# Adaptive predictive approach for emergency voltage control of power networks

Sylvain Leirens, Jean Buisson

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Sylvain Leirens, Jean Buisson. Adaptive predictive approach for emergency voltage control of power networks. IEEE/PES Transmission and Distribution Conference and Exposition - Latin America, Aug 2008, Bogota, Colombia. pp.527-532. [hal-00689900](https://hal.science/hal-00689900v1)

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# Adaptive predictive approach for emergency voltage control of power networks

S. Leirens, J. Buisson

***Abstract***—In this paper, power networks are considered as hybrid non-linear dynamical systems for dealing with emergency voltage control. A hybrid approach based on a symbolic off-equilibrium linearized model and predictive control is presented and illustrated on a 9-bus power system case study.

***Index Terms***—Hybrid systems, predictive control, symbolic linearization, electric power system, voltage collapse.

## I. INTRODUCTION

The human and economic consequences of power outages have shown that the reliable operation of electricity transport and distribution networks plays a crucial role in modern societies, e.g. see recent problems in the US and Canada, as well as in Europe or Latin America [1], [2]. The safe operation has to be carried out not only in regular conditions, but also when the system is close to its limits. A great part of the research effort is devoted to explore new ways to improve power grid operation.

Moreover electric power networks are experiencing rapid and important changes, in particular in the way they are exploited and managed. The development of interconnections between countries (e.g. in Europe) leads to complex large scale dynamical systems. New economic rules due to the growing of energy markets induce unpredicted power flows and demand for reduction of security margins. The number of actors increases as industrial suppliers and households start to feed electricity into the network. In this context, network operators need new control schemes, especially in the case of an emergency, e.g. when a fault occurs such as transmission line drop, loss of generation or any important failure in the system which may lead to partial or complete blackouts.

In general, the system is strongly nonlinear and can work in a wide range of operating conditions possibly far from equilibrium points, particularly in emergency cases. The behavior of power systems is characterized by complex interactions between continuous and discrete dynamics: power systems exhibit a so-called hybrid behavior. Continuous dynamics are mainly driven by components such as generators and loads, and are usually represented by differential algebraic equations. The discrete behavior arises from the nature of connected elements: e.g. capacitor banks, line breakers and limiters in voltage regulators, or the way they are controlled: e.g. load

shedding, on or off switching of generator, load tap changer to control the ratio of a transformer. As a consequence, electric power networks are modeled as non-linear hybrid systems. Power networks moreover tipically span a wide range of time scales and large geographical areas.

Such complex systems are generally controlled in a hierarchical way and control takes place at different layers with a decomposition based on space and time divisions [3]. In first approximation, at the lower layer the controllers act directly on the actuators of the physical system with fast and localized control. At the higher layer, controllers provide set-points and specify constraints to the controllers of the lower layer. The proposed approach is hybrid in the sense that it deals with both continuous and discrete values in an integrated way. Model based approaches and global control for power systems becomes conceivable since wide-area and phasor measurements are available, and utilities increasingly demand for wide-area control, protection and optimization systems.

This paper deals with the problem of voltage instability after disturbances, usually called emergency voltage control. However our approach is not restricted to this particular issue. The current protection schemes against voltage collapse are generally ruled-based and consist in load shedding and reactive power compensation associated with strong operator training. However, the non-linear behavior of power systems makes these rules strongly dependent on the operating conditions. After a disturbance such as breaking of a transmission line, the generation and transmission system may not have sufficient capacity to provide the loads with power and voltage instability may occur. An illustration of this phenomenon is presented in section II. The principal objective of the control system is to minimize the effects of such a disturbance on the quality of the supplied energy. Bus voltages (the voltage plan) must remain into an acceptable range, e.g. more or less 10% of the nominal value. Moreover, the operation of power grids may have to fulfill some other practical requirements such as physical limitations of interconnected elements, minimization of power losses and economic objectives.

In section II we present an illustration of the voltage instability phenomenon with a 9-bus case study. In section III we detail the power system modelling. The section IV presents the steps to build up an accurate hybrid prediction model of power network, which is the main contribution presented in this paper. The basic concepts of predictive control and the special features of the hybrid approach are presented in section V. Due to space limitations, more details about the

S. Leirens is with the Electronic and Electrical Engineering Department, Universidad de Los Andes, Bogotá, Colombia. E-mail: sleirens@uniandes.edu.co

J. Buisson is with the Hybrid System Control Group, Supélec – IETR, Cesson-Sévigné, France.
way to solve the mixed optimization problem associated with
the control scheme can be found in [4], [5]. The results of this
approach applied to the 9-bus network introduced in section
II is presented in section VI. The section VII gives some
concluding remarks and future directions.

