Optimization and Stabilization of Distributed Secondary Voltage Control with Time Delays and Packet Losses Using LMIs

Abstract

The proposed hierarchical secondary voltage control is a spatially distributed control system using communication networks which are disturbed by both a time delays and packet data dropouts. A state feedback integral control is adopted to eliminate the effect of non-zero disturbance and provide exact tracking of the references of the pilot points and alignment of the reactive powers of the generators that participate in the control. The system is modeled as a discrete-time switched system, and the control gains are synthesized by solving LMIs for a stability condition based on a state-dependent Lyapunov function. For that, the cone complementarity linearization (CCL) algorithm is used. The effectiveness of the proposed control strategy in preventing time delays and packet losses is simulated, considering the model of a realistic electric power grid under typical operational conditions using MATLAB.

1. Introduction

In electrical networks, the voltage must be constant, which is often not the case with disturbances. A principal factor that impacts voltage levels is the increased demand for electricity. Because of this, there is a tendency for the electrical networks to be operated close to their stability limits [1]. Large deviations in bus voltages can cause malfunctions or deterioration in performance. Thus, it is desirable that consumers receive energy at a substantially constant voltage. Therefore, electrical networks must be controlled using certain control techniques within the hierarchical control concept to maintain voltage at specified values in certain key buses of the grid, called pilot points in the sequel. The first level of this hierarchy is the Automatic Voltage Regulator (AVR), which is strictly local and works to maintain the terminal voltages of synchronous generators [2]. A secondary reactive power regulator (SQR) is added at a higher hierarchical level of AVR to control the reactive power production of each generator in one zone. Its role is to provide the AVR loop with a reference signal to modify the stator voltage by delivering or absorbing the reactive power of the generator, taking into account these reactive capability limits [3]. When the pilot point voltages deviate from their reference because of either load variations, disturbances, or the connection/disconnection of synchronous generators, the primary level is not able to regulate the voltage of the electric network. Then, secondary voltage control (SVC) is used to remove the pilot point voltage deviation in the steady state caused by the aforementioned problems, and to ensure good performances of the power system [4]. Furthermore, the secondary level also ensures specific reactive power production for each generator. In the literature, several secondary voltage control strategies have been proposed to maintain the pilot point voltage close to the reference value. A classical proportional–integral (PI) controller in a cascade implementation is proposed in [3,5,6] to regulate the pilot point voltage by distributing reactive power requirements among the power plants. In [7,8], a secondary voltage control based on genetic algorithm (GA) was proposed as a single region power system. In [9], the authors proposed a parallel implementation approach with two parallel PI controllers for pilot point voltage regulation and instantaneous reactive power control, generated by participating power plants affected by a participating factor. Model-based Predictive Control (MPC) was proposed in [10,11]; this controller allows the selection of the inputs that will not lead to difficulties in operation [12]. In [13,14], the authors proposed a Distributed Model Predictive Control (DMPC) system for large-scale electric power systems by reducing the control problem into multiple, less complex sub-problems. However, the previous works are limited, and do not meet all the specifications of managers of electric networks, mostly in highly meshed networks for both pilot point voltage control and reactive power alignment, leading to unfeasible solutions [15]. The secondary voltage control is generally placed in a dispatching center to deal with all power plants [16] by using a real-time communication network to close the feedback control system; this concept is called the network control system (NCSs). The closed-loop feedback control system formed by a real-time network, which is called the network control system. This system has many advantages such as low cost, high reliability, and easy installation and maintenance, and is widely used in control generally [17]. However, in the context of large-scale systems this architecture presents several problems due to the limitation of its network communication capabilities. It will inevitably bring about problems such as network delays and data packet loss [18], which tend to degrade the system’s robustness and stability, because it is difficult to model all interactions between power plants [19]. In the conventional feedback control system of SVC, the transmission delays and packet losses are not taken into account in the synthesis of the regulator, which considerably reduces the robustness of the control system and can cause instability in cases of large delays. In [20], a control law built on the state feedback integral control for secondary voltage control and reactive power alignment has been proposed. This control is improved here to achieve robustness against time delays and data packet losses. For that SVC control is synthesized using information received from each power plant at times determined by the event conditions, based on a sparse communication network that closes the loop between the secondary controller and the electric power plants. In other terms, the dynamics of one power plant may directly influence another power plant, but each plant is independently controlled through a control law to achieve common goals. The new proposed SVC controller is synthesized and formulated in a discrete-time switched model, and designed using an iterative algorithm. The stability conditions are derived in terms of linear matrix inequalities (LMIs). The main contributions of this paper are as follows: (i) A generalized formulation of secondary voltage control is designed in discrete-time switched networked control system including state feedback integral control to achieve zero steady-state error and disturbance rejection. (ii) The proposed SVC law improves the performance of the system in terms of the transient and permanent responses of the control loops by keeping the pilot point voltage to a desired value and achieving reactive power alignment among power plants. (iii) It allows us to stabilize the electric power system in the presence of time delays, packet losses, and others disturbances in networked control systems. The rest of this paper is organized as follows. In Section 2, the modelling and control law introduced in [20] are recalled. The discrete-time modelling of NCSs with time delays and packet losses is described in Section 3. The control system of SVC is detailed in Section 4, including the networked controller design of secondary voltage. In Section 5, the proposed control law will be simulated for a realistic transmission network with different cases. Finally, in Section 6, the conclusions are given.

