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# Delayed Feedback Controller for the Power System Stability of the SMIB System

Sami Tliba, Islam Boussaada, Adrien Mercier, Ali El Ati

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## ► To cite this version:

Sami Tliba, Islam Boussaada, Adrien Mercier, Ali El Ati. Delayed Feedback Controller for the Power System Stability of the SMIB System. Joint IFAC Conference 2025 - 9th IFAC Symposium on System Structure and Control (SSSC), 19th IFAC Workshop on Time Delay Systems (TDS), and 2nd IFAC Workshop on Control of Complex Systems (COSY), CentraleSupélec, Jun 2025, Gif-sur-Yvette, France. hal-05149011

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Submitted on 7 Jul 2025

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# Delayed Feedback Controller for the Power System Stability of the SMIB System

Sami Tliba* Islam Boussaada**,* Adrien Mercier***
Ali El-Ati**

* Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des Signaux et Systèmes (L2S), 91190, Gif-sur-Yvette, France,
(e-mails: {sami.tliba,islam.boussaada}@l2s.centralesupelec.fr)

** IPSA Paris, 94200 Ivry-sur-Seine, France
(e-mails: {islam.boussaada,ali.el-ati}@ipsa.fr)

*** Université Paris-Saclay, CNRS, CentraleSupélec, GEEPS, 91190, Gif-sur-Yvette, France, (e-mails: adrien.mercier@geeps.centralesupelec.fr)

**Abstract:** This work concerns the small-signal stabilization of a Synchronous Generator (SG) connected to an ideal network (infinite bus) through a lossless transformer and a transmission line. The addition of classical Proportional-Integral (PI)-based automatic voltage regulators (AVRs) is known to reduce local mode damping, often imposing the deployment of Power System Stabiliser (PSS) as a mitigation measure. This latter is based on a phase-lead controller in series with a Washout Filter (WF). The whole control structure is designed to comply with the Transmission System Operators (TSO) requirements. The case under study is the Single Machine Infinite Bus (SMIB), submitted to small disturbances around a given initial operating point. This is a practical industrial testbed to validate the connection to the real-world grid of new power generation units. To control the SG voltage while damping the electrical power oscillations, the proposed controller is based on a delayed feedback controller *via* realistic measurements including an integral term. Its gains are set thanks to a Partial Pole Placement (PPP) method relying on the Multiplicity-Induced Dominancy (MID) property. Some simulations on the linear and the full nonlinear model are proposed to show the effectiveness of the approach.

**Keywords:** SMIB, PSS, Time-delay, PPP-based control design, MID property.

## 1. INTRODUCTION

TSO typically ensure large-scale power system stability by taking actions at different time frames, from planning to real-time operation. In the short-term, the acceptability of system wide operating points, according to a given risk policy, is often defined by time-domain simulations. In some cases, corrective actions are taken based on the real-time monitoring of specific performance indicators. However, at the planning stage, assessing the instability risk is extremely challenging since accurate dynamic models of the future assets are unavailable. In practice, TSO maintains system stability in the long term through performance-based grid connection requirements.

In particular, the small-signal stability of Synchronous Machines (SM)-dominated power systems has historically been ensured by installing PSS in large power plants Larsen and Swann (1981). They typically act on the Automatic Voltage Regulator (AVR) to improve the damping of low-frequency oscillatory modes (< 2 Hz) proper to the unit (local modes) but also those involving

several units in large interconnected systems (inter-area modes). In general, large SG are required to include a PSS (EU). In addition to classical performance specifications, such as voltage regulation settling time, RTE, the French TSO, applies robustness requirements to assess the acceptability of the implemented solution. These requirements enable the system to cope with modelling uncertainties, as well as the evolving nature of the grid. Consequently, to gain grid access in France, among other compliance criteria, large power plants must demonstrate sufficient stability margins. Although PSS design for local mode damping has been extensively studied over the decades, recent research has focused on enhancing their contribution to interarea mode damping Elices et al. (2004); Sreedivya et al. (2019); Cardozo et al. (2020). Moreover, defining appropriate performance requirements and compliance verification procedures for Power Oscillation Damping (POD) controllers in power electronic-interfaced resources remains an open challenge Demey et al. (2024).