II. POWER SYSTEM EXAMPLE – VOLTAGE STABILITY ISSUE

Consider the power system shown on figure 1. This case
study allows to reproduce most of the phenomena of interest
such as voltage instability and collapse. All the numeric values
are expressed in per unit (p.u.).

$G_\infty$ is a generator modeling a surrounding network which is assumed to be strong. $G_1$ and $G_2$ are generators equipped with average voltage regulator (AVR) and overexcitation limiter (OXL), and represent simplified models of connected sub-networks. $L_1-L_9$ are transmission lines. $C_1$ is a flexible AC transmission system (FACTS). $C_2$ is a capacitor bank. $tr_1$ and $tr_2$ are transformers with fixed ratio. $tr_3$ is a transformer equipped with a load tap changer (LTC). $l_1, l_2$ and $l_3$ are dynamic loads modeling sets of consumers.

<figure><img src="image_2.png" /><figcaption>Fig. 1. 9-bus power network</figcaption></figure>

Now consider a fault on line $L_2$ at time $t = 30$ s which corresponds to the disconnection of both extremities of the transmission line. As illustrated on figure 2, at time $t = 60$ s which is only 30 s later, an example of manual control consisting in load shedding of 10% of all the loads as well as maximum reactive power compensation (FACTS $C_1$ and capacitor bank $C_2$) is not sufficient to prevent voltage collapse. Bus voltages start to collapse when the load power consumption reach the maximum transfer capability of the system ($t \approx 100$ s).

A major source of power outages is voltage instability. This
dynamic phenomenon arises when the controllers of the loads
attempt to restore the consumed power beyond the capacity
(physical limit) of the production-transport system [6]. It may
occur following the outage of one or more components in the
system such that the load demand can not be statisfied with

<figure><img src="image_3.png" /><figcaption>Fig. 2. Voltage instability phenomenon</figcaption></figure>

a physically sustainable profile of the voltage plan, typically
voltage levels becomes very low. The reduced capacity of the
network together with the requested load consumption require
corrective actions to avoid that involved dynamics drive the
system into unwanted or unacceptable states.

The purpose of emergency voltage control is to supply a set of corrective actions to apply to the system following an outage while fulfilling physical and operational constraints. Usually, the control objectives are specified as follows. First, to achieve a steady-state point of operation allowing the voltage plan to stay between 0.9 and 1.1 p.u, that is close to the nominal values to fulfill with safety and quality of energy requirements. Second, the use of control means has to be optimized (economic aspects). For example, the use of capacitor banks has to be prefered in comparison to load shedding which is the ultimate option as it disconnects consumers from the grid.

The need for more accurate operation techniques as well
as computation capabilities of modern computers call for new
methods and algorithms. The work presented in this paper
deals with the non-linear and hybrid aspects of electric power
networks in the case of emergency voltage control. The next
two sections present the modeling framework in details.

III. HYBRID MODEL OF POWER SYSTEMS

The dynamics involved in voltage stability issues are said
to be slow (time constants about 30–60 s) compared with
electromechanical ones involved in frecuency issues. Thus
quasi-static models (QSS) are commonly used to deal with
voltage instability. QSS models assume that frecuency is
constant, that is fast dynamics are neglected. The models are
equivalent one-line diagrams and voltage, current and state
variables are represented by phasors. Phasor magnitudes are
shown as capital letters, e.g. **V**. Column vectors and matrices
are shown as bold characters, e.g. **x** and superscript **T** denotes
transpose. Time derivatives appear with a dot, e.g. **\dot{x}**.