2. Modelling and Control Law of New Generation of SVC

2.1. Problem Formulation

The general structure of the proposed SVC in the network control system is shown in Figure 1. The power plant output voltage is determined by the AVR level. The SQR control scheme is adopted in the primary level to control the reactive power production of the rotating generators. The impact of variations or the connection/disconnection of reactive loads and line impedances greatly influences the pilot point voltage and reactive power sharing among power plants. Therefore, the SVC level is adopted to perform the real-time adjustment of pilot point voltage deviations and improve reactive power alignment through a communication network [21]. The pilot point is selected as the control bus in the electric network, where voltage amplitude is measured. The secondary control signal is added to the voltage and frequency reference of the primary droop control as a supplementary control signal. As the voltage control is carried out in dispatch center in a local region, the data between multiple elements (power plants, sensors, etc.) are exchanged in the communication network system at the same time, which may cause congestion and delays in information exchange, packet loss, and timing errors in transmission, with a connection interruption in some cases. This causes performance deterioration, potential instability, and deteriorated reliability of the system. In this work, we detail a secondary control system used to avoid pilot point voltage deviation and reactive power alignment in NCS shared by multiple power stations at the same time.

2.2. Control Objectives of Proposed SVC

To avoid the problems mentioned above, in this paper, a new kind of secondary voltage control system is proposed to perform voltage regulation of the zone, with good reference tracking, under different real scenarios that can cause system instability [20]. The control objectives include

• The restoration of pilot point voltage ${\mathrm{V}}_{\mathrm{pp}}$ to its desired reference value ${\mathrm{V}}_{\mathrm{pp}}^{*}$;

  $$\underset{t\to \infty }{lim}{\mathrm{V}}_{\mathrm{pp}}\left(\mathrm{t}\right)={\mathrm{V}}_{\mathrm{pp}}^{*}.$$

  (1)
• Reactive power control to ensure the alignment of the reactive power of each power generator by ensuring that each power plant participates in the overall voltage regulation effort of the zone, in proportion with its capacity.

2.3. Mathematical Modeling of an SVC Region/Zone

Secondary voltage control is a form of regional control wherein each region is a predefined decomposition of the overall network used to regulate the voltage at critical nodes and align the relative reactive power output of the power plants. In contrast with the classic SVC, where only var/voltage characteristics are taken into account in the design of the SVC controller, the proposed SVC takes into consideration all measurable quantities of the power generators (stator voltages and reactive powers), and pilot point voltage. As the SVC action is an external loop in a hierarchical controller, its dynamic is much slower than the primary loop (AVR), and its dynamics can be neglected. Thus, the static and dynamic characteristics of the network behavior can be modelled in the state space by considering the stator voltages as states $x\left(s\right)$ of a system, and the sensitivities matrices of reactive powers and pilot point voltage $V_{pp}$ as an output matrix. The transfer function between the SVC control U and the stator voltage V, considering the primary loop AVR, is as follows:

$$\begin{array}{c}\hfill V=R\left(s\right)U,R\left(s\right)=1/(1+s{T}_{\delta }).\end{array}$$

(2)

where ($T_{\delta }=1s$) is the dynamics of the AVR loop of the generator. Based on the above analysis, the general state space modeling of the proposed SVC, considering all characteristics of the whole system and regional zone with only one pilot point $V_{pp}$, is as follows:

$$\left[\begin{array}{c}{\dot{V}}_{s1}\\ {\dot{V}}_{s2}\\ ⋮\\ {\dot{V}}_{sm}\end{array}\right]=-\frac{1}{T_{\delta }}\stackrel{A\in {\mathbb{R}}^{m\times m}}{\overbrace{\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\end{array}\right]}}\left[\begin{array}{c}{V}_{s1}\\ V_{s2}\\ ⋮\\ V_{sm}\end{array}\right]+\frac{1}{T_{\delta }}\stackrel{B\in {\mathbb{R}}^{m\times m}}{\overbrace{\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1\end{array}\right]}}\left[\begin{array}{c}{U}_1\\ U_2\\ ⋮\\ U_m\end{array}\right]$$