This paper provides a method for the design of a state feedback-like controller for the SMIB problem described in Kundur (1994), Chapter 12. This controller is based on delayed measurements and contains an integral action. It aims to perform both AVR and PSS functions while ensuring robustness requirements enforced by the French

<footnote>* This work has been partially funded by Réseau de Transport d'Electricité, RTE. The authors would like to thank the power system stability team of the R&D department, namely Philippe JUSTON, Carmen CARDOZO, and Adrien GUIRONNET, for their valuable contributions.</footnote>
TSO, in particular the one concerning the delay margin. Introducing a delay in a feedback control structure inevitably raises stability issues since this feature pushes the system's dimension to infinity, making control of the resulting dynamics more difficult to tackle.

These last years, a new concept of prescribed stabilization based on a pole placement idea has emerged in the control of linear dynamical systems with delays. See, for instance, Boussaada et al. (2018); Mazanti et al. (2021). Named PPP for Partial Poles Placement, this method is founded on two key spectral properties called respectively the multiplicity-induced dominancy (MID) and the coexistent real characteristic roots induced dominancy (CCRID), as discussed in Amrane et al. (2018); Bedouhene et al. (2020). The MID property simply asserts that, under specific conditions, a characteristic root with sufficiently high multiplicity dominates the remaining characteristic roots. In other words, it becomes the spectral abscissa of the corresponding system prescribing the decay rate of the trivial solution. See Michiels and Niculescu (2014) and the references therein for the definition and a deeper discussion about the properties of the spectral abscissa in delay systems.

The remainder of this paper is organized as follows: Section 2 describes the studied system and the models considered here. Section 3 describes the main requirements expected by the TSO for the SMIB in small signal stability operations. Then, the control design procedure, based on PPP leading to a quasipolynomial as a characteristic function, is developed while including an integral term within the control law. Then, Section 4 analyzes the stability conditions in the frequency domain. It shows numerical results for the closed-loop system, for both linear and nonlinear models of the SMIB, in the time and frequency domains. Performances and robustness requirements are examined, discussed and compared to an instantaneous state feedback controller. Finally, some conclusions are given in Section 5.

## 2. DESCRIPTION OF THE SMIB SYSTEM

The system under consideration is sketched in Fig. 1. It is composed of a SG delivering its electrical power to an ideal network through a transmission line of reactance $X_l$. A transformer, of reactance $X_T$, is positioned between the SG and the transmission line (TL). The ideal network is modeled by a voltage source $E_B$, operating at the nominal frequency $\omega_0 = 2\pi f_0$ where $f_0$ is typically equal to 50 Hz in Europe, and to 60 Hz in the USA. The voltage across the SG terminal is $E_t$.

<figure><img src="image_4.png" /><figcaption>Fig. 1. The SMIB system.</figcaption></figure>

The SMIB is a complex system with a nonlinear dynamic, whose parameters depend on the operating point and the SG physical parameters. For electrical power production purposes, the SMIB is supposed to work around a fixed operating point defined by the voltage and the produced power. The linearization of the model is, therefore, of a non-superfluous necessity. Both models are needed to design the control and to analyze the closed-loop properties.

### 2.1 Nonlinear model of the SMIB

The considered system is described in detail in Example 12.3 of Kundur (1994) book. In the following, the notations used to describe the variables and the parameters are those used in Kundur (1994). The compact nonlinear state-space equation is given in (1), where the state vector is $x(t) := [\Delta\omega_r(t) \delta(t) \psi_{fd}(t)]^T$, $\delta$ and $\psi_{fd}$ are electrical variables, and $\Delta\omega_r$ is the SG's rotor speed variation. The control input is the exciter voltage, $u(t) := E_{fd}(t)$. The exogenous input $w(t) := T_m(t)$ is the mechanical torque applied to the SG's rotor, whose power will be converted into electrical power. The current $I := [i_d \ i_q]^T$ delivered by the SG is given by (2). The SG voltage output $E_t := [e_d \ e_q]^T$ is given by (3).