Generators are modeled using the well known steady-
state approximation of the synchronous machine equations
and associated control devices, and the OXL is modeled by
a saturation element. This mainly results in neglecting the
electromechanical oscillations [6]. Load is a common term for
aggregates of many different devices which are mainly voltage
dependent. Thus load dynamics are considered from the point
of view of power restoration and are described by a smooth
non-linear differential equation [7]:

$$T_p \dot{x}_{lp} + x_{lp} = P_s(V_l) + P_t(V_l) \quad (1)$$

$$P_l = (1 - s_l k_l) (x_{lp} + P_t(V_l)) \quad (2)$$

where $x_{lp}$ is a continuous state variable, $V_l$ the load voltage magnitude, $P_s(V_l) = P_0 V_l^{\alpha_s}$ and $P_t(V_l) = P_0 V_l^{\alpha_t}$ are the steady-state and transient voltage dependencies respectively, $P_l$ is the actual active load power and $T_p$ is the active power recovery time constant. A similar model is used for the reactive load power considering the following variables: $x_{lq}$, $Q_s(V_l) = Q_0 V_l^{\beta_s}$, $Q_t(V_l) = Q_0 V_l^{\beta_t}$, $Q_l$ and $T_q$. $k_l$ is a constant load shedding step and $s_l$ is a discrete control variable which takes its values in a bounded discrete set. $s_l$ and $k_l$ are used in both active and reactive power equations. Let $x = (x_{lp}, x_{lq})$ denote the internal state of the load.

Transformers are modeled using an impedance in series with an ideal transformer whose ratio is denoted $n_t$. The presence of a Load Tap Changer (LTC) allows the ratio to vary tap by tap within bounds. Since the size of a tap step is quite small (usually in the range of 0.5%–1.5% of the nominal ratio), to simplify the control design, $n_t$ is considered as a bounded continuous variable. Two kinds of reactive power compensation devices are considered in this paper: capacitor banks and Flexible AC Transmission Systems (FACTS). The number of connected parts $s_c$ of a capacitor bank can only take its values in a bounded discrete set whereas a FACTS is considered here as a continuously varying capacitor whose capacity value is adjusted by a variable $n_c$ which can takes its value in a bounded continuous set. Transmission lines are modeled using complex impedances. If a line can be (dis)connected in order to modify the power system topology, a boolean control variable $\delta_L$ is used to represent the state of the line, connected (1) or disconnected (0). More details about the modeling of the network components can be found in [4].

The hybrid nature of power systems is particularly characterized by the presence of two kinds of control inputs: discrete ones arise from loads ($s_l$), capacitor banks ($s_c$) and transmission lines ($\delta_L$) whereas continuous ones arise from transformers ($n_t$) and FACTS ($n_c$). In the sequel, the discrete control vector is denoted as $\mathbf{u}_d^T = (\mathbf{s}_l^T \mathbf{s}_c^T \mathbf{\delta}_L^T)$ and the continuous one as $\mathbf{u}_c^T = (\mathbf{n}_t^T \mathbf{n}_c^T)$. A discrete disturbance is considered in this paper: a transmission line is lost, i.e. unintentionally disconnected. This discrete disturbance together with the state of the generators (maximum excitation limitation) and the discrete control inputs defines the discrete operating mode $i$ of the network.

The general power system model relevant to this study takes the form of a set of differential-algebraic equations (DAE), in each mode $i$:

$$\dot{\mathbf{x}} = \phi_i(\mathbf{x}, \mathbf{y}, \mathbf{u}_c) \quad (3)$$

$$\mathbf{0} = \psi_i(\mathbf{x}, \mathbf{y}, \mathbf{u}_c) \quad (4)$$

which is a non-linear and hybrid DAE multimodel relationship, where $\mathbf{x}$ is the load state vector and the output vector $\mathbf{y}$

typically includes bus voltage magnitudes. This model is useful to perform simulations. Nevertheless, it requires dedicated algorithms such as DASSL [8], [9] and a special attention to the interactions between continuous and discrete dynamics [10], [11].

## IV. LINEARIZATION – ADAPTIVE SCHEME

Power systems are strongly non-linear and to predict the system evolution, a feasible approach is to use locally a linear or, more generally, affine model. In [12], several methods are proposed to obtain predictions of the state and output trajectories. But the reduction of equations (3)–(4) to an explicit form is done using the current values of the discrete variables and thus the linearized models obtained that way cannot handle any hybrid aspects. In [13], the continuous state-input space is gridded into polyhedral partitions. Then for each polytope and combination of the discrete inputs, the system response is approximated, in general, by a different affine model. To be accurate, the gridding needs to be tight enough and leads to a great number of affine models. This involves to reach a compromise between the so-obtained combinatorial complexity and needed accuracy.

