stability (Fig. 3.48d)), at the price of temporarily sacrificing voltage tracking quality. This compromise is, however, quite affordable since power loss due to MPPT degradation at too low irradiance values represents in general a rather insignificant percentage of rated power (less than 5 W in this case), due to flatness of P-V curve around maximum.
Fig. 3.48e) gives information about the grid-injected current, whereas Fig. 3.48f) presents the time evolution of total voltage signal $v_{DC}$ (that is, including the ripple component whose magnitude ultimately depends on irradiance).

## 3.5 Summary of contributions and conclusion

This final section aims at summarizing the contributions that we chose to detail throughout this chapter, with respect to **control** of some types of renewable energy conversion systems (RECS). Although not exhaustive, Fig. 3.49 attempts to offer an **unitary vision** of the main features and strong points of our contributions.

As *grid-connected* RECS applications were the ones mainly dealt with, it appears somehow obvious that *maximization of captured power irrespecive of the primary resource variations* to be of *focal interest*. Thus, Maximum Power Point Tracking (MPPT) proved to be the main control objective having been identified as common to all three types of approached RECS: wind energy conversion systems (WECS), micro-hydro energy conversion systems (miHECS) and photovoltaic energy conversion systems (PV).

<figure><img src="image_73.png" /><figcaption>Figure 3.49: Suggested axes towards an unitary vision of the RECS control approaches presented in Chapter 3.</figcaption></figure>

Some comments must specifically be made in relation to grid-connected RECS applications and microgrids, in particular, because having put the focus on this topic equally impacted on **control** problem *statement*, control-oriented **modelling** and control *design*.

Thus, the grid-connected – or, otherwise called, *PQ* operation – is different, including from a control viewpoint,
from the *Vf* operation, when the microgrid is islanded from a "strong" grid (or stand-alone). The two operating modes have been recalled in Section 2.1 in Chapter 2 earlier in this memoir. As its name shows, when in *PQ* operation, a microgrid is required to provide a certain amount of active (*P*) and reactive (*Q*) power to the main grid; this objective is achievable only if the respective power amounts can be *guaranteed*, otherwise said, if the power source (microgrid) is *dispatchable*. Or, taking account of the *irregularity of their primary resources* and also of their *relatively small inertia* (quite fast dynamics), **RECS** are *a priori not dispatchable*. A possibility to increase their regularity and predictability, so their dispatchability, is to complement (hybridize) them with *storage units*.

There are, however, different degrees of dispatchability, as synthetically indicated in Fig. 3.49 for each RECS type. Thus, in view of the combination between the primary source regularity and the inertia, miHECS appear as being the most dispatchable. Indeed, this was an aspect emphasized when proposing a direct power control for such systems. Regarding the other two RECS types, PV systems look rather more dispatchable than WECS. Thus, the two are submitted to quite variable primary resource – with slightly more significant irregularity for the wind, in view of its high-frequency, turbulence component, compared to the solar irradiance – whereas PV systems are clearly characterized by the smallest inertia (fastest dynamics) among the three types of RECS considered here. However, the lack of PV inertia may look easier to handle and compensate than mechanical loads and high-frequency excitation of moving parts in WECS. Hence, PV systems would be rather superior to WECS from the dispatchability viewpoint, although this remains arguable.

As a partial conclusion of the above arguments, since RECS are unlikely dispatchable by themselves when grid connected, then they can be required to provide the maximum – instead of a certain amount – of power available in the resource. Hence, our interest for MPPT methods, which came as a quite complex control challenge, as it has previously been detailed within this chapter.

Instead of being exhaustive and uselessly analytical by attempting conclusions on all the previously briefed approaches, a concluding discussion around the schematic guidelines suggested in Fig. 3.49 is here preferred. This figure suggests how different formal control approaches were employed in order to implement the MPPT, some of which are common to two of the three RECS types, obviously depending on some identified common features, as above argued.

Thus, MPPT had to be integrated within mixed-criteria dynamic optimization when control efforts had to be taken into account – it is the case of rotating-prime-mover-based RECS, like WECS and miHECS, for which MPPT well performance may be paid back by important mechanical loads. Application of frequency-separation principle for WECS led to an LQG optimal control problem being formulated, whereas more practical – less formal, but more intuitive – control solutions were proposed for miHECS, like phase-locked-loop (PLL)-based synchronization, taking inspiration from communications engineering.

An extremely profitable role in achieving *MPPT* based on *minimal knowledge* about the system was played by the Extremum Seeking Control (ESC) techniques, guaranteeing convergence and a certain implicit robustness against the only approximate *a priori* knowledge. Belonging, from a principle viewpoint, to the larger class of so-called *Perturb and Observe* (P&O) methods, application of ESC in its "conventional" form proved beneficial, especially in the case of PV systems. Then, idea of integrating the *already existing turbulence wind speed as a high-frequency searching signal* within an ESC-based strategy for WECS MPPT was an original adaptation and, meanwhile, improvement. That allowed, instead of adding exogenous perturbation, using the already present one: the primary resource turbulence component, thus also implicitly alleviating mechanical efforts.

This new idea of "adapted ESC" further turned out to be again fruitful in the PV case, namely for the class of directly-grid-inverter-connected PV systems. In this case, inverter operation determines *ripple – proportional with the solar irradiance level – of intermediary DC-link voltage*, which in turn can be used as "intrinsic", endogenous searching signal in ESC.

Last, but not least, MPPT as global control method found its place into *more complex management and control strategies* – and this was the case especially for miHECS, as well as for PV systems. In both cases, some *event-driven supervisors* had to be designed in the form of state automata in order to suitably handle switchings between different operating modes. In this context, *PQ-to-Vf-to-PQ* switchings were particularly studied for a
miHECS-based microgrid. As regards PV case, such an approach was necessary for a common-DC-link *cascaded* – instead of a *paralleled* – topology that needed giving up MPPT from time to time, to prevent from safety (over-voltage) limits’ violation.

Before the end, it is appropriate to mention the role of validation, which is of great importance for any control
design. Thourough numerical simulation validations were conducted for all the proposed control methods, most of
which have subsequently also been validated on real-time test benches, according to hardware-in-the-loop simulation
(*HILS*) principles. Validation on a real-world prototype was also possible in the case of miHECS, thus marking
the successful passage from preliminary checking and rapid prototyping towards effective, on-site implementation.
<figure><img src="image_74.png" /></figure>
## 4.1 Formal foundations and mathematical tools

This section aims at overviewing some basic concepts of $\mathcal{H}_\infty$ and LPV control frameworks, as well as the main associated mathematical tools. Indeed, this mixture of control approaches appears pertinent as justified by the analysis and modelling of the considered MSES, revealing their definitive parameter-varying nature, as well as the fact that they are disturbance-driven, having to supply the (a priori uncontrollable) load power demand. In particular, LPV framework is employed in Section 4.2 for a multi-storage instance of MSES. Principles of genetic-algorithm (GA)-based multi-objective optimization are also re-called, as GA were used within the two mentioned frameworks.

Consider in a first place a linear time-invariant (LTI) dynamical system $\Sigma$ of the form:

$$ \Sigma: \begin{bmatrix} \dot{\mathbf{x}} \\ \mathbf{z} \\ \mathbf{y} \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \mathbf{B}_1 & \mathbf{B}_2 \\ \mathbf{C}_1 & \mathbf{D}_{11} & \mathbf{D}_{12} \\ \mathbf{C}_2 & \mathbf{D}_{21} & \mathbf{D}_{22} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{x} \\ \mathbf{\omega} \\ \mathbf{u} \end{bmatrix}, \quad (4.1) $$

where $\mathbf{x}(t) \in \mathbb{R}^n$ is the system's state vector, $\mathbf{u}(t) \in \mathbb{R}^{n_u}$ is the control input vector, $\mathbf{\omega}(t) \in \mathbb{R}^{n_\omega}$ is the exogenous (disturbance) input vector, $\mathbf{z}(t) \in \mathbb{R}^{n_z}$ is the controlled output vector and $\mathbf{y}(t) \in \mathbb{R}^{n_y}$ is the measured output vector, with state ($\mathbf{A}$), input ($\mathbf{B}_1, \mathbf{B}_2$), output ($\mathbf{C}_1, \mathbf{C}_2$) and direct transfer ($\mathbf{D}_{11}, \mathbf{D}_{12}, \mathbf{D}_{21}, \mathbf{D}_{22}$) matrices of appropriate dimensions.

<figure><img src="image_75.png" /><figcaption>Figure 4.1: Closed-loop block diagram supporting $\mathcal{H}_\infty$ control problem formulation, including within the LPV framework.</figcaption></figure>

The $\mathcal{H}_\infty$ control problem is a disturbance-attenuation problem, which consists in finding a dynamical output feedback controller $\mathbf{K}$ of the form:

$$ \mathbf{K}: \begin{bmatrix} \dot{\mathbf{x}}_c \\ \mathbf{u} \end{bmatrix} = \begin{bmatrix} \mathbf{A}_c & \mathbf{B}_c \\ \mathbf{C}_c & \mathbf{D}_c \end{bmatrix} \cdot \begin{bmatrix} \mathbf{x}_c \\ \mathbf{y} \end{bmatrix}, \quad (4.2) $$

having $\mathbf{x}_c \in \mathbb{R}^{n_c}$ as internal state, which ensures the closed-loop minimization of the impact of the exogenous input vector $\mathbf{\omega}(t)$ on the controlled output vector $\mathbf{z}(t)$ (see block diagram in Fig. 4.1). Mathematically, with notation $\| \cdot \|_2$ standing for $\mathcal{L}_2$ signal norm, controller $\mathbf{K}$ ensures that, for a given real positive number $\gamma$:

$$ \sup_{\omega(t) \neq 0} \frac{\| \mathbf{z} \|_2}{\| \mathbf{\omega} \|_2} \leq \gamma, \quad (4.3) $$

or, equivalently:

$$ \| T_{z\omega}(s) \|_{\infty} = \| C(s\mathbf{I} - \mathbf{A})^{-1} \cdot \mathbf{B} + \mathbf{D} \|_{2} \leq \gamma, \quad (4.4) $$

where $\mathbf{I}$ is the identity matrix and $\mathcal{A}, \mathcal{B}, \mathcal{C}$ and $\mathcal{D}$ are the closed-loop state-space matrices, computed respectively as:

$$ \left\{ \begin{array}{l} \mathcal{A} = \begin{bmatrix} \mathbf{A} + \mathbf{B}_2 (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{D}_c \mathbf{C}_2 & \mathbf{B}_2 (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{C}_c \\ \mathbf{B}_c (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{C}_2 & \mathbf{A}_c + \mathbf{B}_c (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{D}_{22} \mathbf{C}_c \end{bmatrix} \\ \\ \mathcal{B} = \begin{bmatrix} \mathbf{B}_1 + \mathbf{B}_2 (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{D}_c \mathbf{D}_{21} \\ \mathbf{B}_c (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{D}_{21} \end{bmatrix} \\ \\ \mathcal{C} = \begin{bmatrix} \mathbf{C}_1 + \mathbf{D}_{12} (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{D}_c \mathbf{C}_2 & \mathbf{D}_{12} (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{C}_c \\ \mathbf{D}_{11} + \mathbf{D}_{12} (\mathbf{I} - \mathbf{D}_c \mathbf{D}_{22})^{-1} \mathbf{D}_c \mathbf{D}_{21} \end{bmatrix} \end{array} \right. \quad (4.5) $$