(3)

$$\left[\begin{array}{c}{Q}_1\\ Q_2\\ ⋮\\ Q_m\\ V_{pp}\end{array}\right]=\stackrel{C_i\in {\mathbb{R}}^{m+p\times m}}{\overbrace{\left[\begin{array}{cccc}{C}_{q11}& C_{q12}& \cdots & C_{q1m}\\ C_{q21}& C_{q22}& \ddots & ⋮\\ ⋮& ⋮& \ddots & ⋮\\ C_{qm1}& \cdots & \cdots & C_{qmm}\\ C_{v1}& \cdots & \cdots & C_{vp}\end{array}\right]}}\left[\begin{array}{c}{V}_{s1}\\ V_{s2}\\ ⋮\\ V_{sm}\end{array}\right]$$

(4)

where $V_{si}$ is the terminal voltages vector of power units; Q is the reactive power deviation vector of power units, $Q_m=C_{qmm}\times V_{sm}$; $V_{pp}$ is the voltage vector of pilot nodes in the control zone, $V_{pp}=C_{vp}\times V_{sm}$; and m is the number of generators that participate to the voltage control in regional zone.

In order to realise the above objectives, both static and dynamic levels will be revisited and presented as follows:

2.3.1. Static SVC Objectives

The static objective is to maintain the tracking the pilot point voltage reference value $V_{pp}^{*}$ in the steady state, and to achieve the reactive power generation objective of all power plants participating in the control of system. This control can be achieved if an alignment of reactive powers on an average level of the zone can be achieved. The average level of the zone is defined as

$$Q_{bal}=\frac{{\sum }_{i=1}^m{Q}_i}{{\sum }_{i=1}^m{Q}_{ri}},$$

(5)

where $Q_{ri}$ is the participation factor of each power plant and its value is constant, and m is the number of power plants. Thus, in order to achieve a reactive power alignment among power plants, it is necessary that the predefined ratio $Q_i^{*}=\frac{Q_i}{Q_i^{ri}}$ follows the average level of the zone Equation (5).

Lemma 1.

Consider a local zone of m power plants and one pilot point, and let $j\in \{1,\dots ,m\}$.

• if $Q_i^{*}=Q_{bal}$, $i\in \{1,\dots ,m\}$ and $i\ne j$ then
•     $Q_j^{*}=Q_{bal}$
• end if

Proof. 

If we Rewrite (5) as

$$Q_{bal}=\frac{Q_j+Q_{bal}{\sum }_{i\ne j}{Q}_{ri}}{Q_{rj}+{\sum }_{i\ne j}{Q}_{ri}}$$

It follows that

$$Q_{bal}{Q}_{rj}+Q_{bal}\sum _{i\ne j}{Q}_{ri}=Q_j+Q_{bal}\sum _{i\ne j}{Q}_{ri}$$

From which

$$\frac{Q_j}{Q_{rj}}=Q_{bal}\Rightarrow Q_j^{*}=Q_{bal}$$

   □

From Lemma 1, we can conclude that to align all the generators of a one pilot point zone $(p=1)$, it is sufficient to regulate only $m-1$ reactive powers. Thus, the control objectives will be achieved. $Tota{l}_{objectives}=m-p+p=m$, because the maximum number of objectives that can be pursued is equal to the number of orders, i.e., m in our case. This is the reason that in the command formalism, only $m-1$ integrators are placed on the errors in tracking reagents (we choose $m-1$ groups from the m). This result will be exploited in the dynamic SVC objectives to choose the output to be regulated.

2.3.2. Dynamic SVC Objectives

The dynamic control objective function is to align on $Q_{bal}$ the production of generators that participate in the pilot point voltage regulation. The mathematical statement that is minimized to find a best pilot point voltage along the desired reference with good transients eliminates errors $(e_1$ and $e_2)\to 0$ when $t⟶+\infty$, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{e}_1\hfill & =Q_i^{*}-Q_{bal}=(C_Q-C_Q^{{}^{\prime }})x,i=1,\dots ,m\hfill \\ e_2\hfill & =V_{pp}-V_{pp}^{*}=C_{V}x-V_{pp}^{*}\hfill \end{array}\right.\end{array}$$

(6)

From Equation (6), we can observe that the number of variables to be tracked is an important factor, because we have m plants and p pilot nodes. In command formalism and based on the analysis presented in the above subsection, only $m-1$ integrators will be placed on the errors in tracking reagents (we choose $m-1$ power plants from the m plants), and the $m^{th}$ reagent will be automatically aligned with the other reactive powers. The $m-1$ choice is based on the power capacity of each plant.