An alternative method is detailed in this paper aiming to get an accurate prediction model and to avoid combinatorial explosion. It consists in symbolically linearize equations (3)–(4) with respect to continuous variables, i.e. the load state $\mathbf{x}$ and the continuous control input $\mathbf{u}_c$. The operating point denoted $(\mathbf{x}_0, \mathbf{u}_{c0})$, which is not necessary an equilibrium, and the discrete mode $i$ are symbolic parameters of the linearized model. Thus the evolution of the network can be predicted with accuracy since the model used to compute the prediction is adapted to the current operating point. Moreover this modeling framework allows to handle the hybrid aspects explicitly. In this study, the operating point is assumed to be measured or estimated.

Nevertheless almost all power systems are large scale systems. The symbolic computation required to directly linearize equations (3)–(4) becomes too complex and rapidly unfeasible when the network size exceeds a few buses. The complexity is reduced by making profit of the particular structure of the model. The generation and consumption part (basically generators and loads) of the network forms a set of non-linear equations whereas the transportation part (basically lines, compensation devices and transformers) form a linear set of equations. The model can then be divided in two subsystems as shown on the figure 3.

The non-linear subsystem is represented by the following set of equations:

$$\mathbf{x} = \mathbf{f}(\mathbf{x}, \mathbf{V}_l) \quad (\text{load}) \quad (5)$$

$$\mathbf{I}_l = \mathbf{g}(\mathbf{x}, \mathbf{V}_l, \mathbf{s}_l) \quad (\text{dynamics}) \quad (6)$$

$$\mathbf{0} = \mathbf{h}(\mathbf{V}_g, \mathbf{I}_g) \quad (\text{generators}) \quad (7)$$

where generator voltages and currents are denoted as $\mathbf{V}_g$ and $\mathbf{I}_g$, and load voltages and currents are denoted as $\mathbf{V}_l$ and $\mathbf{I}_l$.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Decomposition of the power network</figcaption></figure>

The linear subsystem is represented by the following equation:

$$w_{out} = M w_{in} \quad (8)$$

where $w_{out}$ and $w_{in}$ are constituted of generator and load voltages and currents $V_{g,l}$ and $I_{g,l}$, depending on the causality assignment in the setting up of the equations (5)–(8) [4], [14]. The elements of the matrix $M$ depend on the control inputs $u$. To set $\dot{x}_l = 0$ in (5) leads to consider a purely static problem which is similar to the load flow calculation.

The linearized model is computed in two steps. In the first step, the set of equations (5)–(7) is symbolically linearized considering the following formal parameters $x_0$, $u_{c0}$ and $i$, leading to a set of parametrized jacobian matrices. The computations are performed off-line using the software package Mathematica [15]. The second step consists in updating online the so obtained symbolic model with the current values of the operating point and the discrete mode.

Time is discretized with a sample period $T_s$ and using backward-Euler method to approximate the derivative, the following discrete time affine model of the power network is obtained:

$$x(k+1) = A_i x(k) + B_i u_c(k) + a_i \quad (9)$$

$$y(k) = C_i x(k) + D_i u_c(k) + c_i \quad (10)$$

where $x(k) \equiv x(kT_s)$, the matrices $A_i$, $B_i$, $C_i$ and $D_i$ as well as the vectors $a_i$ and $c_i$ depend on the discrete mode $i(k)$ and current operating point $(x_0, u_{c0})$. $u_c(k)$ is the continuous valued control inputs vector applied to the system from instant $k$ to instant $k+1$ and is assumed to be constant during the sample period. Due to the presence of generator excitation limiters, it is relevant to take into account the state of the generators (maximum excitation limitation) in the prediction computation. This is done by adding a set of constraints:

$$F_i x(k) + G_i u_c(k) \le f_i \quad (11)$$

to the model (9)–(10). These constraints in the continuous state-input space (mapped by $x$ y $u_c$) describe the state of the generators with respect to the discrete mode $i$. The resulting model has now a piecewise affine (PWA) form and allows to predict the evolution of the state of the generators. The

linearization procedure is resumed on the figure 4. It should be noticed that the computation effort required by this approach is influenced more by the number of connected generators and loads than by the number of buses.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Prediction computation diagram</figcaption></figure>