In order to take into account the parameter-varying nature of most of dynamical systems, while still keeping linearity assumptions, the more complex framework of linear parameter-varying (LPV) systems was elaborated. LPV systems appear as a bridge between the nonlinear and the well-known LTI systems [125]. The theory of LPV systems offers systematic design methods for guaranteeing robust stability and performance, compared to classical gain-scheduled control [126].
An LPV system is described by the same state-space representation as in (4.1), where at least one of the matrices depends on an N-dimensional parameter vector $\rho(t) = [\rho_1(t) \ \rho_2(t) \ \dots \ \rho_N(t)]^T \in \mathbb{R}^N$, assumed to be measurable (or observable) and bounded ($\rho(t) \in \mathcal{P}$) for all time instants. Next, for sake of simplicity, $\rho(t)$ is simply noted as $\rho$; it is also called scheduling parameter vector. Hence, an LPV system can be described as:

$$ \Sigma(\rho) : \begin{bmatrix} \dot{\mathbf{x}} \\ \mathbf{z} \\ \mathbf{y} \end{bmatrix} = \begin{bmatrix} \mathbf{A}(\rho) & \mathbf{B}_1(\rho) & \mathbf{B}_2(\rho) \\ \mathbf{C}_1(\rho) & \mathbf{D}_{11}(\rho) & \mathbf{D}_{12}(\rho) \\ \mathbf{C}_2(\rho) & \mathbf{D}_{21}(\rho) & \mathbf{D}_{22}(\rho) \end{bmatrix} \cdot \begin{bmatrix} \mathbf{x} \\ \mathbf{\omega} \\ \mathbf{u} \end{bmatrix}, \quad (4.6) $$

where notation meanings are preserved. Depending on the nature of parameter vector $\rho$ variation, different classes of systems result by particularizing (4.6); thus:

* LTI systems are obtained if $\rho$ does not vary with time;

* if $\rho$'s dependence with time is explicit, then it is about linear time-variant (LTV) systems;

* quasi-linear parameter-varying (qLPV) systems result if $\rho$ depends on the state vector – i.e., $\rho(t) = \rho(x(t))$;

* finally, genuine LPV systems are obtained if $\rho$ is external (independent) to the system.

If the system's matrices are affine with respect to the parameter vector $\rho$, then the system is called an affine LPV system. Polytopic LPV systems are a special class of affine LPV systems, whose matrices are convex combinations of some constant matrices:

$$ \mathbf{A}(\rho) = \sum_{i=1}^{2^N} \alpha_i(\rho) \cdot \mathbf{A}_i \quad \mathbf{B}(\rho) = \sum_{i=1}^{2^N} \alpha_i(\rho) \cdot \mathbf{B}_i \quad \mathbf{C}(\rho) = \sum_{i=1}^{2^N} \alpha_i(\rho) \cdot \mathbf{C}_i \quad \mathbf{D}(\rho) = \sum_{i=1}^{2^N} \alpha_i(\rho) \cdot \mathbf{D}_i, \quad (4.7) $$

where $2^N$ is the number of vertices of the polytope corresponding to the extreme values of the N-dimensional parameter vector $\rho$, $\sum_{i=1}^{2^N} \alpha_i(\rho) = 1$ and all $\mathbf{A}_i, \mathbf{B}_i, \mathbf{C}_i$ and $\mathbf{D}_i$ matrices are constant. Hence, a polytopic LPV system is a convex hull of a finite number of LTI systems, each of which is "placed" at a vertex of the polytope. This feature turns out to be of great interest in control design and implementation, as shown next.

Now, the $H_\infty$ framework is extended to LPV systems. Thus, for the LPV system (4.6) the LPV/$H_\infty$ control problem consists this time in finding a dynamical output feedback controller that is also parameter varying:

$$ \mathbf{K}(\rho) : \begin{bmatrix} \dot{\mathbf{x}} \\ \mathbf{u} \end{bmatrix} = \begin{bmatrix} \mathbf{A}_c(\rho) & \mathbf{B}_c(\rho) \\ \mathbf{C}_c(\rho) & \mathbf{D}_c(\rho) \end{bmatrix} \cdot \begin{bmatrix} \mathbf{x}_c \\ \mathbf{y} \end{bmatrix}, \quad (4.8) $$

that guarantees not only the closed-loop stability, but also satisfies the performance (4.3) for all values of parameter vector $\rho$. Fig. 4.1 suggests that such a framework extension can take place somehow naturally. In this case, the optimal $H_\infty$ gain, noted as $\gamma^*$, is the smallest gain for all existent controllers $\mathbf{K}(\rho)$:

$$ \gamma^* = \min_{\mathbf{K}(\rho)} \| T_{z\omega}(s) \|_{\infty} \quad (4.9) $$

As regards the solution of the stated LPV/$H_\infty$ control problem, a parameter-varying controller of form (4.8), having the same order as the LPV system ($n_c = n$), is obtained by solving a theoretically infinite number of linear matrix inequalities (LMIs) – as parameter vector $\rho$ varies continuously – while minimizing $\gamma$ [150], [128]. Different approaches can be used in order to relax this problem and reduce it to solving a finite number of LMIs:

* by suitably defining a grid of the parameter space and selecting basis functions [129];

* by transforming the parameter-dependent system into an uncertain system using the Linear Fractional Transformation (LFT) [130];

* by adopting a polytopic approach, consisting in solving $2^N$ LMIs only, that is, for each vertex of the polytope corresponding to the extreme values of the N-dimensional parameter vector $\rho$ [150] – this is the method having led to the results summarized next in Section 4.2.

Let us resume the definition (4.7) of a polytopic LPV system as a convex combination of multiple LTI systems ($\mathbf{A}_i, \mathbf{B}_i, \mathbf{C}_i, \mathbf{D}_i$) evaluated at each vertex $i$ of the polytope, $i = 1, 2, \dots, 2^N$. For each scheduling parameter $\rho_j$
its minimum and maximum values are noted as $\rho_j$ and $\bar{\rho}_j$, respectively. To each vertex $i$ is associated an $N$-dimensional vector $\zeta_i$, whose $j$-th element is defined as:

$$ \zeta_{ij} = \begin{cases} \bar{\rho}_j, & \text{if parameter } \rho_j \text{ at vertex } i \text{ is } \rho_j \\ \rho_j, & \text{otherwise} \end{cases} \qquad (4.10) $$

The LPV system in (4.7) can be written under the equivalent form:

$$ \begin{bmatrix} \mathbf{A}(\rho) & \mathbf{B}(\rho) \\ \mathbf{C}(\rho) & \mathbf{D}(\rho) \end{bmatrix} = \sum_{i=1}^{2^N} \alpha_i(\rho) \cdot \begin{bmatrix} \mathbf{A}_i & \mathbf{B}_i \\ \mathbf{C}_i & \mathbf{D}_i \end{bmatrix}, \qquad (4.11) $$

where weights $\alpha_i(\rho)$ are defined as:

$$ \alpha_i(\rho) = \frac{\prod_{j=1}^{N} |\rho_j - \zeta_{ij}|}{\prod_{j=1}^{N} |\bar{\rho}_j - \rho_j|} \geq 0, \quad \sum_{i=1}^{2^N} \alpha_i(\rho) = 1, \qquad (4.12) $$

with $\zeta_{ij}$ given by (4.10). At each vertex $i$ of the polytope the associated LMI set is solved off line to yield a controller $\mathbf{K}_i: \begin{bmatrix} \mathbf{A}_{c_i} & \mathbf{B}_{c_i} \\ \mathbf{C}_{c_i} & \mathbf{D}_{c_i} \end{bmatrix}$. The global LPV/$\mathcal{H}_\infty$ controller, $\mathbf{K}(\rho)$, is finally computed on line as the convex combination of the off-line computed local (vertex) controllers, $\mathbf{K}_i$, in analogy with (4.11):

$$ \mathbf{K}(\rho) : \begin{bmatrix} \mathbf{A}_c(\rho) & \mathbf{B}_c(\rho) \\ \mathbf{C}_c(\rho) & \mathbf{D}_c(\rho) \end{bmatrix} = \sum_{i=1}^{2^N} \alpha_i(\rho) \cdot \begin{bmatrix} \mathbf{A}_{c_i} & \mathbf{B}_{c_i} \\ \mathbf{C}_{c_i} & \mathbf{D}_{c_i} \end{bmatrix}, \qquad (4.13) $$

thus resulting in a gain-scheduled form. A comprehensive review concerning this approach can be found in [131].

**Genetic algorithms (GAs)** represent one of the most popular multi-objective optimization methods, being inspired from the natural mechanism of adaptation and evolution. Thus, basic principles of GAs are employed towards developing generations of members (parameters) in a way such as to extremize multi-objective functions [132], [133]. Members in the decision space are classified regarding some suitably defined fitness (objective) functions into the decision space (see Fig. 4.2 [134]).

GAs are very effective in finding a convenient solution where the problem to solve involves large sets of parameters to be determined – evolutionary process principles are formalized and employed to this end. The idea is to use probabilistic, multi-point search, random operations – such as mutation and crossover – and data from the previous generation in order to improve the evaluation of current population.

