Nonlinear Coordinated Passivation Control of Single Machine Infinite Bus Power System with Static Var Compensator

Salma Keskes; Souhir Salleem; Larbi Chrifi-Alaoui; Mohamed Ben Ali Kammoun

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Nonlinear Coordinated Passivation Control of Single Machine Infinite Bus Power System with Static Var Compensator PDF

- ORCID：
Salma Keskes
✉
- ORCID：
Souhir Salleem
✉
- ORCID：
Larbi Chrifi-Alaoui
✉
- ORCID：
Mohamed Ben Ali Kammoun
✉

Electric Machines and Power Networks Control Laboratory from the Electrical Engineering Department, National Engineering School of Sfax, Sfax, Tunisia； Innovative Technologies Laboratory, University of Picardy Jules Verne, Cuffies, France

Updated：2021-11-23

DOI：10.35833/MPCE.2019.000173

Abstract

We propose a nonlinear coordinated control of the generator excitation and the static var compensator (SVC) in order to enhance the transient stability and voltage regulation of power system by the passivation approach. SVC is installed in the middle of the transmission line of power system and consists of a single machine infinite bus (SMIB) system. The design of the proposed controller consists of two parts. On one hand, the generator excitation controller is designed based on a back-stepping controller. On the other hand, the conception of SVC control input is based on the coordinated passivation approach, which can guarantee the asymptotic stability of the closed-loop system. The simulation results show the effectiveness of the proposed controller compared with other methods, which ensures better performance than the uncoordinated control when the system is subjected to a disturbance.

Keywords

Coordinated control; passivation approach; transient stability; control design

I. Introduction

IMPROVING the safety and stability of power systems has become the main concern of electric power companies and suppliers in a global context of industrialization, economic development and rapidly changing energy market. Network specialists and engineers are called upon to face these important challenges by modernizing existing control equipment and developing new control technologies for better operation and performance of power grids. The control systems should have the ability to maintain the stability against the disturbances of power systems, which may not only cause the waste of the system synchronism but also result in short-term voltage dips and sags [

1]. Some research work has been done to achieve these objectives that can be summarized in two important classes in power system operation: transient stability and voltage regulation.

In the first class, the excitation control of synchronous generator is concerned. Thus, new control techniques at the level of power system excitation have been deployed to improve stability performances of voltage regulation and transient stability. In this regard, some researches apply advanced automatic techniques with only excitation control.

Traditionally, the generator excitation is equipped with the conventional regulators including an automatic voltage regulator (AVR), which makes it possible to regulate the terminal voltage of the generator by acting on the speed variation. However, the action of the voltage regulator leads to the oscillations. Therefore, power system stabilizer (PSS) has been developed [

1], [2], which works with AVR by adding an additional input to the excitation system. PSSs are considered as conventional tools for damping electromechanical oscillations since it is easy to install the systems that are practical, efficient and cheap. Consequently, these regulators only ensure the stability and remain limited in the case of major disturbances.

In order to improve the transient stability, modern control based regulators are proposed to replace the conventional regulators which are usually developed for the single machine infinite bus (SMIB) such as nonlinear control [

3]-[7].

Nevertheless, the simultaneous enhancement of transient stability and voltage regulation may be difficult to be achieved by applying the control techniques. In addition, although generator excitation controllers are useful for the enhancement of transient stability or voltage regulation, the stability is no more maintained if a major fault occurs far away from the generator terminals [

8].

The improvements in the power electronic technology include the use of new technologies such as fast-response electronic power devices known as flexible alternating current transmission system (FACTS), which is considered as the second class in power system control. FACTS is essential for power transit control, which will enhance transient stability margin, voltage regulation, and the damping improvement of electromechanical oscillations [

9], [10].

The value of incorporating FACTS devices into power systems is highlighted in several research works.

FACTS devices are classified in three categories depending on the manner of connection to the power grid.

1) Shunt devices connected in parallel with the network known as static var compensator (SVC) and static synchronous compensator (STATCOM).

2) Series devices in the transport lines such as thyristor controlled series capacitor (TCSC), thyristor controlled series reactor (TCSR) and small signal stability region (SSSR).

3) Hybrid devices simultaneously use both couplings such as unified power flow controller (UPFC).

References [

11] and [12] have proposed different FACTS devices (series, shunt and hybrid) for oscillation damping.

The most popular type of FACTS devices in terms of application is SVC [

13], which improves power system properties such as steady-state stability limits, voltage regulation and var compensation, dynamic over-voltage and under-voltage control, and damps power system oscillations. SVC is an electronic generator that dynamically controls power flow through a variable reactive admittance to the transmission network.