A special attention must be given to the introduction of LTC dynamics in the model. A good prediction requires that tap changes be synchronous with control, i.e. sample times. Otherwise the prediction model is not coherent with the system to control. A basic approach is to choose a sample time equal to the tap change delay and to introduce the following equation in the model (9)–(10):

$$n_t(k+1) = n_t(k) + \Delta n_t(k) \quad (12)$$

where $n_t$ is now a state variable and the tap change $\Delta n_t$, a discrete control input with $\Delta n_t \in \{-n_{step}, 0, n_{step}\}$. To control the LTC, a kind of bang-bang control strategy is locally implemented:

$$V_t^{ref}(t) = \begin{cases} V_t^{ref,min} & \text{if } \Delta n_t = -n_{step} \\ V_t^{ref,max} & \text{if } \Delta n_t = n_{step} \\ V_t(t) & \text{otherwise} \end{cases} \quad (13)$$

where $V_t^{ref}$ is the LTC voltage reference, $V_t$ is the secondary voltage of the transformer, $V_t^{ref,min}$ and $V_t^{ref,max}$ are lower and upper bounds on $V_t^{ref}$ respectively and $n_{step}$ corresponds to one tap change of the transformer. The transformer ratio $n_t$ between times $k$ and $k+1$ is then coherent with the optimal value of $\Delta n_t$ computed by the controller at time $k$. An extension of this strategy is described in [4] when sample time and tap change delay differ.

A controller designed for this PWA model is likely to control the behavior of the hybrid nonlinear system efficiently. At each sample time, the model update requires a few computations and the controller only acts in a small region around the operating point that the system was linearized at. The next section is dedicated to a brief presentation of the predictive control strategy and mainly focuses on the way to deal with the hybrid nature of the prediction model.

## V. PREDICTIVE CONTROL TECHNIQUE

Model predictive control (MPC) consists in a control algorithm based on solving on-line a finite time optimal control problem using a receding horizon approach. This is summarized in the following steps [16]: at time $k$ and for the current
state $x(k)$, an open-loop optimal control problem over a future time interval is on-line solved taking into account the current and future constraints; the first step in the optimal control sequence so obtained is applied and then the procedure is repeated at time $k + 1$ using current state $x(k + 1)$. The solution is converted into a closed-loop strategy by using the measured or estimated value of $x(k)$ as the current state. The stability of the resultant feedback system can be established by using the fact that the cost function acts as a Lyapunov function for the closed-loop system [17].

The prediction model (PWA) is time invariant thus the time origin can be used as the current time step in the equations without loss of generality. Let $\mathbf{U}_N$ be the sequence of controls on the prediction horizon, chosen of length $N$:

$$ \mathbf{U}_N = (\mathbf{u}^T(0) \ \mathbf{u}^T(1) \ \dots \ \mathbf{u}^T(N-1))^T \quad (14) $$

with $\mathbf{u} = (\mathbf{u}_c^T \ \mathbf{u}_d^T)^T$ and given the following cost function:

$$ J_N(\mathbf{x}(0), \mathbf{U}_N) = F(\mathbf{x}(N)) + \sum_{k=0}^{N-1} L(\mathbf{x}(k), \mathbf{u}(k)) \quad (15) $$

In practice, the cost function (or performance index to be optimized) includes a term based on the state $\mathbf{x}$ compared to a reference and a term based on the control inputs $\mathbf{u}$. Given:

$$ L(\mathbf{x}(k), \mathbf{u}(k)) = \|\mathbf{x}(k+1) - \mathbf{x}_r\|_{\mathbf{Q}_x} + \|\mathbf{u}(k)\|_{\mathbf{Q}_u} \quad (16) $$

$$ F(\mathbf{x}(N)) = \|\mathbf{x}(N) - \mathbf{x}_r\|_{\mathbf{Q}_f} \quad (17) $$

where $\mathbf{x}_r$ is the state reference. Using 2-norm, the cost function is quadratic, i.e. $\|\mathbf{w}\|_{\mathbf{Q}} \triangleq \mathbf{w}^T\mathbf{Q}\mathbf{w}$. The weighing matrices are such that $\mathbf{Q}_x \ge 0$, $\mathbf{Q}_u > 0$ (by taking account of constraints on $\mathbf{u}$, typically actuator constraints, it is sufficient for the matrix $\mathbf{Q}_u$ to be semi-positive definite) and $\mathbf{Q}_f \ge 0$.