$$ 
\left\{ 
\begin{aligned}
\Delta\dot{\omega}_r(t) &= c_0 w(t) - c_1 \Delta\omega_r(t) - c_2 \psi_{fd}(t)^2 \\
&\quad + (c_3 \cos\delta(t) - c_4 \sin\delta(t)) \psi_{fd}(t) \\
&\quad - c_5 \cos(2\delta(t)) - c_6 \sin(2\delta(t)) - c_7, \\
\dot{\delta}(t) &= \omega_0 \Delta\omega_r(t) \\
\dot{\psi}_{fd}(t) &= c_8 u(t) - c_9 \psi_{fd}(t) \\
&\quad + c_{10} \cos\delta(t) + c_{11} \sin\delta(t),
\end{aligned}
\right. 
\quad (1) 
$$

$$ 
\left\{ 
\begin{aligned}
i_d(t) &= c_{11} \psi_{fd}(t) - c_{12} \sin\delta(t) - c_{13} \cos\delta(t), \\
i_q(t) &= c_{14} \psi_{fd}(t) + c_{15} \sin\delta(t) - c_{12} \cos\delta(t).
\end{aligned}
\right. 
\quad (2) 
$$

$$ 
\left\{ 
\begin{aligned}
e_d(t) &= c_{16} \psi_{fd}(t) - c_{17} \sin\delta(t) + c_{18} \cos\delta(t), \\
e_q(t) &= c_{19} \psi_{fd}(t) - c_{20} \sin\delta(t) + c_{21} \cos\delta(t).
\end{aligned}
\right. 
\quad (3) 
$$

The parameters $c_j \in \mathbb{R}^*$, $j = 0, \dots, 21$, depends on the physical parameters of the SG, the SMIB features ($X_l, X_T$ and $E_B$), and the considered power production operating point. In a more realistic model, the physical parameters of the SG themselves can also depend on the SG state variables, influenced by the power production operating point. For simplicity purposes, they are assumed here to be not dependent to the state variables.

The expressions of the active power $P_t$ and reactive power $Q_t$, produced by the SG, are derived from (2) and (3) and are given by

$$ 
\left\{ 
\begin{aligned}
P_t(t) &= e_d(t) i_d(t) + e_q(t) i_q(t), \\
Q_t(t) &= e_q(t) i_d(t) - e_d(t) i_q(t).
\end{aligned}
\right. 
\quad (4) 
$$

The main controlled output is $y(t) := E_{t,m}(t)$, where $E_{t,m}(t) := \sqrt{e_d(t)^2 + e_q(t)^2}$. In the sequel, the state variables in $x$ will be assumed available by an indirect set of sensor measurements. The effective measured outputs are indeed given by $z(t)^T := [E_{t,m}(t) \Delta\omega_r(t) P_t(t)]$. The equations in (3) and (4) relate these measurements to the state variables.
## 2.2 Linearized model

Let us consider the desired operating point of the SMIB, given by the triple $P_{t_0}$, $Q_{t_0}$, $E_{t,m_0}$, making it on a steady state equilibrium $x_0^T = [\Delta\omega_{r_0} \ \delta_0 \ \psi_{fd_0}]$, where $\Delta\omega_{r_0} = 0$, with the inputs $u_0 := E_{fd_0}$ and $w_0 := T_{m_0}$. Let us denote the variations of the input, output and state variables around the operating point by putting the $\Delta$ symbol before the corresponding variable name, i.e. $\Delta\delta := \delta - \delta_0$, $\Delta\Psi_{fd} := \Psi_{fd} - \Psi_{fd_0}$, ..., $\Delta u := E_{fd} - E_{fd_0}$, $\Delta w := T_m - T_{m_0}$, $\Delta P_t := P_t - P_{t_0}$, $\Delta Q_t := Q_t - Q_{t_0}$, $\Delta E_{t,m} := E_{t,m} - E_{t,m_0}$.

The linearized state equation, with zero initial conditions, is given by

$$ \begin{cases} \dot{\Delta x}(t) = A_3 \Delta x(t) + B_{31} \Delta w(t) + B_{32} \Delta u(t), \\ A_3 := \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & 0 & 0 \\ 0 & a_{32} & a_{33} \end{bmatrix}, \quad B_{31} := \begin{bmatrix} b_{11} \\ 0 \\ 0 \end{bmatrix}, \quad B_{32} := \begin{bmatrix} 0 \\ 0 \\ b_{32} \end{bmatrix}. \end{cases} \quad (5) $$

The linearized measured outputs are given by

$$ \begin{aligned} \Delta z(t) &:= [\Delta E_{t,m}(t) \ \Delta \omega_r(t) \ \Delta P_t(t)]^T \\ &:= N \Delta x(t), \end{aligned} \quad (6) $$

where $N \in \mathbb{R}^{3 \times 3}$ is an invertible matrix derived from the linearization of (3), (2) and (4). Let us denote by $N_i := [N_{i_1} \ N_{i_2} \ N_{i_3}]$, the $i$th row of the matrix $N$. Then, $\Delta y(t) = N_1 \Delta x(t)$. \quad (7)

It is worth mentioning that, for synchronous machines operating at realistic conditions, that matrix $A$ has one real eigenvalue that can be stable or unstable, depending on the chosen operating point. The two other eigenvalues are stable but weakly damped complex conjugate values for all the considered operating points in the TSO requirements.