Some GA basic concepts are:

* **individual or member**: it refers to any possible solution in the decision space where any acceptable combination of the parameters can form an individual;

* **population or generation**: it refers to a group of individuals;

* **chromosome**: it refers to one parameter among the individual parameters;

* **genome**: it refers to a group of chromosomes in an individual.

<figure><img src="image_76.png" /><figcaption>Figure 4.2: Multi-objective optimization process as performed by GAs [134].</figcaption></figure>

<figure><img src="image_77.png" /><figcaption>Figure 4.3: Illustration of two important GA operations: a) cross-over; b) mutation.</figcaption></figure>
A GA-based optimization process consists of *several steps*, starting by randomly choosing the first generation; the generations are further developed through four main steps that are *evaluated sequentially in a continuous loop*. The process is terminated when the desired number of generations is achieved. The main four steps are as follows.

* The *individual fitness functions* are computed; individuals better "fitted" are more probable to be selected for the next steps.

* **Selection** is performed at this step, i.e., individuals are sorted and copied in descending order of their fitness function values. The most widely used selection methods are *proportionate selection* [132] and *tournament selection* [135].

* The main operation acting on the population of parents is the *cross-over*, suggested in Fig. 4.3a). It is based on exchanging parts between two selected individuals (parents) to form two new individuals (children). This exchange process is applied either to a single or to multiple chromosomes.

* At this step *mutation* is applied by changing randomly either a chromosome or a genome in the individual (child), as suggested in Fig. 4.3b). Mutation is an important operation since it allows creating new members, different from all developed generations.

## 4.2 Design of robust Power Management Systems (PMSs) for multi-storage systems

This section focuses on a certain class of MSES, namely the multi-storage MSES, where several (two or more) technologically complementary storage units are coordinated in order to effectively operate complementarily. To fix ideas, a *multi-storage power supply on board on an electric vehicle (EV)* was considered. The approach and results briefed next continue and generalize our work reported in this regard in [18]–[21], previously introduced in Section 2.2 in the motivation Chapter 2 of this memoir.

Since it is mainly about *storage* units, then it is about *DC* microgrids, which appear to have gained popularity in the last years vs. AC ones, due to their more simplified operation (e.g., no need for synchronization). Stand-alone applications in avionic, automotive and marine industries, but also power supply of remote areas may benefit from this advantage, among others [136], [137].

In MSES, the roles of Energy Management System (EMS) and Power Management System (PMS) mainly concern sources' coordination towards ensuring a *desired power sharing* between sources to satisfy the load demand, while *preserving reliability of all sources* and extending their remaining useful life [138].

Efficient PMS can be designed using a variety of control approaches. Model Predictive Control (MPC) – with its nonlinear and/or stochastic versions – may be applied at coordinated multivariable control strategies for stand-alone microgrids and/or MSES, with or without renewable sources, as it successfully deals with multiple constraints [139], [14], [140]. However, need of an adequate plant mathematical model and of a *priori* knowledge of load variation to predict the behaviour may be regarded as drawbacks when employing this framework. Indeed, in the particular case of EVs, this results in driving-cycle-dependent PMS solutions. Nonlinear methods [15], [16] or even more sophisticated approaches, like multi-agent-based and games theory [141], proved to be of interest to explore.

PMS design reported here is related to taking into account the *requirements of reliable operation characteristic of each source*, under a suitable form, for example, "translated" and/or adapted from its data sheet specifications. Such an approach leads to considering complex techno-economical criteria in the MSES PMS design. In particular, *reliability-aware design* habitually relies on complex degradation and ageing models of storage units, along with optimization methods [142]. The multi-time-scale character of MSES justify hierarchical control structures [143], with dynamically separated layers for enabling operation of sources in their "specialization" range of variations – according to Ragone's taxonomy, recalled in Fig. 2.3 in Chapter 2 – to improve their reliability [144]. In this way, high-power-density sources (like ultracapacitors), specialized in providing high-amplitude fast-varying currents, are able to protect high-energy-density sources (like fuel cells or batteries), whose current variations must be smoother. As already stated previously in the motivation Section 2.2, filtering is the most straightforward way
of dynamic separation, where variations of current of each source are confined in a predefined frequency interval
[18], [145], [19], whereas LQG approach may also be applied, but restricted to only two sources [20], [21].

<figure><img src="image_78.png" /><figcaption>Figure 4.4: Electrical schematic of a three-storage – *fuel cell / battery / ultracapacitor* – power supply system, detailing the equivalent electrical models of the storage units [146].</figcaption></figure>

To further fix ideas, a three-
storage – *fuel cell, battery and
supercapacitor* – MSES in an
electromobility use case, i.e., as
a power supply on board of an
electric vehicle is here considered.
The number of storage units was
chosen such as to illustrate the
generality of the employed ap-
proach – whereas having consid-
ered only two sources would have
appeared like a particular case.
Thus, the multi-storage parallel
configuration in Fig. 4.4 is con-
sidered, having the advantage of
flexibility (sources’ independent
operation, facility to replace/add
power sources, etc.)

The *fuel cell* acts here as the
*main* source, it is *unidirectional*,
being thus connected to a one-

quadrant DC-DC converter. The two auxiliary sources – the battery and the supercapacitor, both bidirectional – have the role of responding to relatively fast power demand variations and to collect the reversed power (during braking phase). To this end, each auxiliary source is connected to a two-quadrant converter allowing charging/discharging. The electrical motor together with its converter plays the role of load for this DC supply system.

Among the tasks of the PMS is to guarantee that each source, independently controlled by means of its DC-DC converter, operates in its dynamic specialization range, thus ensuring its most rational exploitation.

The results presented next were obtained within collaboration with Ph.D. student Jean-Marc Waleed NWE-SATY, working under the supervision of Professor Olivier SENAME and myself, and made the object of the following main publications:

* W. Nwesaty, A.I. Bratcu, O. Sename (2016). Power sources coordination through multivariable LPV/H∞ control with application to multi-source electric vehicles. *IET Control Theory and Applications*, 10(16), 2049-2059

- summarized in Subsection 4.2.1 – in which the approach of casting the PMS design into the LPV/H∞ framework methodology is briefed, leading to a **generic PMS solution** for multi-storage systems with potentially **any number of sources**, that guarantees closed-loop robust stability;

* J.-M.W. Nwesaty, A.I. Bratcu, A. Ravey, D. Bouquain, O. Sename (2020). Robust energy management system for multi-source DC energy systems – real-time setup and validation. *IEEE Transactions on Control Systems Technology*, 28(6), 2591–2599

- briefed in Subsection 4.2.2 – which reports about the **PMS preliminary proof of concept** achieved in collaboration with colleagues (Associate Professors David BOUQUAIN and Alexandre RAVEY) from FEMTO-ST Laboratory in Belfort, France, by hardware-in-the-loop-simulation (**HILS**) validation on a test bench comprising real battery and supercapacitor, as well as an entirely emulated fuel cell system. A dSPACE™ MicroAutoBox®II device was chosen to embed the designed control, due to its flexibility and ease of programming with MATLAB®.
Taking into account the significant potential of the results obtained during J.-M. W. Nwesaty's Ph.D. thesis of being transferred to the industry, between 2017 and 2018 we pursued activities related to the designed robust PMS *proof of concept* in an EV use case. Funding received from the Institut Carnot *Logiciels et Systèmes Intelligents* (LSI) within the *technology pre-transfer project*

* **K-INF**: Embedded real-time PMS for multi-source electrical energy systems allowed PMS embedding on a general-purpose Texas Instruments microcontroller and closed-loop real-time demonstrations on a test bench built around a rapid-prototyping system dSPACE™ MicroAutoBox®II, thus advancing the technology readiness level (TRL) of our PMS solution. This made the object of an APP (*Agence pour la Protection des Programmes – Software Protection Agency*) software application deposit (alphabetical order of authors):

* A.I. Bratcu, I. Munteanu, J.-M.W. Nwesaty, O. Sename (2018). *K-INF Multi-source management system robust design*. Grenoble Institute of Technology, Grenoble Alpes University. RT2018013, version 1.1 May 2018.

Some relevant results and conclusions of K-INF technology pre-transfer project are summarized in Subsection 4.2.3, to offer an overall glimpse of practical feasibility of a well-performing, yet quite complex, control solution.

### 4.2.1 Robust PMS design for a three-storage DC power supply microgrid on board of an electric vehicle

<figure><img src="image_79.png" /><figcaption>Figure 4.5: Global two-level control block diagram with the LPV/$H_\infty$ controller on the upper level [147]. The PMS design for the above-described on-board EV MSES-based power supply begins with adopting a classical global two-level control structure, as shown in Fig. 4.5: the DC-DC converters' current control loops on a lower level and the LPV/$H_\infty$-based PMS on an upper level, in charge with providing the references to the lower level. The dynamic (frequency)-separation operation of the three storages should take place while meeting some other constraints, leading to the following control objectives being formulated:</figcaption></figure>

* maintain the DC-link voltage constant, around $V_{dc}^*$ setpoint within an error of ±10% regardless of the load current variations. This requirement is very general in the sense that it applies to all configurations where different sources are paralleled on a common DC link. It is this requirement that allows the control problem being stated as a disturbance-attenuation problem and cast into the $H_\infty$ framework;

* ensure frequency separation of storages, i.e., each storage supplies power with respect to its characteristic frequency according to Ragone's plot – see (2.1) and Fig. 2.3 in Section 2.2. To this end, *suitable shaping of weighting functions associated to $H_\infty$ control design* plays a key role;

* maintain the supercapacitor state of charge (SoC) slowly around 50%, which allows continuity in absorbing/providing power to fulfill instantaneous load power demand;

* impose a desired steady-state behaviour for the fuel cell and the battery, that corresponds to some desired power sharing between sources in steady state. This allows to operate the fuel cell at a desired working point, e.g., the one corresponding to maximum efficiency. Steady-state behaviour could be used to determine battery long-term charging cycle, depending on its technology – in particular, battery charging could be achieved by using the main power source, the fuel cell.
Concerning the low-level control design, by supposing that the DC-link voltage $V_{dc}$ is maintained constant at its reference value $V_{dc}^*$ by the PMS, converters' currents have linear dynamics, which justifies their control by classical PI controllers. Being a tracking control level, it is designed to be fast enough to be neglected at the slower-dynamic upper level. Hence, the following equalities hold in the sequel: $I_{fc}^*=I_{fc}$, $I_{bat}^*=I_{bat}$ and $I_{sc}^*=I_{sc}$.