In voltage regulation mode, SVC has no effect on the damping of the power oscillations. However, by adding an additional damping loop (auxiliary) known as power swing damping control (PSDC), it contributes effectively to the damping of power oscillations and voltage regulation. Many research works have been devoted to establish new control structures for SVCs in order to increase their efficiency in improving the transient stability and damping low-frequency power oscillations.

Besides, the excitation controller and SVC controller are developed separately in many research works, which are not the overall configuration. The resulting control laws are developed individually. The coupling of the dynamics of these different controllers is not considered. A proportional integral (PI) controller has been used as an additional command for SVC in [

8]. As for the excitation, it is equipped with the conventional regulators such as AVR and PSS.

The result of the non-coordination of these two control structures is the performance inability and the deterioration of the overall stability of power systems.

It is imperative to coordinate the excitation regulators and SVC controllers since several research works have already underlined the interaction between the two control levels.

In [

14], the design of the coordinated control of SVC and generator excitation is based on the feedback linearization technique and robust control to achieve both the enhancement of transient stability and voltage regulation for power systems. The nonlinearity of power system, the variation in the system structure, the parameter uncertainties of the plant and the interconnections between generator and SVC are considered.

In [

15], the coordinated control of SVC and the excitation of generators in power systems with nonlinear loads are studied using the feedback linearized technique and the control of nonlinear differential and algebraic systems. Thus, both the power angle stability of generators and the voltage stability at the location of SVC are improved.

A robust nonlinear coordinated excitation and SVC controller are proposed in [

16] to enhance the transient stability of power systems. SVC is located at the midpoint of the transmission line. Nonlinear feedback law for the generator is found, which linearizes and decouples the power system model. The coordinated controller design is then carried out based on the linearized and decoupled model. Robust non- linear control theory is employed to design the coordinated controller. The proposed coordinated controller consists of three controllers, an excitation controller and two SVC controllers, which are designed separately only based on local measurements.

In [

17], a back-stepping nonlinear controller for an SMIB power system with a nonlinear generator excitation and SVC has been proposed to effectively improve the transient stability, power angle stability as well as frequency and voltage regulations. In this approach, simultaneous transient stabilization can be achieved and a good regulation of SVC terminal voltage can be accomplished. In [18], a coordinated control for SVC and generator excitation is proposed based on sliding-mode dynamic surface method and disturbance attenuation technique.

The uncertainties in the infinite bus voltage and the internal and external reactance to the generation station are considered in [

19]. First, the excitation and the SVC control inputs are obtained via the immersion and invariance (I&I) technique. Then, the controller is redesigned using a parameter update law, a filter with indirect I&I adaptive control and a two-time-scale technique.

Over the past decade, feedback passivation [

20]-[22] has become a popular approach to design a nonlinear controller, especially for multi-input multi-output (MIMO) systems. The passivity theory presents a physical insight for the analysis of nonlinear systems, and the passivation control approach can be regarded as an improvement of the passivity-based ones.

We investigate the control problem of generator excitation and SVC system by the coordinated passivation approach. The problems of transient stability and voltage regulation are simultaneously considered. The design procedure consists of two parts. First, the excitation voltage input is obtained by back-stepping and Lyapunov methods to achieve the stability of rotor angle, speed, and voltage. Then, the input of SVC susceptance regulator input is designed to ensure the feedback passivity of the whole system, which presents a stabilizing controller for the whole closed-loop system.

The remainder of this paper is organized as follows. Section II presents the nonlinear model of SMIB power system and the dynamic of the SVC is identified. Section III presents an outline of the coordinated passivation approach. The coordinated passivation control design is presented in Section IV. Section V presents simulation results. Conclusions are presented in Section VI.

II. Modelling of Power System

The proposed SMIB power system with SVC is given in Fig. 1.

Fig. 1 SMIB power system with SVC.

We adopt a procedure of systematic simplification. A simplified third-order power system model is obtained, which is used for the design of various excitation controllers. Experimental results prove that the simplified third-order model is suited for stability analysis and control design of SMIB power system [

23]. The essential dynamics of the system are given in the third-order model of SMIB.

A. Mechanical Equations

\dot{δ} = ω

(1)

\dot{ω} = - \frac{D}{H} ω - \frac{ω_{s}}{H} (P_{e} - P_{m})

(2)

where $δ$ is the power angle of the generator; $ω_{s}$ is the synchronous machine speed and $ω_{s}$ =314.16 rad/s; $ω$ is the relative rotor speed of the generator ( $ω = ω_{g} - ω_{s}$ with $ω_{g}$ being the angular speed of the generator); $H$ is the inertia constant; $D$ is the damping constant; $P_{e}$ is the active electrical power; and $P_{m}$ is the mechanical power input.