# 3. CONTROLLER DESIGN

## 3.1 TSO requirements

The French TSO, RTE, prescribes robustness requirements on the voltage regulators to address issues on modeling approximations during control design due to the perpetual changing nature of the power grid. To secure grid access, large power plants are expected to have sufficient stability margins with respect to some physical parameters. In particular, in addition to settling-time within a 5% band of the voltage reference in less than 10 seconds and meeting steady-state error limits, stability margin thresholds are also enforced RTE (2024):

* Modulus margin $M_m \ge 0.34$,

* Complementary Modulus margin $M_{mc} \ge 0.33$,

* Delay margin $\Delta\tau \ge 34$ ms.

The Modulus and Complementary margins are known as the S-based and T-based margins respectively, $S$ and $T$ denoting the closed-loop Sensitivity and Complementary Sensitivity transfer functions. See Skogestad and Postlethwaite (2005); Seiler et al. (2020) for definitions and interpretations of these quantities.

## 3.2 Prerequisites on Delayed Differential Equations (DDE)

Consider the retarded LTI delay differential equation (DDE)

$$ y^{(n)}(t) + \sum_{k=0}^{n-1} \alpha_k y^{(k)}(t) + \sum_{k=0}^{n-1} \beta_k y^{(k)}(t-\tau) = 0, \quad (8) $$

where $y(\cdot)$ is a real-valued unknown function, $\tau$ is a positive time-delay, and $(\alpha_k)_{0 \le k \le n-1}$, $(\beta_k)_{0 \le k \le n-1}$ are some real sequences of coefficients. For a deeper discussion on DDEs, we refer to Hale and Verduyn Lunel (1993); Gu et al. (2003); Michiels and Niculescu (2014).

The characteristic function associated with (8) is the quasipolynomial $\Gamma: \mathbb{C} \to \mathbb{C}$ defined by

$$ \Gamma(s) = P_0(s) + P_\tau(s)e^{-s\tau}, \quad (9) $$

where $P_0$ and $P_\tau$ are the polynomials with real coefficients

$$ P_0(s) = s^n + \sum_{k=0}^{n-1} \alpha_k s^k, \quad P_\tau(s) = \sum_{k=0}^{n-1} \beta_k s^k. \quad (10) $$

Roots of $\Gamma$ are called the spectral values of (8), and they are infinite in number. The exponential stability of the trivial solution of (8) holds true if, and only if, there exists $\gamma > 0$ such that $\mathfrak{R}\{s\} \le -\gamma$ for every root $s$ of $\Gamma$ (see, e.g., Bellman et al. (1963)). When the coefficient $\alpha_m$ is not zero, the degree of the quasipolynomial $\Gamma$ from (9) is the integer $\deg\{\Gamma\} = n + m + 1$ (see for instance Berenstein and Gay (1995); Boussaada and Niculescu (2016).

## 3.3 Controller design for both AVR and PSS functions

Let us denote by $y_{ref}$ the reference desired voltage signal and $\Delta y_{ref}$ its variations around a nominal reference voltage corresponding to a given operating point. To cope with the precision requirement on the voltage regulation, an integral action is included in the proposed control law. This last is

$$ \left\{ \begin{aligned} \varepsilon(t) &= \eta \Delta y_{ref}(t) - \Delta y(t), & \varphi &:= \int_{0}^{t} \varepsilon(\nu) \, d\nu, \quad (11) \\ \Delta u(t) &= K \Delta z(t - \tau) + \gamma \varphi(t - \tau), \end{aligned} \right. $$

which is nothing but a type of state-feedback-plus-integral control law, excepting the measured states are delayed. The control parameters are the delayed state feedback gains $K \in \mathbb{R}^{1 \times 3}$, $\gamma \in \mathbb{R}$, the reference gain $\eta \in \mathbb{R}^*$ and the delay $\tau > 0$. In total, there are five parameters to be determined to assign the dynamics of the closed-loop system, noting that the delay is considered as a parameter during the design procedure.