<figure><img src="image_80.png" /><figcaption>Figure 4.6: Closed-loop P–K form used in the LPV/$H_\infty$ design, where the plant is extended by adding the weighting functions [146].

*K form, where the plant G(ρ) is extended with the weight-*

ing functions* – denoted by W – suitably selected such as to represent the above-listed performance requirements.</figcaption></figure>

Whereas G(ρ) plant's state vector is x = [Vdc V1 V2 V0]T – see Fig. 4.4 – composition of the other different vectors involved in the design is suggested in Fig. 4.6: the exogenous vector, w, contains the disturbance signal I_load, all the other being reference signals; u is the control vector; y is the output vector containing the errors in relation to references and, finally, z is the controlled output vector, composed of the "conditioned"-by-weighting-function errors and control inputs.

The following choices of weighting functions proved to be appropriate in this case, as explained below.

* A first-order $W_{eV_{dc}}$ function is suitable to ensure both desired time response and acceptable DC-link voltage, $V_{dc}$, tracking error range.

* A first-order $W_{eV_{sc}}$ function is used to maintain supercapacitor SoC slowly around about 50%, that is, in the middle of its variation range, where the supercapacitor's voltage, $V_{sc}$, is a direct image of its SoC;

* First-order $W_{uI_{fc}}$, fourth-order $W_{uI_{bat}}$ and fourth-order $W_{uI_{sc}}$ shape the dynamic behaviour of current references of the fuel cell, the battery and the supercapacitor, respectively, according to some prespecified frequency ranges, thus implementing dynamic-specialization-based separation of power sources, i.e., with respect to their respective characteristic frequencies.

* Fuel cell and battery steady-state behaviours are conditioned by means of $W_{eI_{fc}}$ and $W_{eI_{bat}}$, respectively, both constant; a desired steady-state power sharing can further be imposed by using $I_{fc_{steadystate}}$ and $I_{bat_{steadystate}}$ exogenous inputs.

Since the parameter vector ρ has the three converters’ duty cycles as elements, bounded within [0.1;0.9] for practical reasons of avoiding the extreme values, the generalized LPV MIMO system is represented under a polytopic form with 2$^{N=3}$ = 8 vertices:

The three tracking low-level loops have the converters' duty cycles, $\alpha_{fc}$, $\alpha_{bat}$ and $\alpha_{sc}$, as control inputs. It is thus reasonable to identify the varying parameter vector as being composed of the three duty cycles: $\rho = [\rho_1 \ \rho_2 \ \rho_3]^T \equiv [\alpha_{fc} \ \alpha_{bat} \ \alpha_{sc}]^T$.

The sought-for LPV controller **K**(ρ) in Fig. 4.5 must therefore ensure closed-loop stability, meanwhile minimizing the effect of load current, $I_{load}$, acting as a disturbance, namely on the DC-link voltage. Control design is thus tackled in the $H_\infty$ framework applied to LPV systems. The control design diagram is given in Fig. 4.6 under the so-called P–
$$ \begin{bmatrix} \dot{\mathbf{x}} \\ \mathbf{z} \\ \mathbf{y} \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \mathbf{B}_1 & \mathbf{B}_2(\rho) \\ \mathbf{C}_1 & \mathbf{D}_{11} & \mathbf{D}_{12} \\ \mathbf{C}_2 & \mathbf{D}_{21} & \mathbf{D}_{22} \end{bmatrix} \cdot \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \\ \mathbf{u} \end{bmatrix}, \qquad (4.14) $$

where, using circuit element notations in Fig. 4.4 and block diagram in Fig. 4.6, the different matrices are:

$$ \mathbf{A} = \begin{bmatrix} -\frac{1}{C_{dc}R_{dc}} & 0 & 0 & 0 \\ 0 & -\frac{1}{C_1R_1} & 0 & 0 \\ 0 & 0 & -\frac{1}{C_2R_2} & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \quad \mathbf{B}_1 = \begin{bmatrix} 0 & 0 & 0 & 0 & -\frac{1}{C_{dc}} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \quad \mathbf{B}_2(\rho) = \begin{bmatrix} \frac{1-\rho_1}{C_{dc}} & \frac{\rho_2}{C_{dc}} & \frac{\rho_3}{C_{dc}} \\ 0 & 0 & -\frac{1}{C_1} \\ 0 & 0 & -\frac{1}{C_2} \\ 0 & 0 & -\frac{1}{C_0} \end{bmatrix} $$

$$ \mathbf{C}_2 = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}, \quad \mathbf{D}_{21} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{bmatrix}, \quad \mathbf{D}_{22} = \begin{bmatrix} 0 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$

with the other matrices – $\mathbf{C}_1$, $\mathbf{D}_{11}$ and $\mathbf{D}_{12}$ – depending on parameters of the seven weighting functions in Fig. 4.6. In this case it is about finding nineteen such parameters, which is not a trivial task from a practical viewpoint. As heuristic methods appeared as a pertinent solution, **GAs** as evolutionary-inspired optimization method were preferred to facilitate selection of these parameters. Since convergence to optimality is guaranteed, optimization progress can be stopped when an arbitrarily chosen maximum number of iterations is reached, the solution being considered sufficiently good in relation to some predefined objective functions.

The two GA objective functions were originally proposed in our conference paper [148], as follows:

* ensuring sufficient closed-loop stability by imposing the real parts of closed-loop eigenvalues to be smaller than a certain desired value $\delta$:

$$ \min \left\{ f_1 = \max_i \{ \mathrm{Re}(\lambda_i) \} < -\delta, \delta > 0 \right\}, \quad (4.15) $$

where Re($\lambda_i$) is the real part of the eigenvalue $\lambda_i$;

* ensuring the frequency-splitting "ability" of the weighting functions by minimizing the $H_\infty$ norm corresponding to the transfer from disturbance $I_{load}$ for each power source outside the desired – i.e., its specialization – working frequency interval; this goal can be expressed mathematically as:

<figure><img src="image_81.png" /><figcaption>Figure 4.7: Illustrative representation of objective function $f_2$ of the GA with frequency intervals corresponding to configurable dynamic separation of sources [149].</figcaption></figure>

$$ \min \{f_2 = (J_1 + J_2 + J_3)/3\}, \qquad (4.16) $$

with:

$$ J_1 = \frac{\left\| \frac{I_{fc}}{I_{load}} \right\|_{\infty, (\omega_1, \omega_2)}}{\left\| \frac{I_{fc}}{I_{load}} \right\|_{\infty}}, \quad J_2 = \frac{1}{2} \cdot \frac{\left\| \frac{I_{bat}}{I_{load}} \right\|_{\infty, (\omega_3, \omega_4)}}{\left\| \frac{I_{bat}}{I_{load}} \right\|_{\infty}} + \frac{1}{2} \cdot \frac{\left\| \frac{I_{bat}}{I_{load}} \right\|_{\infty, (\omega_5, \omega_6)}}{\left\| \frac{I_{bat}}{I_{load}} \right\|_{\infty}}, \quad J_3 = \frac{\left\| \frac{I_{sc}}{I_{load}} \right\|_{\infty, (\omega_7, \omega_8)}}{\left\| \frac{I_{sc}}{I_{load}} \right\|_{\infty}} $$
where $||\cdot||_{\infty, (\omega_i, \omega_j)}$ is the $H_{\infty}$ norm computed within the $(\omega_i, \omega_j)$ frequency interval and the involved $\omega_i$'s are frequency values ordered like roughly suggested in Fig. 4.7 to correspond to the user-imposed relative frequency separation between power sources.

The LPV/$H_{\infty}$ controller is found in the polytopic approach [128], according to the quadratic-stabilization framework [150], [151]. Thus, a set of LMIs are solved off line at each vertex of the polytope as a convex-optimization problem using a single Lyapunov function, which leads to $2^3 = 8$ vertex controllers of the form $K_i: \begin{bmatrix} A_{ci} & B_{ci} \\ C_{ci} & D_{ci} \end{bmatrix}$, $i = 1, 2, \dots, 8$. The methodology re-called in the previous Section 4.1 is further applied: a unique, global LPV controller $K(\rho)$ is computed on line as a convex combination of the vertex controllers $K_i$, according to (4.13), where expressions (4.12) of $\alpha_i(\rho)$ are here particularized by taking into account that:

$$ \begin{aligned} \bar{\rho}_j &= \max \rho_j = 0.9, \\ \underline{\rho}_j &= \min \rho_j = 0.1, \\ j &= 1, 2, 3 \end{aligned} $$

Each of the eight vertex controllers is an eighteenth-order LTI system, thus suggesting potential difficulties with the practical implementation. Indeed, HILS validation of the developed PMS – as a step towards proving its practical feasibility – pointed out the need for a possibly more "alleviated" version of

<figure><img src="image_82.png" /><figcaption>Figure 4.8: MATLAB®/Simulink® closed-loop performance assessment of LPV/$H_{\infty}$-based PMS for a three-storage MSES on board of an electric vehicle [147]: a) $I_{load}$ variation corresponding to IFSTTAR driving cycle; b) DC-link voltage variation ensured by the LPV/$H_{\infty}$ PMS against the one obtained by classical filtering; c) dynamically-separated time variations of currents of the three sources.</figcaption></figure>
the controller, thus motivating the results briefed in the next Subsection 4.2.2. **LPV/H<sub>∞</sub> controller order reduction** was originally investigated in our conference paper [152], in order to reduce the controller complexity. To this end, MORE toolbox [153] was used to find a reduced-order model that fits the original controller for a certain bounded frequency range. More precisely, the iterative singular value decomposition tangential Krylov algorithm was applied on each vertex controller for the whole closed-loop bandwidth, to finally yield tenth-order vertex controllers. Values of both H<sub>2</sub> and H<sub>∞</sub> norms of full-order and reduced-order closed-loop systems are close in view of the chosen criterion, even though the complexity is highly reduced [147], thus suggesting realistic feasibility of embedding the obtained controller into a numerical device.

This subsection ends with some representative numerical simulation results resumed from [147]. The **IFST-TAR** (Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux – French Institute of Science and Technology for Transport, Development and Networks) **driving cycle** profile (Fig. 4.8a)), representing the **urban** driving conditions – comprising acceleration, deceleration, fixed speed and full brake – was chosen for being rich in frequency content, thus sufficiently challenging to allow assessing performance of DC-bus voltage regulation and the way the three sources are coordinated to satisfy the power demand.