B. Generator Electrical Dynamics

{\dot{E}}_{q}^{'} = \frac{1}{T_{d 0}^{'}} (E_{f} - E_{q})

(3)

where $E_{q}^{'}$ is the transient electromotive force (EMF) in q-axis of the generator; $E_{f} = k_{c} u_{f}$ is the equivalent EMF in the excitation coil of the generator, and $k_{c}$ is the gain of the excitation amplifier, $u_{f}$ is the input to the Silicon controlled rectifier (SCR) amplifier of the generator; $T_{d 0}^{'}$ is the d-axis transient open-circuit time constant; and $E_{q}$ is the q-axis voltage of the generator.

C. SVC Model

A schematic representation of the SVC configuration is shown in Fig. 2. It is a shunt compensation device based on power electronic components and consists of a thyristor-controlled reactor (TCR) in parallel with a fixed capacitor (FC) bank [

13].

Fig. 2 SVC equivalent circuit.

The variable effective susceptance of TCR is given by:

B_{T C R} = B_{L} = f (α) = \frac{2 (π - α) + s i n 2 α}{π X_{L}}

(4)

where $α$ is the class-K function; $X_{L}$ is the reactance of the fixed inductor of SVC; and $B_{L}$ is the susceptance of TCR.

The total susceptance of the SVC is given by:

B_{S V C} = B_{C} - B_{L}

(5)

where $B_{C}$ is the susceptance of the FC bank.

The dynamic model of SVC can be expressed as:

{\dot{B}}_{L} = \frac{- 1}{T_{S}} (B_{L} - B_{L 0}) + \frac{K_{S}}{T_{S}} u_{B}

(6)

where $T_{S}$ is the time constant; $K_{S}$ is the gain; and $u_{B}$ is the input of SVC regulator. $T_{S}$ and $K_{S}$ represent the firing control system of the thyristor.

Equations (4) and (6) show that SVC is able to control the bus voltage magnitude at the point of connection following the control of the thyristor firing angle.

D. Electrical Equations

As shown in Fig. 1, the electrical relationships of the generator considering the SVC device installed in the middle of the transmission line in the SMIB system are given by:

P_{e} = \frac{V_{s} s i n δ}{X_{d Σ}} (\frac{X_{d Σ}}{X_{d Σ}^{'}} E_{q}^{'} - \frac{x_{d} - x_{d}^{'}}{X_{d Σ}^{'}} V_{s} c o s δ)

(7)

E_{q} = \frac{X_{d Σ}}{X_{d Σ}^{'}} E_{q}^{'} - \frac{x_{d} - x_{d}^{'}}{X_{d Σ}^{'}} V_{s} c o s δ

(8)

X_{d Σ} = X_{1} + X_{2} - X_{1} X_{2} B_{S V C}

(9)

X_{d Σ}^{'} = X_{1}^{'} + X_{2} - X_{1}^{'} X_{2} B_{S V C}

(10)

where $x_{d}$ is the d-axis reactance of the generator; $x_{d}^{'}$ is the d-axis transient reactance of the generator; $X_{1} = x_{d} + x_{T} + x_{L 1}$ , $X_{1}^{'} = x_{d}^{'} + x_{T} + x_{L 1}$ , $X_{2} = x_{L 2}$ , and $x_{L i}$ is the reactance of the transmission line $(i = 1, 2)$ , $x_{T}$ is the reactance of the transformer.

The terminal voltage of the generator is given by:

V_{t} = \frac{1}{X_{d Σ}} \sqrt[]{X_{s}^{2} E_{q}^{2} + V_{s}^{2} x_{d}^{2} + 2 X_{s} x_{d} V_{s} E_{q} c o s δ}

(11)

The voltage at the SVC terminal in the middle of the transmission line is given by:

V_{m} = \frac{1}{X_{d Σ}^{'}} \sqrt[]{(V_{s} X_{1}^{'})^{2} + (E_{q}^{'} X_{2})^{2} + 2 V_{s} X_{1}^{'} X_{2} E_{q}^{'} c o s δ}

(12)

The nonlinear differential equations for SMIB power system with SVC are:

\{\begin{array}{l} \dot{δ} = ω \\ \dot{ω} = - \frac{D}{H} ω + \frac{ω_{s}}{H} (P_{m} - P_{e}) \\ {\dot{E}}_{q}^{'} = \frac{1}{T_{d 0}^{'}} (k_{c} u_{f} - E_{q}) \\ {\dot{B}}_{L} = - \frac{1}{T_{S}} (B_{L} - B_{L 0}) + \frac{K_{S}}{T_{S}} u_{B} \end{array}

(13)

III. Overview on Coordinated Passivation Approach

The coordinated passivation approach provides the opportunity to design the dual input controller, which is presented as:

\dot{x} = f (x) + g_{1} (x) u_{1} + g_{2} (x) u_{2}

(14)

where $x \in ℝ^{n}$ ; $u_{1} \in ℝ;; a n d u_{2} \in ℝ$ .