In order to realize the aforementioned control objectives, the internal model principle [22,23] is applied for the proposed SVC. The feedback controller of the proposed secondary control must include an external disturbance model (6), so that the closed-loop system can withstand disturbances and changes in system parameters, and adjust the output of SVC to allow the system to progressively track the pilot point voltage reference signal $V_{pp}^{*}$ by reaching zero steady-state errors and ensure the reactive power alignment among the power plants. For that, we choose the tracking of the reference objective (4), and now we define the tracking error, based on Equation (6) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}e=\left[\begin{array}{c}{e}_1\\ e_2\end{array}\right]\to 0whent\to +\infty \hfill \\ e\triangleq y-y^{*}=\left[\begin{array}{c}Q\\ V_{pp}\end{array}\right]-y^{*}\hfill \end{array}\right.\end{array}$$

(7)

where y is a vector of the chosen outputs and $y^{*}$ is a constant reference to them. Therefore, a new extended-state vector $X={\left[\begin{array}{c}\dot{x}e\end{array}\right]}^T$ is used to form an equivalent extended-state system as follows:

$$\begin{array}{c}\hfill \dot{X}=\left[\begin{array}{c}\ddot{x}\\ \dot{e}\end{array}\right]=\underset{\mathcal{F}}{\underbrace{\left[\begin{array}{cc}A& 0\\ C& 0\end{array}\right]}}X+\underset{\mathcal{G}}{\underbrace{\left[\begin{array}{c}B\\ 0\end{array}\right]}}\dot{u}.\end{array}$$

(8)

If the pair $(\mathcal{F},\mathcal{G})$ is stabilizable, there exists controller $\mathcal{K}$, and we can work out the equivalent control law that stabilizes the closed-loop system, such that

$$\dot{u}=-\mathcal{K}X$$

(9)

The necessary and sufficient conditions for the existence of such a control are given by the internal model principle [22]:

Proposition 1.

The pair $(\mathcal{F},\mathcal{G})$ is stabilizable if, and only if, the following two conditions hold: (*) $r\le m$;  (**) $s=0$ is not an invariant zero of {A,B,C}.

In the secondary voltage control context, objective (6) is to track $m+p$ variables in local zone with m generators. However, in our system, we have m actuators (m is a dimension of u in (3)), and the number variables of outputs to track is limited by a condition (*). Thus, in order to ensure the steady-state reactive powers’ alignment, we use Lemma 1, and the new objective $e_1$ in (6) becomes $m-1$ among the m available. From this proposition, the output y for reactive powers’ alignment and pilot point regulation should be chosen as follows:

$$y={[Q_i-Q_{bal},Q_j-Q_{bal},\dots Q_k-Q_{bal},V_{pp}]}^T$$

(10)

Thus, conditions (*) and (**) are satisfied.

2.4. Control Law of Proposed SVC

The design of the controller is a combination of the pilot point voltage regulation problem and the reactive power alignment problem. Thus, through the application of the internal model principle, the control law in Equation (12) fully conforms to the structure of the robust output regulator required to achieve the control objectives, so that the closed loop system can follow the reference $V_{pp}^{*}$ and ensure robustness and stability in the face of disturbances and parametric variations. The controller structure derived from the principle of the internal model is a multi-variable (i.e., matrix) proportional–integral (PI) control law.

$$\begin{array}{cc}\hfill u\left(t\right)& =-{\mathcal{K}}_{P}x\left(t\right)-{\mathcal{K}}_I{\int }_0^{t}e\left(t\right)dt\hfill \end{array}$$

(11)

The detailed control law for for each generator is

$$u_i=-\sum _{i=1}^m{\Lambda }_i{V}_i-{\int }_0^t\sum _{j=1}^{m-1}{\Gamma }_{ij}(\frac{Q_i}{Q_{ri}}-\frac{Q_j}{Q_{rj}})-{\int }_0^t{\Theta }_{ii}({\mathrm{V}}_{\mathrm{pp}}^{*}-{\mathrm{V}}_{\mathrm{pp}})$$

(12)

where ${\Lambda }_{ij}={\mathcal{K}}_P$ is the matrix that contains the stator voltage gains for each power plant; ${\Gamma }_{ij}$ and ${\Theta }_i$ are in matrix  ${\mathcal{K}}_I$, which contains, respectively, the reactive power alignment gains and tracking error pilot point gains for each power plant. A flow chart diagram of the secondary voltage controller with revisited static and dynamic objectives (12) is shown in Figure 2. In the classical control, the computation of gains for the proposed control law is the result of a pole placement. However, this technique does not take into account the network data packets and delays that can cause system instability. In order to solve this problem, in the following section, the static gains matrix is obtained using a network control system based on LMIs.

3. Discrete-Time Modeling of NCSs with Time Delay and Packet Losses

In the previous section, the mathematical modeling of the proposed new generation of SVC is presented in the continuous time domain. However, SVC is implemented in practice in discrete time. As consequence, the model given by (3) is discretized as follows:

$$\mathit{x}(k+1)=\mathcal{A}\mathit{x}\left(k\right)+\mathcal{B}\mathit{u}\left(k\right)$$

(13)

where

$$\mathcal{A}={\mathrm{e}}^{\mathit{A}h},\mathcal{B}={\int }_0^h{\mathrm{e}}^{\mathit{A}\tau }\mathrm{d}\tau \mathit{B}$$

(14)

3.1. Modeling of SVC with Time Delays

From Figure 1, it can be seen that the control gain $\mathcal{K}$ of the distributed secondary voltage controller depends on the round-trip time (RTT) delay ${\tau }_k$, which is the sum of the transmission delay from the sensors (the stator voltage and the reactive power of each power generator and the pilot point voltage) to the controller ${\tau }_k^{sc}$ (regional dispatch of electric system), and the transmission delay from the controller to the actuator (power plant) ${\tau }_k^{ca}$, and the computation time of the control ${\tau }_k^c$.

$${\tau }_k={\tau }_k^{sc}+{\tau }_k^c+{\tau }_k^{ca}$$

(15)

Generally, ${\tau }_k^c$ is ignored, in some cases because it is a relatively small compared to ${\tau }_k^{sc}$ and ${\tau }_k^{ca}$.