## 3.4 Closed-loop equations and characteristic function

By combining the control law (11) with the SMIB's linear model given by (5), (6) and (7), it comes

$$ \left\{ \begin{aligned} \dot{\Delta x}(t) &= A_3 \Delta x(t) + B_{32} K N \Delta x(t - \tau) \\ &\quad + \gamma \varphi(t - \tau) + B_{31} \Delta w(t), \\ \dot{\varphi}(t) &= \eta \Delta y_{ref}(t) - N_1 \Delta x(t). \end{aligned} \right. \quad (12) $$

In more compact form, (12) becomes

$$ \left\{ \begin{aligned} \dot{\tilde{x}}(t) &= A \tilde{x}(t) + B \Delta u(t) + B_r \eta \Delta y_{ref}(t) \\ &\quad + M \Delta w(t), \\ \Delta u(t) &= \kappa \tilde{x}(t - \tau), \end{aligned} \right. \quad (13) $$

where $\tilde{x}(t) := \begin{bmatrix} \Delta x(t) \\ \varphi(t) \end{bmatrix} \in \mathbb{R}^4$, $\kappa := [K_x \ \gamma] \in \mathbb{R}^{1 \times 4}$, with $K_x := KN = [K_{x_1} \ K_{x_2} \ K_{x_3}]$, is the augmented gain of the delayed state-feedback control law,
and $A := \begin{bmatrix} A_3 & 0_{3 \times 1} \\ -N_1 & 0 \end{bmatrix}$, $B := \begin{bmatrix} B_{3_2} \\ 0 \end{bmatrix}$, $B_r := \begin{bmatrix} 0_{3 \times 1} \\ 1 \end{bmatrix}$,
$M := \begin{bmatrix} B_{3_1} \\ 0 \end{bmatrix}$.

The characteristic function is

$$ \Gamma(s) := \det(s \mathbb{1}_4 - A - B\kappa), \\ = P_0(s) + P_\tau(s) e^{-\tau s}, \quad (14) $$

where $P_0(s)$ and $P_\tau(s)$ are polynomials with degree $n = 4$ and $m = 3$ respectively, with

$$ \begin{aligned} P_0(s) &:= s^4 + \alpha_3 s^3 + \alpha_2 s^2 + \alpha_3 s + \alpha_0, \\ P_\tau(s) &:= \beta_3 s^3 + \beta_2 s^2 + \beta_1 s + \beta_0, \end{aligned} \quad (15) $$

whose coefficients depend linearly on the system's and the control law parameters, given by

$$ \left\{ \begin{array}{ll} \alpha_0 = 0, & \beta_0 = \gamma (a_{13} a_{21} b_{32} N_{12} - a_{12} a_{21} b_{32} N_{13}), \\ \alpha_1 = a_{12} a_{21} a_{33} - a_{13} a_{21} a_{32}, & \beta_1 = -\gamma a_{11} b_{32} N_{13} - a_{13} a_{21} b_{32} K_{x_2} + a_{12} a_{21} b_{32} K_{x_3}, \\ \alpha_2 = a_{11} a_{33} - a_{12} a_{21}, & \beta_2 = \gamma b_{32} N_{13} - a_{13} b_{32} K_{x_1} + a_{11} b_{32} K_{x_3}, \\ \alpha_3 = -(a_{11} + a_{33}), & \beta_3 = -b_{32} K_{x_3}, \end{array} \right. $$

Because the present case is such that $m = n - 1$, the characteristic equation $\Gamma(s) = 0$ is of retarded type. This specific case allows using the latest main results on PPP with the MID framework, published in Boussaada et al. (2025).

### 3.5 MID property and PPP: the key role played by the Elimination-Produced Polynomial

A characteristic root $s_0$ of $\Gamma$ is said to satisfy the MID property if (1) its algebraic multiplicity, denoted by $M(s_0)$, is greater than one, and (2) it is dominant, which means that all the characteristic roots $\lambda_k$ (an infinite number) of the spectrum are located to the left of $s_0 \in \mathbb{C}_-$ The multiple root $s_0$ is then the rightmost root of the spectrum and corresponds to the spectral abscissa of the quasip polynomial $\Gamma$.