At every time step, the following optimization problem $\mathcal{P}_N$ has to be solved, where superscript $o$ means optimality:

$$ \mathcal{P}_N(\mathbf{x}(0)) : J_N^o(\mathbf{x}(0)) = \min_{\mathbf{U}_N} J_N(\mathbf{x}(0), \mathbf{U}_N) \quad (18) $$

fulfilling the model constraints (9)–(11). Additional constraints may allow to include some knowledge about the system which is not captured by the model. Input constraints take account of actuators limits over the prediction horizon and thus are considered as hard constraints. On the contrary, outputs ones are not generally considered as hard constraints since the optimization problem $\mathcal{P}_N$ could be infeasible due to the impossibility for the outputs to fulfill the constraints. Nevertheless these hard constraints can be softened by adding slack variables $s$ representing the amount of constraint violation:

$$ y_{inf} - s(k) \le y(k) \le y_{sup} + s(k) \quad (19) $$

and adding the penalty term $\|s(l)\|_{Q_s}$ to (16). A final state constraint $\mathbf{x}(N) = \mathbf{x}_f$ can be added too, especially to guarantee the stability of the closed loop system [17].

A characteristic of the optimization problem which is formulated above is its mixed nature. The presence of continuous inputs $(\mathbf{u}_c)$ and discrete ones $(\mathbf{u}_d)$ leads to consider the searched control sequence $\mathbf{U}_N$ as a sequence of continuous

inputs denoted $\mathbf{U}_{cN}$ and a sequence of discrete inputs denoted $\mathbf{U}_{dN}$ on the prediction horizon $N$. Then:

$$ J_N^o(\mathbf{x}(0)) = \min_{\mathbf{U}_{dN}} \left( \min_{\mathbf{U}_{cN}} J_N(\mathbf{x}(0), (\mathbf{U}_{cN}, \mathbf{U}_{dN})) \right) \quad (20) $$

subject to the model constraints, for $k = 0, 1, \dots, N-1$:

$$ \mathbf{x}(k+1) = \mathbf{A}_i \mathbf{x}(k) + \mathbf{B}_i \mathbf{u}_c(k) + \mathbf{a}_i \quad (21) $$

$$ \mathbf{y}(k) = \mathbf{C}_i \mathbf{x}(k) + \mathbf{D}_i \mathbf{u}_c(k) + \mathbf{c}_i \quad (22) $$

$$ \mathbf{F}_i \mathbf{x}(k) + \mathbf{G}_i \mathbf{u}_c(k) \le \mathbf{f}_i \quad (23) $$

where $i(k)$ is a function of $\mathbf{x}(k)$, $\mathbf{u}_c(k)$ and $\mathbf{u}_d(k)$.

Let $\mathbf{I}_N = (i(0) \ i(1) \ \dots \ i(N-1))^T \in \mathbb{I}^N$ be a sequence of modes on the horizon $N$. It defines a sequence $\mathbf{U}_{dN}$ and $N$ sets of constraints (21)–(23) for $k = 0, 1, \dots, N-1$. The problem $\mathcal{P}_N$ can be rewritten this way:

$$ J_N^o(\mathbf{x}(0)) = \min_{\mathbf{I}_N} \left( \min_{\mathbf{U}_{cN}} J_N(\mathbf{x}(0), (\mathbf{U}_{cN}, \mathbf{I}_N)) \right) \quad (24) $$

Then for a given sequence of modes $\mathbf{I}_N$, the cost:

$$ J_N^*(\mathbf{x}(0), \mathbf{I}_N) = \min_{\mathbf{U}_{cN}} J_N(\mathbf{x}(0), (\mathbf{U}_{cN}, \mathbf{I}_N)) \quad (25) $$

is the optimal cost that is found by solving a continuous optimization subproblem subject to constraints. This subproblem can easily be reformulated as a standard problem of quadratic programming. The superscript $*$ means optimality with respect to a given sequence of modes $\mathbf{I}_N$. However, the constrained optimization subproblem (25) is not necessarily feasible, i.e. for a given sequence $\mathbf{I}_N$, no solution can satisfy the model constraints (21)–(23) over the prediction horizon.