A natural assumption for the round-trip time delay ${\tau }_k$ is that

$$\check{\tau }⩽{\tau }_k⩽\widehat{\tau }$$

(16)

where  $\check{\tau }$ is the lower limit and $\widehat{\tau }$ is the upper limit of ${\tau }_k$. Let $N_{delay}=\lceil \widehat{\tau }/h\rceil$ be the ceiling operator, which returns the smallest integer h greater than or equal to a specified number $\widehat{\tau }$. According to Equation (16), the value range of $\lceil {\tau }_k/h\rceil$ is

$${\Omega }_1\triangleq \{\lceil \check{\tau }/h\rceil ,\lceil \check{\tau }/h\rceil +1,\cdots ,\lceil \widehat{\tau }/h\rceil \}$$

3.2. Modeling of SVC with Packet Losses

In addition to delays, the packet loss and timing disorders in network transmission may occur between regional dispatch and power plants, which may make the system unreliable and may result in inadequate operation in real time. In the proposed SVC, if the sampled data packets are successfully sent either on the sensor–controller side or on the controller–actuator side, then the data packets are valid in the control system, and its control signal is used to regulate the pilot point voltage and the reactive powers’ alignment. The time series corresponding to valid sampled data packets is denoted $\mathcal{S}\triangleq \left\{{\rho }_1,{\rho }_2.\cdots \right\}$. The sampled data packets between two adjacent valid sampled data packets can be regarded as lost sampled data packets caused by network packet loss [24]. Based on the above ideas, packet loss in NCSs can be defined as

$$\left\{\eta \left({\rho }_m\right)\triangleq {\rho }_{m+1}-{\rho }_m,{\rho }_m\in S\right\}$$

(17)

The packet losses defined by Equation (17) are as follows: from the instant ${\rho }_m$ to the instant ${\rho }_{m+1}$, the number of consecutive packet losses of NCSs is $\eta \left({\rho }_m\right)-1$. Without loss of generality, let the maximum number of consecutive packet losses in NCSs be $\widehat{\eta }-1$, that is

$$\widehat{\eta }\triangleq \underset{{\rho }_m\in S}{max}\left\{\eta \left({\rho }_m\right)\right\}$$

(18)

It is not difficult to see that the range of $\eta \left({\rho }_m\right)$ is ${\Omega }_2\triangleq \left\{1,2,\cdots ,\widehat{\eta }\right\}$. It is worth noting that the packet loss defined by (17) is a generalized packet loss, which essentially comprehensively characterizes the effects of packet loss and data packet timing disorder. Further, considering the effects of delay, packet loss, timing disorder, and zero-order hold of the control signal, we can see from discretization of Equation (11) $(u(k+\lceil {\tau }_k/h\rceil )=\mathcal{K}X\left(k\right))$ that when $k\in \left[{\rho }_m+⌈{\tau }_{{\rho }_m}/h⌉,{\rho }_{m+1}+⌈{\tau }_{{\rho }_{m+1}}/h⌉\right)$, the SVC control input of the controlled power plants is

$$\mathit{U}\left(k\right)=\mathcal{K}\mathit{X}\left({\rho }_m\right)$$

(19)

where $\left({\rho }_m+⌈{\tau }_{{\rho }_m}/h⌉\right)⩽k<\left({\rho }_m+{\eta }_{i_m}+⌈{\tau }_{{\rho }_{m+1}}/h⌉\right)$.

For $k\in \left[{\rho }_m+⌈{\tau }_{{\rho }_m}/h⌉,{\rho }_{m+1}+⌈{\tau }_{{\rho }_{m+1}}/h⌉\right)$ with $⌈{\tau }_{{\rho }_m}/h⌉\le \lceil \widehat{\tau }/h\rceil$ and ${\eta }_{{\rho }_m}\le \widehat{\eta }$, one can infer from control law Equation (12) that at time step k, the control signal of SVC for pilot point voltage regulation is no older than $k-\lceil \widehat{\tau }/h\rceil -\widehat{\eta }$. In view of this, the following augmented state vector is introduced in the closed-loop system:

$$\mathit{z}\left(k\right)={\left[{\mathit{X}}_k^{\mathrm{T}}{\mathit{X}}_{k-1}^{\mathrm{T}}\cdots {\mathit{X}}_{k-[\widehat{\tau }/h]-\widehat{\eta }}^{\mathrm{T}}\right]}^{\mathrm{T}}$$