The input-output $(u_{1},$ $y$ ) is chosen for which the relative degree is one. The zero dynamics associated with input-output will be stabilized with the input $u_{2}$ . If zero dynamics is stable, $u_{2}$ can be used to improve performance.

Equation (14) can be rewritten as:

\dot{z} = q (z, y) + p (z, y) u_{2}

(15)

\dot{y} = α (z, y) + β_{1} (z, y) u_{1} + β_{2} (z, y) u_{2}

(16)

where $y$ is a set of order 1 which defines the output of the system; and $z$ is the set of order $n - 1$ . The zero dynamics system is given by (17). It is assumed to be stable by the control law $u_{2}$ .

\dot{z} = q (z, 0) + p (z, 0) u_{2}

(17)

The zero dynamics is defined by the dynamics of the nonlinear system with $y = 0$ .

The coordinated passivation control approach is primarily based on two steps: the zero dynamics stabilization and the feedback passivation.

Step 1: define $W (z)$ as a control Lyapunov function (CLF) for the zero dynamics of (17) in which there exists a stabilizing control law $u_{2} = γ (z)$ :

\dot{W} = \frac{\partial W}{\partial z} {\dot{z}|}_{y = 0} = \frac{\partial W}{\partial z} (q (z, 0) + p (z, 0) γ (z)) < - α | | z | | \forall z \neq 0

(18)

Step 2: achieve the construction of the command law $u_{1}$ with the feedback passivation approach which allows the stabilization of the complete system (15) and (16). Equation (15) can be rewritten with $u_{2} = γ (z)$ as:

\dot{z} = q (z, y) + p (z, y) γ (z) = \tilde{q} (z) + \tilde{p} (z, y) y

(19)

\begin{array}{l} w h e r e \tilde{q} (z) = q (z, 0) + p (z, 0) γ (z) \tilde{p} (z, y) y = q (z, y) - q (z, 0) + p (z, y) \cdot \\ γ (z) - \end{array}

p (z, 0) γ (z)

The derivative of the storage function $V = W (z) + y^{2} / 2$ along the trajectory of (15) and (16) is:

\dot{V} = \frac{\partial W}{\partial z} \dot{z} + y (α (z, y) + β_{1} (z, y) u_{1} + β_{2} (z, y) u_{2})

(20)

Then we design the control law $u_{1}$ as:

u_{1} = β_{1}^{- 1} (z, y) (- β_{2} (z, y) - α (z, y) - \frac{\partial W}{\partial z} \tilde{p} (z, y) + v)

(21)

Thus, $\dot{V} = \partial W / \partial z \tilde{q} + v y \leq - α | | z | | + v y \leq 0$ , which means the theory of passivity. Additional output feedback $v = - ϕ (y),$ where $ϕ (y)$ is the sector nonlinearity satisfying $y ϕ (y) > 0$ for $y \neq 0$ and $ϕ (0) = 0$ , then $\dot{V} \leq - y ϕ (y) \leq 0$ , which ensures the stability of the closed-loop system. Moreover, if the system is detectable in zero state, asymptotic stability is also guaranteed.

IV. Design of Coordinated Controller for SMIBPower System with SVC

In this section, we focus on coordinated passivation approach, through which we can design the control law that coordinates the generator excitation control and the control of the SVC.

We aim to ensure the transient stability and the voltage regulation of the SMIB power system by designing the control law ( $u_{f}, u_{B}$ ).

The SMIB power system with SVC can be classified to MIMO system. The system is divided into two parts and the controller design is realized into two steps by coordinated passivation approach. On one hand, we choose the input-output pairs where relative degrees are one or zero. On the other hand, the zero dynamic of the system will be stabilized by the other inputs.

Figure 3 depicts the structure of the proposed coordinated controller.

Fig. 3 Structure of proposed coordinated controller.

SMIB power system with SVC is considered as nonlinear dynamic system with two inputs:

\{\begin{array}{l} \dot{x} = f (x) + g_{1} (x) u_{1} + g_{2} (x) u_{2} \\ y = h (x, y) \end{array}

(22)

The relative degree is the number of derivatives of the output $y$ until the input $u_{1}$ appears.