For the case under study, the MID property can be met when $M(s_0) = n + 1$. Then, the obtaining of the unknown parameters is done by following the procedure, which consists of solving sequentially the following set of equations in the controller's parameters,

$$ \Gamma^{(k-1)}(s) \Big|_{s=s_0} = 0, \quad k = 1, \dots, M(s_0), \quad (16) $$

where $\Gamma^{(j)}(s)$ stands for the $j$th derivative of $\Gamma(s)$ in terms of $s$. This is achieved thanks to an elimination procedure throughout the resolution of the equation set (16).

Some useful results to derive the analytical expressions of the controller's parameters rely on the Elimination Produced Polynomial (EPP) defined in Proposition 3.1. The following propositions are due to Boussaada et al. (2025).

**Proposition 3.1.** Consider the quasip polynomial $\Gamma$ from (14)-(15) (with $m = n - 1$). If the real number $s_0$ is a root of multiplicity at least $n + 1$ of $\Gamma$, then $\mathbb{P}(s_0, \tau) = 0$, where $\mathbb{P}$ is called the EPP, given by

$$ \mathbb{P}(s, \tau) = \sum_{k=0}^{n} \binom{n}{k} P_0^{(k)}(s) \tau^{n-k}. \quad (17) $$

**Proposition 3.2.** Consider the quasip polynomial $\Gamma$ from (14)-(15) (with $m = n - 1$). The real number $s_0$ is a root of multiplicity at least $n + 1$ of $\Gamma$ if, and only if,

$$ \mathbb{P}(s_0, \tau) = 0, \quad (18) $$

and we also have

$$ \Gamma(s) = (s - s_0)^{n+1} \int_0^1 \frac{\tau}{n!} \mathbb{P}(s_0, \tau t) e^{-t(s-s_0)\tau} dt, \quad (19) $$

where $\mathbb{P}$ is defined in (17).

Those propositions, proved in Boussaada et al. (2025), allow to find an admissible pair $(s_0, \tau)$ corresponding to a root $s_0$ of $\Gamma$, with multiplicity $M(s_0) = n + 1$. Once the integral form in (19) computed, developed and identified to (14)-(15), it permits the computation of the controller's parameters.

**Remark 3.3.** The last algebraic computations can be very tedious. The use of a symbolic computation program can be helpful. The very large expressions of $K_{x_1}$, $K_{x_2}$, $K_{x_3}$, and $\gamma$ are voluntary omitted by lack of necessary space.

## 4. NUMERICAL APPLICATION

### 4.1 System data

This section shows the results of the application of the proposed control law to the SMIB system described in the Example 12.3, Chapter 12 of the book Kundur (1994). In particular, the same physical parameters for the SG, the line, and the transformer are used. However, the operating point is different. We have considered here the nominal operating condition case specified by RTE in its national grid code provided to power production companies, given in RTE (2024). Hence, for this case study, the following input data are considered for the controller design.

- The line has a reactance value $X_l = 0.6$ p.u..

- The transformer has a reactance value $X_l = 0.15$ p.u..

- For the operating point, the power and voltage requirements at the connection point of the line are $P_{R_0} = 0.9$ p.u., $Q_{R_0} = 0.0$ p.u., $E_{R_0} = 1.0$ p.u.. It corresponds to the power and voltage requirements at the SG output $P_{t_0} = 0.9$ p.u., $Q_{t_0} = 0.1215$ p.u., $E_{t_0} = 1.0091$ p.u..

Here, p.u. stands for per unit, the system of units to express the numerical values after an appropriate normalization of the variables and parameters.

Omitting, for brevity, giving the numerical values of the parameters $c_j$, $j = 0, \dots, 21$, of the nonlinear model in (1), (2), (3), and (4), the ones of the linearized model are given hereafter

$$ \begin{align*} A_3 &= \begin{bmatrix} -0.00071429 & -0.10861 & -0.1306 \\ 376.99 & 0 & 0 \\ 0 & -0.20516 & -0.39512 \end{bmatrix}, \\ B_{3_1} &= \begin{bmatrix} 0.14286 \\ 0 \\ 0 \end{bmatrix}, \quad B_{3_2} = \begin{bmatrix} 0 \\ 0 \\ 0.13659 \end{bmatrix}, \\ N &= \begin{bmatrix} 0 & -0.16645 & 0.39532 \\ 1 & 0 & 0 \\ 0 & 0.75519 & 0.9106 \end{bmatrix}, \\ N_1 &= \begin{bmatrix} 0 & -0.16645 & 0.39532 \end{bmatrix} \end{align*} $$
<figure><img src="image_5.png" /><figcaption>Fig. 2. Real (right) and imaginary (left) parts of the 4 roots of the obtained *Elimination-Produced Polynomial*.</figcaption></figure>