Based on Equations (13) and (19), we can obtain the closed-loop augmented:

$$\mathbf{z}(k+1)=\left(\overline{\mathcal{A}}+\overline{\mathcal{B}}\mathcal{K}{\tilde{\mathit{E}}}_{k-{\rho }_m}\right)\mathbf{z}\left(k\right)$$

(20)

$$\overline{\mathcal{A}}=\left[\begin{array}{ccccc}\mathcal{A}& 0& \cdots & 0& 0\\ I& 0& \cdots & 0& 0\\ 0& I& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & I& 0\end{array}\right],\overline{\mathcal{B}}=\left[\begin{array}{c}\mathcal{B}\\ 0\\ 0\\ ⋮\\ 0\end{array}\right]$$

(21)

${\tilde{\mathit{E}}}_{k-{\rho }_m}=\left[\begin{array}{ccccc}0\hfill & \cdots \hfill & I\hfill & \cdots \hfill & 0\hfill \end{array}\right]$. It is worth pointing out that except for the $k-{\rho }_m+1$ element in ${\tilde{\mathit{E}}}_{k-{\rho }_m}$ as the identity matrix, the remaining elements are all zero matrices. It is not difficult to see that the NCSs model shown in Equation (20) is a discrete-time switching model, where ${\delta }_k=k-{\rho }_m$ is the switching signal; it transfers from mode i to mode j in the switching matrix $\left[{\delta }_{ij}\right]$, and its value set is ${\Omega }_3$:

$${\Omega }_3\triangleq \{\lceil \check{\tau }/h\rceil ,\lceil \check{\tau }/h\rceil +1,\cdots ,\lceil \widehat{\tau }/h\rceil +\widehat{\eta }\}$$

The proposed SVC control in the NCSs model shown in Equation (20) considers the effects of delays, packets loss based on $\lceil \widehat{\tau }/h\rceil +\widehat{\eta }+1$ subsystems without making assumptions about the probability distribution of delay, and packet loss, which can be completely random or conform to a certain probability distribution using different ${\delta }_k$ sequences.

4. Networked Controller Design of Secondary Voltage

4.1. Stabilization of SVC in Networked Control System

The stability condition of the proposed distributed secondary voltage control in NCS system to manage delays and packet loss will be used in the development of the following work in order to maintain stability when switching from one mode to another one based on a switching matrix $\left[{\delta }_{ij}\right]$.

Theorem 1.

The SVC closed-loop in NCS (20) with the presence of time delays and packet losses is asymptotically stable, if there exist positive definite matrices ${\mathcal{P}}_i>0,(i\in {\Omega }_3)$ that satisfy

$${\delta }_{ij}\left({\left(\overline{\mathcal{A}}+\overline{\mathcal{B}}\mathcal{K}{\overline{E}}_i\right)}^{\mathrm{T}}{\mathcal{P}}_j\left(\overline{\mathcal{A}}+\overline{\mathcal{B}}\mathcal{K}{\overline{E}}_i\right)-{\mathcal{P}}_i\right)<0$$

(22)

where $i,j\in {\Omega }_3$ and ${\delta }_{ij}\in \left[{\delta }_{ij}\right]$.

Proof. 

The proof in this paper is omitted because of space limitations; please refer to [25].    □

4.2. Stabilizing Controller Design for the SVC in NCS

Theorem 1 gives the stability condition of NCSs. However, it is difficult to obtain a stabilising controller directly from this theorem, because a trade-off between conservatism and complexity should be obtained. Morever, the sufficient stability conditions (20) are non-linear in matrices $\mathcal{K}$ and ${\mathcal{P}}_i$. However, the following theorem proposes an equivalent stability condition to Equation (20), in order to circumvent the synthesis problem.

Theorem 2.

The closed-loop (20) in the presence of time delays and packet losses is asymptotically stable, if for $(i,j\in {\Omega }_3)$, there exist positive definite matrices ${\mathcal{P}}_i>0\in {\mathbb{R}}^{n\times n}$ and ${\mathcal{Q}}_i>0\in {\mathbb{R}}^{n\times n}$ that satisfy

$${\delta }_{ij}\left[\begin{array}{cc}-{\mathcal{P}}_i& *\\ \overline{\mathcal{A}}+\overline{\mathcal{B}}\mathcal{K}{\overline{E}}_i& -{\mathcal{Q}}_j\end{array}\right]<0$$

(23)

$${\mathcal{P}}_j{\mathcal{Q}}_j=I$$

(24)

$${\mathcal{P}}_{\mathit{i}}={\mathcal{Q}}_{\mathit{i}}^{-\mathbf{1}}$$

(25)

Proof. 