The DC-link voltage well regulation at its reference value, 150 V, within accepted tracking error of ±10%, is a prioritary goal, adequately achieved as Fig. 4.8b) shows. Time evolutions in Fig. 4.8c) confirm the desired suitable dynamic separation of the three power sources: their currents are provided to the system in the manner of fuel cell supplying average current and supercapacitor handling peak variations, while the battery provides the midrange current variations.

The PI-based filtering PMS was also run in simulation for benchmarking purposes; results are skipped here, they can be found in [147]. Being characterized by a set of parameters to be tuned manually and different design constraints that prevent control objectives from being independently achieved, this baseline PMS strategy is difficult to handle. The comparison with the baseline PMS solution suggests a broader conclusion: the LPV/H<sub>∞</sub> PMS can be viewed as a **generalized filtering strategy** in a **MIMO** context, i.e., completed with certain degrees of freedom to meet a multitude of control goals and **optimized** according to some user-defined criteria.

## 4.2.2 HILS-assessed PMS real-time performance

<figure><img src="image_83.png" /><figcaption>Figure 4.9: Schematics of the rapid-prototyping test bench built at FEMTO-ST Laboratory in Belfort, France, used to validate the LPV/H<sub>∞</sub> PMS [146].</figcaption></figure>

This section refers to some further validation results, namely those obtained on a dedicated test-bench designed in collaboration with FEMTO-ST Laboratory in Belfort, France. Real battery and supercapacitor are used in the test bench, whereas a complete real-time emulator developed at FEMTO-ST Laboratory represents the fuel cell [154]. A reduced-order version of the controller is employed in real time. Fig. 4.9 shows an expanded view of the interconnections between the different parts used in the assembled test bench. The active load variation is controlled to represent the desired driving cycle scenario by means of a rapid-prototyping platform based on the dSPACE™ MicroAutoBox®II device.

Running hardware-in-the-loop simulations required some *adaptations* having been made. Thus, application of the proposed methodology was resumed, this time on the test bench model, and yielded a reduced-order controller suitable for real-time implementation. Saturations of sources' currents were taken into account in the design. Comparison between real-time and numerical simulation results was performed
under the same scenario. Thus, samples of the measured load current were used to reperform numerical simulation.

Adaptations regarding selection of control references and weighting functions were also necessary, as detailed in [146]. As a global result, some differences between numerical simulation and real-time results were obtained, which are normal since they confirm reasonable expectations. Fig. 4.10 offers a view of real-time evolutions of the three currents of interest, where dynamic separation can again be confirmed and qualitatively compared with homologue numerical simulation results in Fig. 4.8c).

The modeling uncertainty of sources was expected to be the main source of differences here, that is, differences between nominal and real values of parameters characterizing each of the three sources. While datasheet values of parameters were used in numerically-implemented models and

simulations, real values of these parameters – resistances, inductances, capacities – vary in time and may not any longer be equal to the datasheet ones. Therefore, currents of sources had slightly different time evolutions in real time compared to numerical simulation. Meanwhile, as expected, the proposed control strategy performed robustly against these parameter variations.

<figure><img src="image_84.png" /><figcaption>Figure 4.10: Real-time results obtained on the test bench in Fig. 4.9 [146]: the three power sources' currents corresponding to IFSTTAR driving cycle and zoom.</figcaption></figure>

### 4.2.3 Some results of embedding robust PMSs in general-purpose microcontrollers

<figure><img src="image_85.png" /><figcaption>Figure 4.11: Photo of the K-INF demonstrator at GIPSA-lab, built within the technology pre-transfer project funded by Institut Carnot LSI.</figcaption></figure>

To this end, the K-INF PMS design code, written in MATLAB® for the electric vehicle application described in the previous two sections, was translated into executable code, then effectively embedded into the *general-purpose Texas Instruments (TI) C2000 microcontroller* – chosen as target real-time computing device – by using dedicated libraries of MATLAB®.

Funding obtained between 2017 and 2018 from Institut Carnot *Logiciels et Systèmes Intelligents* (LSI) in the framework of the **K-INF: Embedded real-time PMS for multi-source electrical energy systems** technology pre-transfer project allowed further advancing the technology readiness level (TRL) of our PMS solution. Efforts were oriented towards achieving a *laboratory proof of concept*.
<figure><img src="image_86.png" /><figcaption>Figure 4.12: K-INF technology pre-transfer project results: oscilloscope capture of the total time the TI C2000 microcontroller spends with K-INF PMS computations, $T_c$, against sampling time, $T_s$.</figcaption></figure>

The mathematical model of the three-source power supply on board of EV was embedded into dSPACE™ rapid-prototyping system, which was further coupled in closed loop with the microcontroller, in order to support real-time tests. In this way, the **K-INF demonstrator** in Fig. 4.11 – whose functional design is mainly due to Iulian MUNTEANU, who assumed the role of technical expert in this project – was built and put into service at GIPSA-lab, with the help of R&D engineers Gelu IONESCU and Christophe SAVARIAUX, heads of GIPSA-lab Technical Service, and R&D engineers Sophie MIN-PICAULT and Julien MINET.

One of the objectives of this project was to check practical implementation feasibility of the new PMS, from the viewpoint of necessary computation resources (computation time and allocated memory). Indeed, given its complexity, the new control structure appeared to challenge computation

performance of some supporting device. Fig. 4.12 allows concluding that the total computation time $T_c \approx 25$ µs, that is, 25% of the sampling time, $T_s = 100$ µs, programmed in the TI C2000 microcontroller. Also, the total memory used is approximately 20 kB, that is, only 10% of the available memory. These evaluations indicate that there is still sufficient room for possibly accommodating an even more complex PMS, if needed – conversely, a less powerfull (also cheaper) microcontroller can be employed instead.

The demonstrator allowed extensive tests having been performed, under a plethora of driving cycle scenarios, including some standard driving cycles, all of which were user-configured by means of a monitoring console under ControlDesk® (software tool provided by dSPACE™). Rapid-prototyping code allowed suitable replication of real-environment phenomena, like noises, pure delays, filtering, etc. Another monitoring console, this time under MATLAB®, allowed completing visualization of results (time evolutions and frequency analysis).

<figure><img src="image_87.png" /><figcaption>Figure 4.13: K-INF technology pre-transfer project results: time evolutions of variables of interest for Artemis Urban standard driving cycle.</figcaption></figure>

Effectiveness of K-INF PMS was concluded under the whole set of scenarios. Fig. 4.13 gives a synthetic image of some relevant results, where *Artemis Urban driving cycle* was chosen as excitation scenario. The corresponding
electric vehicle speed profile can be seen in Fig. 4.13a), whereas the associated amplitude spectrum of the load current demand is shown in Fig. 4.13d). Performance of DC-bus voltage regulation and time variation of the untracapacitor state of charge are exhibited in Figs. 4.13 b) and e), respectively. Figs. 4.13 c) and f) illustrate convenient dynamic separation of sources in time and frequency domain, respectively.

## 4.3 Robust PMS design for frequency and voltage regulation in multi-source AC microgrids

While remaining in the nowadays "fashionable" domain of microgrids (MGs), the focus is in this section shifts to **AC MGs**, more precisely, to those having a high penetration rate of renewables. In this context, to ensure MG stability and suitable overall performance is a challenge of crucial importance, especially during **islanded** – otherwise said, **isolated** or **autonomous**, or again **off-grid** or **stand-alone** – mode, where low inertia, uncertainties, and intermittent nature of distributed energy resources play an even more constraining role [2], [3].

Thus, **compelling frequency and voltage deviations** to stay within some maximally admissible ranges as imposed by stiff **grid codes** is of focal importance in stand-alone operation mode. Finding pertinent technological solutions to this problem has determined a large research effort being deployed lastly, out of which **use of short-and medium-term storage systems** – e.g., batteries, flywheels, or supercapacitors – has proved to open a promising way towards feasible implementations. The resulted new grid configurations have again emphasized the need for advanced control structures to deal with unexpected disturbances and model uncertainties. Results briefed in this section remain thus focused on how to ensure a judicious management of storage units, in order to enhance the mixture of classical and renewable sources. They are coherent with those in the literature, where it is shown that, if suitable dynamical coordination of storages with other generation sources is achieved and saturations are avoided, then relatively small storages can significantly reduce both frequency [155] and voltage [156] variations.

To fix ideas, the chosen storage system is a **supercapacitor**, which is further inserted into a **Diesel-PV hybrid MG** rated at MW and operating in stand-alone mode, as depicted in Fig. 4.14. The different power sources are connected in parallel to a point of common coupling (PCC) and feed a common load.

The $H_\infty$ robust control design formalism was proposed as solution to both frequency and voltage regulation problems, in a coherent systematic approach based on a suitable control-oriented modelling and rigorous formal translation of various engineering specifications, among which the (micro)grid codes.

<figure><img src="image_88.png" /><figcaption>Figure 4.14: Functional schematics of the stand-alone AC MG considered for $H_\infty$ robust frequency and voltage regulation purposes [157].</figcaption></figure>
The results presented next were obtained within collaboration with Ph.D. student Quang-Linh LAM, working under the supervision of Professor Delphine RIU and myself, and made the object of the following main publications:

* Q.L. Lam, A.I. Bratcu, C. Boudinet, M. Thomas, A. Labonne, D. Riu (2020). Primary frequency $H_\infty$ control in stand-alone microgrids with storage units: a robustness analysis confirmed by real-time experiments. *International Journal of Electrical Power and Energy Systems*, 115, art. no. 105507

– summarized in Subsection 4.3.1 – in which the primary **frequency** regulation problem is first solved within a robust $H_\infty$ approach, then uncertainty in the steady-state value of the supercapacitor SoC is taken into account in a μ-analysis to determine its maximum variation range for which the imposed closed-loop performances are still respected for the considered operating point. With the help of R&D engineers Cédric BOUDINET, Martin THOMAS and Antoine LABONNE from Grenoble Electrical Engineering Laboratory (G2ELab), a rapid-prototyping test bench was configured to support power hardware-in-the-loop simulations under meaningful scenarios;

* Q-L. Lam, A.I. Bratcu, D. Riu (2021). Multi-variable $H_\infty$ control approach for voltage ancillary service in autonomous microgrids: design and sensitivity analysis. *IEEE Access*, 9, 140212–140234

– presented in Subsection 4.3.2 – which deals with an $H_\infty$-based **voltage** controller that robustly forces the voltage magnitude of a PCC, such as to satisfy design requirements.