Due to the presence of discrete variables, $\mathcal{P}_N$ is qualified as a NP-hard problem. A kind of branch and bound algorithm is used to solve the problem $\mathcal{P}_N$ efficiently. It basically consists in a best first descent strategy to reach suboptimums and a branch cutting strategy to make profit of parcial horizon cost evaluation and unfeasible subproblems. More details about this algorithm are in [4], [5].

## VI. APPLICATION EXAMPLE

We now deal with the application of the proposed approach to the 9-bus power system presented in section II. The control inputs are the following ones: $(\delta_{L7}, \delta_{L8}, \delta_{L9}) \in \{0, 1\}^3$ allow to connect (1) or disconnect (0) lines $L_7, L_8$ and $L_9$ respectively, $n_{c1} \in [0, 2]$ is the control input of $C_1$ (reactive power injection up to 0.1 p.u.), $s_{c2} \in \{0, 1, 2, 3\}$ is the capacitor bank $C_2$ control input (3 parts of 0.05 p.u.), $V_9^{ref} \in [0.8, 1.2]$ is the voltage reference of the LTC of $tr_3$ (ratio $n_{t3}$) and $(s_{l1}, s_{l2}, s_{l3}) \in \{0, 1, 2\}^3$ are load shedding inputs (0, 5 or 10% of load).

Then $s_{c2}, s_{l1}, s_{l2}, s_{l3}, \delta_{L7}, \delta_{L8}, \delta_{L9}$ and $\Delta n_{t3}$ are discrete valued control inputs whereas $n_{c1}$ is continuous valued. This case study has a strong combinatorial nature as almost all the control variables are discrete ones and aims to demonstrate the ability of the approach presented in this paper. Indeed, increase the number of continuous variables doesn't significantly increase the complexity of the control problem. $V_i$ with
$i = \{1, \dots, 9\}$ are the voltage magnitude of corresponding buses and are considered as the outputs of the system.

To simulate the power system, a full non-linear model has been implemented and simulations have been performed using the software Matlab. The figures 5–6 shows results obtained using adaptive predictive control in the case of a transmission line drop ($L_2$) at $t = 30$ s. The sample period is $T_s = 30$ s and the prediction horizon is $N = 3$ to exceed the time constants of the load dynamics ($T_{p,q} = 60$ s). The controller is tuned by using weighing matrices as described in section V. Resort to reactive power compensation devices and transformer ratio change is slightly penalized. On the contrary, resort to line reconfiguration and load shedding is more penalized. According to the viewpoint of an electric energy provider, voltage deviations of the buses where consumers (loads) are connected to the network are the most penalized.

On the figure 5, 30 s after the fault occurrence (worst case), voltage stability is recovered and all the voltages still remain above 0.9 p.u. A new steady-state point of operation has been reached without resort to load shedding. The solution given by the controller consists in injecting all the remaining reserve of reactive power ($C_1$ and $C_2$) and to connect the transmission line $L_9$ at time $t = 60$ s. The figure 6 presents the same emergency case but an error (increase) of 10% in load parameters $P_0$ and $Q_0$ has been introduced in the prediction model to check the robustness of the controlled system. The stability is recovered but to maintain the bus voltages above 0.9 p.u. is not physically possible without resort to load shedding. Beyond 10%, the collapse is too fast and to stabilize the network, the sample time $T_s$ has to be decreased.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Bus voltage magnitudes (nominal model)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Bus voltage magnitudes (load paremeter increase of 10%)</figcaption></figure>

## VII. CONCLUSION

This paper presents an adaptive predictive approach to solve the emergency voltage control problem of power networks.

A symbolically off-equilibrium linearized prediction model is used to deal efficiently with the hybrid and non-linear characteristics of power network models. The symbolic linearization is performed off-line to minimize the computation effort needed to update on-line the hybrid prediction model at each sample time. Simulation results illustrate the implementation on a 9-bus power network case study. Extension of the approach to other outages is straightforward, e.g. generation loss.

Future research focuses on interconnected networks and decentralized control since the increase of interconnections between countries leads to consider very large power networks. Furthermore, for industrial politics and security reasons, only a partial knowledge of the system may be available, e.g. subnetwork structure and dynamics, and control systems have to be robust with respect to unpredictable disturbances such as unpredicted power flows.

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