The proof in this paper is omitted because of space limitations, please refer to [26].    □

In [27], the switching matrix $\left[{\delta }_{ij}\right]$ with $(i,j\in {\Omega }_3)$ is constructed based on the following analysis:

$$\Delta \left[{\delta }_{ij}\right]=\left[\begin{array}{cc}{\Delta }_{11}& {\Delta }_{12}\\ {\Delta }_{21}& {\Delta }_{22}\end{array}\right]$$

(26)

With ${\Delta }_{11}\in {\mathbb{R}}^{(\lceil \widehat{\tau }/h\rceil -\lceil \check{\tau }/h\rceil +1)\times (\lceil \widehat{\tau }/h\rceil -\lceil \check{\tau }/h\rceil +1)}$ and ${\Delta }_{22}\in {\mathbb{R}}^{\widehat{\eta }\times \widehat{\eta }}$. The switching signal ${\delta }_{ij}$ transfers from mode i to mode j with a feasibility ${\delta }_{ij}$. Then, if

• ${\delta }_{ij}=0$, then that means the transition is feasible; and
• ${\delta }_{ij}=1$, then that means the transition is unfeasible

4.3. Cone Complementarity Linearization Approach

It is worth noting that the Equation (24) in Theorem 2 cannot be solved directly using LMI toolbox in MATLAB software because it is not a convex set due to matrix equality constraints. To solve this non-convex problem, several approaches exist in the literature. The Cone Complementarity Linearisation (CCL) approach was used here to transform the non-convex feasibility problem in Theorem 2 into the following minimization problem of linear matrix inequality constraints [28].

$$minTrace\left(\sum _{i=\lceil \tau /h\rceil }^{\lceil \widehat{\tau }/h]+\widehat{\eta }}{\mathcal{P}}_i\left(k\right){\mathcal{Q}}_i\left(k\right)\right)$$

(27)

$$s.t:formula\left(23\right);$$

The basic idea of the CCL algorithm is to consider the problem of static feedback design by iteratively minimizing trace involving LMIs conditions for any matrix variables $\mathcal{P}\in {\mathbb{R}}^{n\times n}>0$ and $\mathcal{Q}\in {\mathbb{R}}^{n\times n}>0$. Thus, based on the idea of the cone complementarity linearization strategy, the following equality $\mathcal{P}\mathcal{Q}=I$ can be relaxed with the following LMI [29]:

$$\left[\begin{array}{cc}\mathcal{P}& I\\ I& \mathcal{Q}\end{array}\right]\ge 0$$

(28)

Although it is impossible to find a global optimal solution, this minimization problem is much easier and very efficient in numerical implementation than the original non-convex feasibility problem [30]. In this work, we apply the CCL approach to calculate the stabilizing controller using the following iterative algorithm for solving the optimization problem [31]. The controller design based on Theorem 2 and the above linearization method is given in Algorithm 1 (N is the maximum number of iterations allowed).

Algorithm 1: Finding the optimal $\mathcal{K}$ with the CCL Algorithm

5. Simulations Studies

The control is tested on a zone of 410 kV using a real benchmark system, shown in Figure 3, in MATLAB/Simulink environment. It is composed of four generators and one pilot point. All generators have the same power, and each one is supporting an auxiliary local load. The generator remains connected all the time, except the auxiliary load, which is connected sequentially. In order to examine the performance and stability of the proposed secondary voltage control in the networked control systems, several numerical scenarios are studied, such as nominal performance with a reactive load, the presence of synchronous and asynchronous time delays and packet losses in communication systems, and finally robustness against uncertainty and disturbance. In all scenarios, the proposed secondary voltage controller attempts to realize alignment of reactive power, sharing among the generators with pilot point voltage regulation. Some scenarios of the proposed secondary voltage control scheme are compared with the control method [20], which employs a linear quadratic regulator approach.

5.1. Nominal Performance Evaluation

The objective of this section is to test the performance of the proposed control strategy for different scenarios. Firstly, the reference of pilot point voltage is increased to 1.2 pu, at t = 1000 s; then, a load is suddenly connected to the bus of the pilot point voltage at t = 2000 s and removed at t = 3000 s. Finally, the reference of pilot point voltage is decreased to its initial reference value. Figure 4a,b show the simulation curves of the $V_{pp}$ and reactive power. From these results, we can see that the voltage magnitude converges to the desired value under the reference step within a finite time of 350 s. Afterwards, the load variation in the voltage is well restored, and returns to the specified range after the static deviation is removed thanks to the action of the proposed distributed secondary control within a setting t = 350 s, as shown in Figure 4a. The reactive powers are shared with precision among power plants in spite of load variations, as depicted in Figure 4b. As can be observed, the proposed controller is able to regulate the pilot point voltage and enhance the precision of reactive power alignment, which is beneficial to maintain the stability of the power system.