TRL advancing of K-INF PMS solution developed in the **technology pre-transfer** project – presented previously in Subsection 4.2.3 – was pursued, but this time by changing the use case. Thus:

* **technology transfer** project K-INF: Robust PMS for multi-source electrical energy systems (Gestionnaire d'énergie robuste embarqué des systèmes multi-sources),

funded by Linksium Grenoble Alpes Technology Transfer Office (TTO) between 2018 and 2020, adapted K-INF PMS design methodology – including microcontroller embedding – to the use case of **frequency regulation in MGs with renewable energy sources and hybrid storage units**; more precisely, a **battery-supercapacitor** storage tandem was considered. This made the object of an APP software application deposit, accompanied by a know-how documentation, as follows (alphabetical order of authors):

– A.I. Bratcu, I. Munteanu (2019). *K-INF Multi-storage energy management system robust design for frequency regulation in AC microgrids*. Grenoble Institute of Technology, CNRS, Grenoble Alpes University. *APP software application*;

– A.I. Bratcu, I. Munteanu (2020). *Design procedure of a robust multi-storage energy management for frequency regulation in AC microgrids*. Grenoble Institute of Technology, CNRS, Grenoble Alpes University. *Know-how patent application*.

Some interesting results and conclusions of K-INF technology transfer project are summarized in Subsection 4.3.3 – among them, *quantitative assessment of hybridization benefit* in terms of sources' lifetime savings. Website http://www.k-inf.fr was created for presenting details about this project's history and K-INF PMS technical outcomes and economical benefits in the MG primary frequency regulation use case.

### 4.3.1 Frequency-regulation PMS for an ultracapacitor-based AC microgrid: robust control design and PHILS validation

When high penetration of renewables is aimed at, especially in stand-alone operation, frequency variations can lead to global instabilities [158]. In the MG in Fig. 4.14, primary frequency regulation relies upon participation of both the energy storage system (ESS – the supercapacitor in this case) and the classical source (the Diesel generator), where the task assigned to the former is a faster recovery of the MG frequency, $f_{grid}$, in response to either load or production power variation.

Thus, *dynamic performance* of the frequency response – overshoot, response time, steady-state error – is imposed by the *grid codes*, whose requirements may slightly vary in different countries or regions. These grid
codes can be pertinently "translated" based on the time-domain response in the case where only the classical source participates in primary control – in this way, a template of the MG small-signal frequency deviation can result, like shown in Fig. 4.15. Another dynamic specification concerns the DC-bus voltage regulation (here, the reference value is chosen at $V_{dc}^{ref}=150$ V). Note that in this case it is not about some stiff grid codes to be met, but just to define some suitable performance specification like, for example, indicated in Fig. 4.16 [159], [160].

Time-domain templates in Figs. 4.15 and 4.16 are important, as they further help at deducing the corresponding templates in the *frequency* domain – namely, in the form and parameters of the *weighting functions* – coherently with the $H_\infty$ control design.

<figure><img src="image_89.png" /><figcaption>Figure 4.15: Translation of *frequency grid codes*: time-domain performance specification on the MG frequency variation in response to a step disturbance of ±5% of the load rated active power [161].</figcaption></figure>

<caption>Figure 4.16: Some suitable time-domain performance specification on the DC-bus voltage variation in response to a load step disturbance [161].</caption>

Fig. 4.17 depicting the MG electrical circuit diagram serve at further obtaining the MG *linearized per-unit model* around a typical steady-state operating point. Indeed, per-unit modelling was here preferred for its generally better numerical conditioning, thus improved numerical computation stability, especially when it is about high-order, possibly multi-scale systems [162], [163].

<figure><img src="image_90.png" /><figcaption>Figure 4.17: Electrical circuit diagram of the studied MG for modelling purposes, where notations of interest variables for both frequency and voltage regulation are introduced [157].</figcaption></figure>

Some simplifying, still realistic, assumptions are first adopted. Thus, since frequency control is at this time aimed at, it is sufficient to only consider the *mechanical dynamic* of the Diesel generator – of $T_{diesel}$ time constant – whereas, for example, its local voltage regulation is much faster and not intervening in the frequency control. As such, the circled part of the diagram in the figure is neglected for frequency-regulation modelling purposes, being taken into account for voltage-regulation modelling only (as briefed in the next Subsection 4.3.2). Also, supposing that active power variations – responsible for the MG frequency variations – do not significantly influence the PCC voltage, then PCC voltage direct and quadrature components remain constant (i.e., $\Delta V_{rd}=0$ and $\Delta V_{rq}=0$).
For sake of brevity, detailed state-space modelling is skipped here – it can be found in [161] and [164] – whereas
allowing to drive a conclusion for control purposes, that is, adoption of a **two-level hierarchical control strategy**.

Thus, by adopting the underline notation "$\Delta x$" for denoting linearized per-unit variations of variable $x$ and
following the same justification and guidelines as in the case of multi-storage (DC) MGs considered in previous
Section 4.2, the outer (upper) control level deals with output regulation imposing low-frequency dynamics (e.g.,
supercapacitor and DC-link voltage per-unit variations, $\Delta V_{sc}$ and $\Delta V_{dc}$, respectively, Diesel power per-unit varia-
tions, $\Delta P_{diesel}$, and MG frequency per-unit variations, $\Delta f_{grid}$) and the inner (lower) level concerns current reference
tracking of high-frequency dynamics (e.g., supercapacitor current per-unit variations, $\Delta I_s$, and inverter current $d$-
and $q$-component per-unit variations, $\Delta I_d$ and $\Delta I_{rq}$, respectively).

In this way, the classical principle of cascade control is here illustrated: the **outer loop** results from demand-
ing to reject disturbances coming from the aggregated load power per-unit variations, $\Delta P_{load} - \Delta P_{PV}$ – this
**disturbance-rejection** problem is cast into the $\mathcal{H}_{\infty}$ formalism to yield an $\mathcal{H}_{\infty}$ controller that further provides
references $\underline{\Delta I}_{s}^{ref}$, $\underline{\Delta I}_{rd}^{ref}$ and $\underline{\Delta I}_{rq}^{ref}$ to the **inner tracking loops**. In particular, as the quadrature current $I_{rq}$ does
not participate to frequency control, its variation is set to 0 ($\Delta I_{rq}=0$), hence only two control inputs are effectively
used: $\underline{\Delta I}_s$ and $\underline{\Delta I}_{rd}$. Moreover, this strategy is coherent with considering the power sources as current sources –
whose currents are explicitly controlled by lower-level PI controllers – thus being preferable from an application
viewpoint. Fig. 4.18 shows the proposed global control structure, with some newly introduced notations: $s_{diesel}$ is
the speed regulator droop value, $\underline{H}$ is the MG equivalent inertia constant and $\underline{D}_{load}$ is the load damping constant.

<figure><img src="image_91.png" /><figcaption>Figure 4.18: Block diagram of the $\mathcal{H}_{\infty}$-based two-level control structure dedicated to stand-alone MG primary frequency regulation [164].</figcaption></figure>

Based on the general control configuration in [165], the primary frequency regulation problem can now be
classically cast into an $\mathcal{H}_{\infty}$ formalism, as depicted in Fig. 4.19, where the so-called **P–K** form can be identified,
making appear vector $\underline{\Delta z}$ of **performance outputs**, which must be suitably "conditioned" by the **weighting functions**, $W$, such as to comply with the performance specifications. For purpose of brevity, this form appears
in this figure already completed with the uncertainty $\underline{\Delta}$ blocks, further required by the robustness $\mu$-analysis, as
explained later.
Because disturbance rejection is here aimed at and tracking is not an issue, the S/KS mixed-sensitivity optimization in the standard regulation form must typically be solved [165], with sensitivity functions being defined as

$$S_1(s) = \frac{\Delta V_{dc}(s)}{\underline{\Delta P_{load}}(s) - \underline{\Delta P_{PV}}(s)} \qquad S_2(s) = \frac{\Delta f_{grid}(s)}{\underline{\Delta P_{load}}(s) - \underline{\Delta P_{PV}}(s)} \qquad (4.17)$$

and complementary sensitivity functions

$$KS_1(s) = \frac{\Delta I_s^{ref}}{\underline{\Delta P_{load}}(s) - \underline{\Delta P_{PV}}(s)} \qquad KS_2(s) = \frac{\Delta I_{rd}^{ref}}{\underline{\Delta P_{load}}(s) - \underline{\Delta P_{PV}}(s)} \qquad (4.18)$$

allowing limitation of the $H_\infty$ controller size and bandwidth, with direct impact on the control energy used. This design method requires an appropriate choice of weighting functions $W_{perf}(s)$ to formalize the disturbance rejection performance specifications and weighting functions $W_u(s)$ to translate the constraints imposed to the control inputs. Thus, first-order weighting functions $W_{perf}$, band-pass weighting function $W_{u1}$ and first-order weighting function $W_{u2}$ are used, as detailed in [161].

The optimization problem aiming at minimizing the norm $|| W_{perf}S ||_\infty$ is numerically solved based upon the model directly instantiated with the test bench parameter values, resulting in a multi-variable eight-order $H_\infty$ controller, with an optimal value $\gamma$ of 2.1 and an $H_\infty$ norm of 0.47.

<figure><img src="image_92.png" /><figcaption>Figure 4.19: P–K block diagram for $H_\infty$ primary frequency control design, completed with the N–Δ form for robust performance analysis [160], [161]. To this end, the extended general control configuration in the so-called N–Δ form [165] is included in Fig. 4.19, where N(s) is defined as the connection of the plant together with the weighting functions and the previously designed $H_\infty$ controller. The linear time-invariant uncertain matrix $\Delta(s) = \text{diag}\{\Delta_f(s) \Delta_s(s)\}$, where $\Delta_f(s)$ is the fictive full-block complex uncertain matrix representing the $H_\infty$ norm specifications and $\Delta_s(s)$ is the block-diagonal real uncertain matrix representing the parametric uncertainty in $V_{sc_e}$.</figcaption></figure>

Next, the robustness analysis in the presence of modelled parameter uncertainties is carried out using the μ-analysis. Here, the parameter chosen to answer the robust performance test is the steady-state value of the supercapacitor voltage, $V_{sc_e}$, as it is directly representative of its state of charge (SoC). The $H_\infty$ design initially considers this value as being constant, i.e., in the middle of its variation range (36 V in this case, corresponding to the real supercapacitor used in the test bench).
Note that the weighting functions used for the robust performance analysis and appearing in Fig. 4.19 bear subscript "CL", as they are different from the ones used for the nominal design in the sense of "normalization" by the previously obtained value of $\gamma$: $W_{perf_{CL}}(s) = W_{perf}(s)/\gamma$ and $W_{u_{CL}}(s) = W_u(s)/\gamma$.