## 4.2 PPP for the control law's gains

Figure 2 shows the real and imaginary parts of the four roots of the degree-4 EPP, derived from $\Gamma$ in (14)-(15). One can observe that, until $\tau \approx 0.25$ s, there is only one root of multiplicity 4. Beyond this characteristic delay value, there are two complex-conjugate roots coexisting with one real root of multiplicity 2. But one can observe the perpetual dominance of a negative real root, for any delay $\tau$. These diagrams allows to select an appropriate real root $s_0$ of multiplicity 5 to cope with the settling time requirement. After several trials with different values of $s_0$, the better choice for the multiple root, offering a good trade-off between performances and robustness was found to be $s_0 = -5$. As shown by Fig. 2, several possible delays give this value but only one makes the EPP having only real roots. In that case, $\tau \approx 73.862$ ms, the corresponding matrix gain $K \approx [-24.894 \ 0 \ -34.856]$, the integral action gain $\gamma \approx 26.696$. The reference gain $\eta$ is taken unitary for graphics readability purposes. But it can be adjusted with convenience, depending on the control interface. It is worth noting that one gain of $K$ is zero, meaning the uselessness of the corresponding measurement, say $\Delta\omega_r$ here, which can state an advantage in practical implementation.

For comparison purposes, an instantaneous control law, ie having $\tau = 0$, with an integral action has been designed to place the same root but with a multiplicity equal to 4. A simple computation shows that the corresponding gains would be $K \approx [-147.91 \ 5544.4 \ -93.404]$ and $\gamma \approx 187.67$, which are very greater than the ones with the delayed measurements. Moreover, all the measurements are now required. Figure 3 shows the closed-loop time responses performed with the SMIB's nonlinear model (1)-(2)-(3)-(4) for both the instantaneous and the delay-based control law. The voltage reference tracking responses are pretty similar. Still, the rejection of a step disturbance on the mechanical torque input leads to more pronounced overshoots for the delay-based control law compared to the instantaneous one. The control input responses in closed-loop for both cases are shown in Figure 4. Their magnitudes are quite similar, although the delay-based

control law has lower gains. Lastly, for both control laws,

<figure><img src="image_6.png" /><figcaption>Fig. 3. SG voltage $E_{t,m}$ (top), active power $P_t$ (middle), reactive power $Q_t$ (bottom). The closed-loop system inputs are in blue, responses in red for the delayed case, in green for the instantaneous case.</figcaption></figure>

the obtained input modulus and complementary modulus margins largely satisfy the requirements for the design model, with $M_m \approx 0.647$ and $M_{mc} \approx 0.76106$ for the delay-based case, and $M_m = 1$ and $M_{mc} \approx 0.872$ for the instantaneous one. It has also appeared that the delay-based control law gives a very good delay margin of
443 ms, larger than the one with the instantaneous one ($\simeq 62.6$ ms).

<figure><img src="image_7.png" /><figcaption>Fig. 4. SG Exciter Voltage (control input) in closed-loop. Delayed case in red, instantaneous case in green.</figcaption></figure>

## 5. CONCLUSION AND FUTURE WORKS

This paper has shown the efficiency of PPP method relying on the MID property, applied to the voltage control problem of a SMIB power system. The classical two-step control for such a system, say the design of an AVR and a PSS, has been replaced by a single controller described by a state-feedback control law with an integral action, using delayed measurements. The delay has been considered as a control parameter in addition to the other gains of the control law. The PPP approach has allowed to obtain in one step, all the needed gains for the delay-based control law, with the substantial benefits of a greater delay margin and a reduced number of needed measurements than the instantaneous one. Some future works could address the conversion of the proposed delayed state feedback plus integral action control law into the standard AVR+PSS form, which is more suitable for the power system community, following a similar method to the one proposed in Elices et al. (2003).

## REFERENCES

* Amrane, S., Bedouhene, F., Boussaada, I., and Niculescu, S.I. (2018). On qualitative properties of low-degree quasipolynomials: further remarks on the spectral abscissa and rightmost-roots assignment. *Bull. Math. Soc. Sci. Math. Roumanie (N.S.)*, 61(109)(4), 361–381.