5.2. Impact of Communication Time Delays

In this case study, the proposed distributed SVC in network control system based on the CCL algorithm is tested with a communication delay. Based on the specifications of the electricity transmission system operator, we know that the allowable delay bound due to measurements of data is $3\times h$, and for reception of data is ${\tau }_{ca}=3\times h$. Due to its small negligible value in comparison with the network delay, the computation delay of the controller is not considered in the simulation, and is equal to zero. h = 10 s is the sampling time. In order to show the impacts of time delay on the performance of the secondary voltage control in cyber-physical systems, the following two sub-cases are investigated for different delays.

5.2.1. With Synchronous Time Delays

In this subcase, we set the synchronous time delays $\tau$ = 20 s and $\tau$ = 30 s for the reception of measurements and transmission of control signals. Figure 5 illustrates the simulation results to see how delay affects the pilot point voltage regulation and reactive power alignment responses. In the first half of the simulation test, the pilot point voltage reference is increased in the meshed electrical grid, and in the second half, this is decreased to its nominal value when the delay is set to 20 s and 30 s for each canal. From Figure 5a,b, we see that the distributed SVC based on the proposed control technique exhibits a very good response, with small overshoots compared to linear quadratic control (LQR). The pilot point voltage value converges to the corresponding optimal reference value with good reactive power alignment. Figure 5a,b, show that the proposed control method for the distributed SVC ensures good robustness and maintains the power system’s stability when the delay increases by successfully following the reference value. The response based on a classical optimal controller is very sensitive to instability when the time delay becomes larger; it has a bigger settling time, and leads to serious oscillatory behavior compared to the proposed controller.

5.2.2. With Asynchronous Time Delays

Based on the same tests as in the above section, this sub-case proposes a test of the secondary voltage control with diverse time delays in reception and transmission networks; the simulation results are presented in Figure 6a,b. From the results, we observe that the proposed controller ensures good performance with diverse time delays, and it converges to the reference value without significant oscillatory behavior, in contrast to the LQR optimal controller which presents much more serious oscillatory behavior.

5.3. Impact of Packet Losses

In order to further verify the effectiveness and usability of the proposed control algorithm, this paper performs pilot point voltage tracking control on the actual regional electric power system. The measurements and control signals of the network for offline simulation are sampled with period h = 10 s. The proposed pilot point voltage controller has been simulated in different operating conditions with different packet loss information statistics, which are $0.1$, $0.3$ and $0.5$ (genuine packet loss), and interferences in the network, which may be disturbances and measurement noises. By using empirical trials in simulation, the obtained critical packet loss rate, which destroys the system’s stability, is estimated at about $0.75$. Figure 7 shows the time evolution of pilot point voltage for the three simulation tests, and the effects of the packet losses on the transmission/reception formations. The blue stripes in Figure 7 correspond to instant packet transmission, while the red stripes corresponds to instant packet reception (when there is no packet loss, these stripes overlap). We can observe that the system is not affected by disturbances and measurement noises, but it is affected in the time interval within which it is experiencing packets losses, specifically when the probability is higher than $0.5$, as shown in the red stripes in Figure 7. However, in the remaining time instants, the proposed secondary voltage control works under normal operations. The above statistical information shows the effectiveness of the delay model and packet loss model used in this work.

5.4. Robustness against Parameter Uncertainty and Disturbance

In this test, the robustness of the proposed secondary voltage control in NCS is investigated against line parameter uncertainty by modifying the sensitivity coefficients of the pilot point voltage and reactive power. Each coefficient is decreased by −25% and −50% and increased by $25\%$ and $50\%$ in the presence of a reactive load disturbance of 100 MVAR, connected to the pilot point voltage of the meshed grid. The upper and lower bounds for the variation range of the sensitivity coefficients of reactive power and pilot point voltage are set at −50% and $50\%$ of the nominal values; these sensitivity coefficients are influenced by the transmission lines. Figure 8a,b show the pilot point voltage and reactive power sharing among the power plants. From these figures, we can see that the voltage is maintained with very small fluctuations during the parametric variation within a specific range, with the connection and disconnection of the reactive load between t = 2000 s and 4000 s, and the addition of a disturbance at the output of the $SVC$ at t = 3000 s. These results are caused by the inaccurate estimated measurements due to the line parameters uncertainties. However, the reactive power alignment among the power plants has zero steady-state error, since the closed loop is still stable. From these results, we can observe that the proposed controller is robust, and the system remains stable after small transients and even in the presence of disturbance, which can be caused by different events such as a fault in the actuator or excitation voltage in the power generator, or an offset that disturbs the output of the secondary controller.

6. Conclusions

In this paper, an integral state feedback controller of secondary voltage control is designed and implemented in a manner such that each power generator uses communication networks to communicate with its control station. This control law allows us to take into account maximum delay bounds and packet losses, which can disrupt the system in real time, by using Lyapunov stability theory and a CCL approach. The proposed controller was tested and studied for communication delays and packet losses, with a rigorous analysis of their impacts in cyber-physical networks for pilot point voltage control. The simulation results show that the proposed SVC in NCS is effective, and makes the system robustly stable, showing the improved performance of cyber–physical energy systems in the presence of delay and packet losses.