Finally, results of computing an upper bound and a lower bound of the structured singular value $\mu$ for the considered controlled MG led to the conclusion that the closed-loop system remains robust in performance up to 37% uncertainty in $V_{sc_e}$ around its design value, which indicates a "robust" variation range theoretically covering the whole supercapacitor's SoC variation domain [161]. This result confirms that the chosen supercapacitor can effectively be employed for robust primary frequency control, as its *SoC can vary from empty to full without resulting in closed-loop stability or performance degradation.*

This subsection ends with some representative *power* hardware-in-the-loop simulation (PHILS) results resumed from [161]. This kind of validation technique supposes using a *power amplifier* in addition to a basic HILS structure, in order to get even closer to the real-world system.

The test bench built for PHIL validation (see Fig. 4.20) – configured in the framework of collaboration with G2ELab – is composed of two main subsystems: the real supercapacitor-based ESS including a PWM-controlled two-quadrant chopper and a PWM-controlled three-phase voltage-source inverter, and a second subsystem aiming at emulating the remainder of the studied MG – that is, the Diesel engine generator, the PV system and the aggregated static load – by means of a real-time digital simulator and a power amplifier.

<figure><img src="image_93.png" /><figcaption>Figure 4.20: PHIL test bench for validation of the stand-alone MG robust control [164].</figcaption></figure>

Two real-time digital simulators, RT-LAB® and dSPACE™, are used in parallel for controlling the hardware parts of the test bench. As regards the software layer of the emulated subsystem, it is composed of MATLAB®/Simulink® models of the Diesel generator, the PV system and the aggregated load, as well as the RT-LAB®-supported control architecture. The real supercapacitor-based ESS is rated at kW-level, a power level much lower than that of the considered MG; therefore, the analogous scaling law was used for dynamic replication [161].

Results in Fig. 4.21 confirm predictions of the robustness analysis: the imposed closed-loop overshoot performance of the DC-bus voltage, $V_{dc}$ (i.e., maximum overshoot of 7.5 V) is preserved regardless of the initial value of supercapacitor SoC, $SoC_{sc}$, meanwhile employing a reasonable control effort (the supercapacitor current, $I_s$, remains below its maximum admissible value – here, 32 A).
<figure><img src="image_94.png" /><figcaption>Figure 4.21: Closed-loop PHIL results for the MG in Fig. 4.17, suggesting robust performance for the two extreme values of the supercapacitor SoC [161]: time-domain responses of supercapacitor voltage $V_{sc}$ (channel 1, 20 V/div), supercapacitor output current $I_s$ (channel 2, 2 A/div), DC-bus voltage variation $\Delta V_{dc}$ (channel 3, 2.5 V/div) and chopper PWM duty cycle $\alpha_c$ (channel 4, 0.2/div) under small load step disturbances of ±5% of load rated active power (± 150 W), where: a) $SoC_{sc_e}=25\%$; b) $SoC_{sc_e}=100\%$ – at the right, zooms in a time interval.</figcaption></figure>

### 4.3.2 Voltage-regulation PMS for an ultracapacitor-based AC microgrid: robust control design and numerical simulation validation

Recalling the stand-alone MG topology case study in Fig. 4.14 in the previous Subsection 4.3.1, the interest moves now to MG robust voltage control. Whereas the majority of works in the literature usually neglect MG interaction dynamics or assume them as lump-sum external disturbance, explicitly including accurate modelling of overall MG dynamics obviously lead to designing a control robustified to interaction dynamics. To this end, in contrast with frequency-control-oriented modelling in the previous subsection, voltage-control-oriented model will now include the circled part in Fig. 4.17, i.e., the detailed model of the Diesel generator together with its primary voltage control.

The supercapacitor is again assigned the role of complementing the Diesel generator in the high-frequency domain, but this time for guaranteeing faster recovery of the line-to-line voltage magnitude at the point of common coupling (PCC), $U_{PCC}$, consequent to a load or PV output reactive power variation. Thus, a template for the PCC line-to-line voltage magnitude deviation in response to a load reactive power step disturbance can be defined as in Fig. 4.22 to formalize grid code requirements in

terms of dynamic performance (overshoot, response time, steady-state error). This template, together with the one for the DC-bus voltage, $V_{dc}$, already defined in Fig. 4.16 in relation to the frequency control, further serve as input data for the robust control design procedure.

A thirteen-order per-unitized state-space model results in this case – with three-phase variables repre-

<figure><img src="image_95.png" /><figcaption>Figure 4.22: Translation of voltage grid codes: time-domain performance specification of PCC voltage magnitude in response to a load step disturbance [157].</figcaption></figure>
sented by their *d* and *q* components – which is submitted to $\Delta I_{load} - \Delta I_{PVd}$ and $\Delta I_{loadq} - \Delta I_{PVq}$ as disturbances corresponding to aggregated-load-PV reactive power variations. The PCC line-to-line voltage magnitude variation, $\Delta U_{PCC}$, exhibits **non-minimum-phase behaviour and oscillations** of its *d* and *q* components due to presence of right-half-plane complex-conjugate zero pairs. For sake of brevity, modelling details are skipped here, they can be found in [157] and [164].

Following the same idea as for the frequency control design, a **hierarchical two-level control architecture** can be again put in place, as shown in Fig. 4.23, where notations of interest variables are visible.

This cascade system is composed of a regulation outer level and a tracking inner level. Being in charge with rejecting disturbances coming from $\Delta I_{load} - \Delta I_{PVd}$ and $\Delta I_{loadq} - \Delta I_{PVq}$ and imposing low-frequency dynamics (e.g., $\Delta V_{dc}$, $\Delta V_{PCCd}$, $\Delta V_{PCCq}$), the outer loop is based on an $\mathcal{H}_{\infty}$ controller and provides $\Delta I_s^{ref}$, $\Delta I_{rd}^{ref}$ and $\Delta I_{rq}^{ref}$ as references to the high-frequency inner PI-based control loops.

<figure><img src="image_96.png" /><figcaption>Figure 4.23: $\mathcal{H}_{\infty}$-based two-level control block diagram dedicated to stand-alone MG primary voltage regulation [157], [164].</figcaption></figure>

$\omega_{grid}$ is considered a time-invariant parameter in voltage regulation, i.e., $\Delta\omega_{grid} = 0$ and thus $\omega_{grid} = \omega_{gride} = 2\pi \cdot 50$ rad/s. As in the previously discussed $\mathcal{H}_{\infty}$-based two-level frequency regulation case, the outer $\mathcal{H}_{\infty}$ plant "sees" $\Delta I_s \equiv \Delta I_s^{ref}$, $\Delta I_{rd} \equiv \Delta I_{rd}^{ref}$ and $\Delta I_{rq} \equiv \Delta I_{rq}^{ref}$, because inner control loops ensures much faster dynamics. Also, the supercapacitor voltage $V_{sc}$ is considered time invariant in the $\mathcal{H}_{\infty}$ controller design, hence $\Delta V_{sc} = 0$. It is also assumed that the Diesel engine generator output voltage is well regulated at its setpoint value, so $\Delta V_{gd} = 0$ and $\Delta V_{qq} = 0$.

Under the above assumptions, the **P–K** form in Fig. 4.24 is obtained to further guide the $\mathcal{H}_{\infty}$ primary voltage control design as an *S/KS mixed-sensitivity optimization problem* [165].

The generalized plant **P** has five inputs, namely: the *d*- and *q*-component of the aggregated-load-PV current variations, $\Delta I_{load} - \Delta I_{PVd}$ and $\Delta I_{loadq} - \Delta I_{PVq}$, respectively, acting as disturbance inputs, $\Delta w$, and the current reference variations $\Delta I_s^{ref}$, $\Delta I_{rd}^{ref}$ and $\Delta I_{rq}^{ref}$, composing the control input vector, $\Delta u$. The measured output vector $\Delta y$ is composed of the DC-bus voltage variation, $\Delta V_{dc}$, and the *d*- and *q*-component of the PCC voltage variation, $\Delta V_{PCCd}$ and $\Delta V_{PCCq}$, respectively. Finally, $\Delta V_{dc}$, $\Delta V_{PCCd}$ and $\Delta V_{PCCq}$, as well as $\Delta I_s^{ref}$, $\Delta I_{rd}^{ref}$ and $\Delta I_{rq}^{ref}$, are chosen as performance outputs, thus making the object of further conditioning by means of some suitable weighting functions, such as to express the performance requirements. Their vector is noted $\Delta z$.

Again, the key of this design method is the appropriate definition of weighting functions "applied" to the performance output vector, $\Delta z: W_{perf}(s)$ to guarantee the performance specifications and $W_u(s)$ to formalize the control input constraints.
<figure><img src="image_97.png" /><figcaption>Figure 4.24: **P–K** block diagram for $\mathcal{H}_\infty$ primary voltage control design [157], [164].</figcaption></figure>

The performance template corresponding to grid requirements imposed in Fig. 4.22 for $\Delta U_{PCC}$ is identically transferred to its $d$-component, $\Delta V_{PCCd}$, because its $q$-component is much smaller than its $d$-component. First-order forms were chosen for both $W_{perf}(s)$ and $W_u(s)$, as detailed in [157].

After having computed a fifteen-state $\mathcal{H}_\infty$ controller as a solution of the S/KS mixed-sensitivity optimization problem, numerical-simulation validations were performed under small-signal-disturbance scenarios. Further, a MATLAB®/Simulink®-based sensitivity analysis was performed to give an idea about the obtained closed-loop robustness. To this end, uncertainties in the steady-state value of the supercapacitor SoC, $SoC_{sc_e}$ (or, equivalently, supercapacitor voltage, $V_{sc_e}$) and the length $l$ of the transmission line connecting the Diesel engine generator to the PCC were chosen as varying parameters.

Fig. 4.25 and Fig. 4.26 present time evolutions of voltages of interest, when taking into account the uncertainty in supercapacitor SoC and in the transmission line length, respectively.