* Bedouhene, F., Boussaada, I., and Niculescu, S.I. (2020). Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay. *Comptes Rendus. Mathématique*, 358(9-10), 1011–1032.

* Bellman, R., Cooke, K.L., Bellman, R., and Gillis, J. (1963). Differential-difference equations.

* Berenstein, C.A. and Gay, R. (1995). Exponential polynomials. *Complex Analysis and Special Topics in Harmonic Analysis*, 198–259.

* Boussaada, I., Tliba, S., Niculescu, S.I., Ünal, H.U., and Vyhlídal, T. (2018). Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system. *Linear Algebra Appl.*, 542, 589–604.

* Boussaada, I., Mazanti, G., Niculescu, S.I., and Michiels, W. (2025). Decay Rate Assignment through Multiple Spectral Values in Delay Systems. *IEEE Transactions on Automatic Control*, 70(2), 830–844.

* Boussaada, I. and Niculescu, S.I. (2016). Characterizing the codimension of zero singularities for time-delay systems: a link with Vandermonde and Birkhoff incidence matrices. *Acta Appl. Math.*, 145, 47–88.

* Cardozo, C., Diaz-Garcia, A., Giannuzzi, G., Torresan, G., Xavier, F., Cordón, A., Coronado, L., Zaottini, R., and Pisani, C. (2020). Small signal stability analysis of the angle difference control on a hvdc interconnection embedded in the ce synchronous power system. In *2020 IEEE/PES Transmission and Distribution Conference and Exposition (T&D)*, 1–5.

* Demey, P., Noupowo Kenne, J.L.R., Cardozo, C., Guironet, A., Torresan, G., and Tliba, S. (2024). Compliance criteria for power oscillation damping functions in power electronic interfaced resources. In *23rd Wind & Solar Integration Workshops 2024. energynautics, Helsinki, Finland*.

* Elices, A., Rouco, L., Bourles, H., and Margolin, T. (2003). Conversion of state feedback controllers to the standard avr+pss form. In *2003 IEEE Bologna Power Tech Conference Proceedings.*, volume 2, 7 pp. Vol.2–.

* Elices, A., Rouco, L., Bourles, H., and Margotin, T. (2004). Physical interpretation of state feedback controllers to damp power system oscillations. *IEEE Transactions on Power Systems*, 19(1), 436–443.

* (EU), C.R. (2016). 2016/631 of 14 april 2016 establishing a network code on requirements for grid connection of generators.

* Gu, K., Kharitonov, V., and Chen, J. (2003). *Stability of time-delay systems*. Control Engineering. Birkhäuser Boston, Inc., Boston, MA. doi:10.1007/978-1-4612-0039-0.

* Hale, J.K. and Verduyn Lunel, S.M. (1993). *Introduction to functional-differential equations*, volume 99 of *Applied Mathematical Sciences*. Springer-Verlag, New York.

* Kundur, P.S. (1994). *Power system stability and control*. Power Syst. Eng. Series. McGraw-Hill Education.

* Larsen, E.V. and Swann, D.A. (1981). Applying power system stabilizers part i: General concepts. *IEEE Transactions on Power Apparatus and Systems*, PAS-100(6), 3017–3024.

* Mazanti, G., Boussaada, I., and Niculescu, S.I. (2021). Multiplicity-induced-dominancy for delay-differential equations of retarded type. *J. Differ. Equ.*, 286, 84–118.

* Michiels, W. and Niculescu, S. (2014). *Stability, control, and computation for time-delay systems: An eigenvalue-based approach*, volume 27 of *Advances in Design and Control. Soc. Ind. Appl. Math, Philadelphia, PA*, second edition.

* RTE (2024). *Documentation technique de référence*. RTE. URL https://www.services-rte.com/files/live/sites/services-rte/files/documentsLibrary/20241225_DTR_7401_fr.

* Seiler, P., Packard, A., and Gahinet, P. (2020). An introduction to disk margins [lecture notes]. *IEEE Control Systems Magazine*, 40(5), 78–95.

* Skogestad, S. and Postlethwaite, I. (2005). *Multivariable Feedback Control: Analysis and Design*. Wiley.

* Sreedivya, K., Jeyanthy, P.A., and Devaraj, D. (2019). An effective avr-pss design for electromechanical oscillations damping in power system. In *2019 IEEE Int. Conf. on Clean Energy and Energy Efficient Electronics Circuit for Sustainable Development (INCCES)*, 1–5.