<figure><img src="image_98.png" /><figcaption>Figure 4.25: Time-domain responses to a small (5%) load step disturbance of the load rated reactive power (45 kVAr for the considered MG), when taking into account the uncertainty in supercapacitor SoC, the transmission line length being fixed at $l=1$ km [157]: a) PCC line-to-line voltage magnitude, $U_{PCC}$; b) DC-bus voltage, $V_{dc}$.</figcaption></figure>

Fig. 4.25a) shows that the five curves of the PCC line-to-line voltage magnitude – corresponding to different SoC levels between empty (here taken as 25%) and full (100%) – are practically superposed, which suggests
that performance is unaltered irrespective of the supercapacitor SoC value varying within the considered domain. Fig. 4.25b) exhibits some slight variations of the DC-bus voltage with the SoC value, however still complying with the performance template proposed in Fig. 4.16. These results allow concluding that the designed $\mathcal{H}_\infty$ controller is robust in performance for the whole considered domain of supercapacitor SoC variation, i.e., [25; 100] %.

<figure><img src="image_99.png" /><figcaption>Figure 4.26: Time-domain responses to a small (5%) load step disturbance of the load rated reactive power (45 kVAr for the studied MG), when taking account of the uncertainty in the transmission line length, $l$, the supercapacitor SoC being fixed at $SoC_{sc_e}=56\%$ ($V_{sc_e}=585$ V) [157]: a) PCC line-to-line voltage magnitude, $U_{PCC}$; b) DC-bus voltage, $V_{dc}$.</figcaption></figure>

At their turn, results synthesized in Fig. 4.26 also suggest the closed-loop robustness in performance, this time when $l$ varies between 1 and 10 km. Both sets of results, a) and b), emphasize that the larger $l$ is, the more damped the oscillations of $U_{PCC}$ and $V_{dc}$ are, respectively, because the line resistance increases, determining the system damping coefficient increasing.

The above-reported results of the numerical-simulation-based sensitivity analysis confirm that the chosen supercapacitor can effectively fulfill its role in the robust primary voltage control – as it was also the case for the frequency control, discussed in the previous Subsection 4.3.1 – because its *SoC can vary from empty to full all by maintaining the closed-loop stability and imposed performance*. These results also suggest relatively good flexibility in sizing the transmission line length without trespassing the primary voltage regulation grid codes.

### 4.3.3 Frequency-regulation PMS for a battery-ultracapacitor AC microgrid: some results of microcontroller embedding

Results reported within this subsection were obtained in the framework of a technology transfer project funded between 2018 and 2020, whose aim was to prove the practical feasibility of an $\mathcal{H}_\infty$-based PMS solution and quantify its economical benefits, in order to advance its technology readiness level (TRL) as close as possible to market commercialization. The use case considered to this end was that of the primary frequency regulation in a stand-alone MG equipped with both classical and renewable (PV) sources and with a hybrid energy storage system (HESS) consisting of a battery and a supercapacitor.

The important penetration rate of the renewables justifies in the first place the need of *hybridizing* the storage, that is, to consider two complementary storage technologies. Here, this complementarity is represented by the battery for supplying the load slow, long-term, low-frequency variations and the supercapacitor able to supply fast, instantaneous, high-frequency (and possibly high-magnitude) load variations, respectively. Thus, the supercapacitor preserves its obvious role of "fast" backup storage – as in MG considered in the previous subsections – moreover, it protects the "slow" storage, the battery, from being sollicited by abrupt load variations. In this reliability-aware context, battery lifetime saving results in some quantifiable economical benefits, like, for example, reduction of initial investment costs.
MG configuration in Fig. 4.27 was considered to fix the ideas. Contribution of **K-INF PMS** is to optimally manage this complementarity, such as to result in a "fusion" of the two storages. The used methodology is generalizable to any number of storages/sources, applying the same principle as described in the electric-vehicle use case reviewed in Section 4.2. Some design and implementation details are purposely omitted here, for reasons of intellectual property protection.

<figure><img src="image_100.png" /><figcaption>Figure 4.27: Block diagram of a stand-alone MG with large penetration of renewable energy (PV) and hybrid (battery-supercapacitor) storage, controlled by K-INF PMS.</figcaption></figure>

Funding obtained to pursue K-INF TRL advancing allowed employing the demonstrator built in the technology pre-transfer phase – whose photo has been presented in Fig. 4.11 in Subsection 4.2.3 – within a more complex test bench presented in Fig. 4.28, where a **real** – instead of only simulated – **battery pack** together with its **boost DC-DC converter** were used.

Such a setup enabled real-time tests aimed at submitting the real battery pack to cycles relevant to the frequency-regulation use case, in order to further assess its aging parameters, in particular, the *residual capacity of discharge*.

To this end, the K-INF PMS version adapted to the frequency-regulation use case was embedded on the same TI C2000 microcontroller. In the end, *more than 50% increase of battery lifetime* was predicted based on post-cycling measurements, which suggests significant economical benefits of hybridization combined with K-INF PMS – more details can be found on the project website, http://www.k-inf.fr.

<figure><img src="image_101.png" /><figcaption>Figure 4.28: Photo of the HILS test bench at GIPSA-lab, replicating configuration of K-INF-controlled MG in Fig. 4.27 – here the test bench is used for *battery cycling* in the primary-frequency-regulation use case.</figcaption></figure>

Fig. 4.29 gives an idea about K-INF closed-loop dynamic performance under most unfavourable load power scenario, i.e., consisting of a series of step variations. Fig. 4.29a) presents the time evolutions of load power, $P_{LD}$, Diesel generator power, $P_{GD}$, and supercapacitor power, $P_{SK}$, with the corresponding time evolutions of d-component of inverter current, $i_{rd}$, battery current, $i_{bt}$, and supercapacitor current, $i_{uc}$, shown in Fig. 4.29b).
<figure><img src="image_102.png" /><figcaption>Figure 4.29: Time-domain illustration of the two storage technologies’ dynamic complementarity in the MG primary-frequency-regulation case: a) step-variation load power scenario; b) evolutions of concerned currents.</figcaption></figure>

<figure><img src="image_103.png" /><figcaption>Figure 4.30: Histogram of MG primary-regulated frequency values, with and without storage. Values appear to be closer to the rated one (here 50 Hz) if the storage is present.</figcaption></figure>

Dynamic separation of battery and supercapacitor currents can be concluded based on time evolutions of the concerned currents in Fig. 4.29b), where battery usage is "smoothed" due to presence of supercapacitor, able to supply high-amplitude high-frequency power variations. Hence, promises of **storage hybridization** resulting in more reasonable, less extreme solicitation of sources can be effectively kept. This further implies **aging phenomena being slowed down**, with positive economical effects, like both initial investment cost (CAPital EXpenditure – CAPEX) and operation cost (OPERational EXpenditure – OPEX) reduction.

Finally, a more general viewpoint – namely, concerning the storage beneficial role in renewable-based MG control – is expressed in Fig. 4.30 visualizing a histogram-based comparison of no-storage- *vs*. storage-based PMS: average of the MG frequency

values appear to be closer to the rated one (here 50 Hz) if the storage is present.

## 4.4 Summary of contributions and conclusion

At the end of this chapter a concluding regard is aimed at, with focus on main features and strengths of our contributions.

The research effort reported here concerns the microgrids (MGs) operating in **grid-forming** (stand-alone) mode, namely by considering – without loss of generality – two main use cases: the so-called "mobile" and "stationary" applications, respectively. The feature of disposing of **large penetration of renewables** adds further precision to the studied MG characterization, as well as consequent use of **storage units** as a mean of "taming" the resulting stochasticity and unpredictability.

From a control viewpoint, **heterogeneity** of the plant – that is, the MG to control – is in a first place expressed by the classical-*vs*.renewable-source taxonomy, which can further be extended to storage units too, that is, by employing at least two different storage technologies. It is precisely at this point where our contributions come in, namely by proposing **to transform an initial drawback** – the heterogeneity, mainly expressed by **co-existence of different dynamic natures** – into an advantage: the **complementarity**. A systematic manner of dealing with this is proposed in the **robust $H_\infty$ control framework** and applied to Power Management Systems (PMSs) design, mainly focused on optimally exploiting MG’s sources complementarity.
Indeed, it is its **dynamic** substrate that enables complementarity to justify and meanwhile be exploited for control and coordination purposes.

More precisely, our proposed PMS design procedures are fed with input data characterizing the **dynamic specialization** of sources, as extracted from their datasheets and according to their placement on the energy-density-*vs*.power-density Ragone plot – recalled earlier in Fig. 2.3 in Section 2.2 of motivation Chapter 2 – enabling definition of **own frequencies** (see (2.1) in the same section) as a precise quantification of their dynamic properties. Such an approach proved to be coherent in a broader sense with a **reliability-aware**, thus **sustainable** design.

$H_\infty$ formal framework allowed the PMS design being cast as an *S/KS mixed-sensitivity problem*, with further possibility of performing **systematic robustness analysis**. The use of "fast" storages – like supercapacitors in our case – was proven as beneficial and robustly well-performing in both improving closed-loop dynamic behaviour – in primary regulation of frequency, as well as of voltage – and also in protecting other "slower" storages – like batteries – from being over-sollicited outside their "specialization" operation range, with significant gains of service time extension.

The role of hardware-in-the-loop simulation (HILS) validation was again focal, allowing experiments on dedicated test benches and hence going deeper within the proof of concept. Results of the two technology pre-transfer and transfer projects practically *demythized the fear of a "too complex and computationally heavy" control algorithm, difficult to run with good performance on an ordinary numerical device*.

Thus, **practical feasibility of embedding $H_\infty$-based PMS solutions on general-purpose microcontrollers was largely confirmed.** Moreover, our PMS solution, called K-INF, employs a MATLAB®/Simulink®-based automatic design procedure, easily adaptable to specific use cases and their various needs of multiple design iterations.

<figure><img src="image_104.png" /><figcaption>Figure 4.31: Intuitive representation of PMS design, handling heterogeneity and complementarity towards optimal coordination in multi-source/multi-storage MGs.</figcaption></figure>

As a corollary projection on a more philosophical plane, a suggestive representation of **our PMS control design vision** is given in Fig. 4.31, where the idea of **harmonious fusion** between different power sources naturally emerges as an instance of a more general consensus principle.