( $δ_{0}$ , $ω_{0}$ , $E_{q 0}^{'}$ , $B_{L 0}$ ) represents an operation point of (13). Define new state variables as $x_{1} = δ - δ_{0}$ , $x_{2} = ω - ω_{0}$ , $x_{3} = E_{q}^{'} - E_{q 0}^{'}$ and $x_{4} = B_{S V C} - B_{S V C 0}$ . The inputs of the system are $u_{1} = u_{B}$ and $u_{2} = u_{f}$ . The output is chosen as $y = x_{4} = B_{S V C} - B_{S V C 0}$ . Then the system represented by (13) can be presented by (23)-(26).

{\dot{x}}_{1} = x_{2}

(23)

\begin{array}{l} {\dot{x}}_{2} = - \frac{D}{H} x_{2} + \frac{ω_{s}}{H} P_{m} + \frac{ω_{s}}{H} \frac{x_{d} - x_{d}^{'}}{2 X_{d Σ}^{'} X_{d Σ}} V_{s}^{2} s i n (2 (x_{1} + δ_{0})) - \\ \frac{ω_{s} V_{s} s i n (x_{1} + δ_{0})}{H X_{d Σ}^{'}} (x_{3} + E_{q 0}^{'}) \end{array}

(24)

{\dot{x}}_{3} = - \frac{x_{d} - x_{d}^{'}}{T_{d 0}^{'} X_{d Σ}} V_{s} c o s (x_{1} + δ_{0}) - \frac{1}{T_{d 0}^{'}} \frac{X_{d Σ}^{'}}{X_{d Σ}} (x_{3} + E_{q 0}^{'}) +

\frac{k_{c}}{T_{d 0}^{'}} u_{2}

(25)

\dot{y} = - \frac{1}{T_{S}} y + \frac{K_{s}}{T_{S}} u_{1}

(26)

It is clearly that the relative degree of the input-output pair ( $u_{1}, y$ ) is 1.

The conception of the controller is divided into two parts. On one hand, the design of the zero dynamics is ensured by the backstepping controller to obtain $u_{2}$ . On the other hand, passivation approach will be applied to design $u_{1}$ to make the whole system asymptotically stable.

A. Design of Backstepping Controller for x-subsystem

From (23)-(26), the zero dynamics system can be presented as:

\dot{x} = [\begin{matrix} x_{2} \\ - \frac{D}{H} x_{2} + \frac{ω_{s}}{H} θ + μ_{1} (x) - μ_{2} (x) (x_{3} + E_{q 0}^{'}) \\ μ_{3} (x) \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ \frac{k_{c}}{T_{d 0}^{'}} \end{matrix}] u_{2}

(27)

In the following procedure, we use the backstepping method to design the control input $u_{2}$ . For simplicity, we denote $μ_{1} (x) = \frac{ω_{s}}{H} \frac{x_{d} - x_{d}^{'}}{2 X_{d Σ 0}^{'} X_{d Σ 0}} V_{s}^{2} s i n (x_{1} + δ_{0})$ , $μ_{2} (x) = \frac{ω_{s} V_{s} s i n (x_{1} + δ_{0})}{H X_{d Σ 0}^{'}}$ , and $μ_{3} (x) = - \frac{x_{d} - x_{d}^{'}}{T_{d 0}^{'} X_{d Σ 0}} V_{s} c o s (x_{1} + δ_{0}) - \frac{1}{T_{d 0}^{'}} \frac{X_{d Σ 0}^{'}}{X_{d Σ 0}} (x_{3} + E_{q 0}^{'})$ .

Step 1: define the error variables $z_{1} = x_{1}$ , $z_{2} = x_{2} - α_{1}$ .

In (27), $x_{2}$ is considered as the virtual control variable. The dynamic of $z_{1}$ is given by:

{\dot{z}}_{1} = {\dot{x}}_{1} = z_{2} + α_{1}

(28)

$V_{1}$ is the corresponding Lyapunov function and $V_{1} = z_{1}^{2} / 2$ . The time derivate along the system trajectory is:

{\dot{V}}_{1} = z_{1} {\dot{z}}_{1} = z_{1} (z_{2} + α_{1})

(29)

We choose the virtual control as $α_{1} = - k_{1} z_{1}$ , where $k_{1} > 0$ is a design constant, which ensures ${\dot{V}}_{1} \leq 0$ when $z_{2} = 0$ .

Step 2: augment the Lyapunov function of the previous step as:

V_{2} = V_{1} + \frac{1}{2} z_{2}^{2}

(30)

Note that:

{\dot{z}}_{2} = {\dot{x}}_{2} - {\dot{α}}_{1} = (k_{1} - \frac{D}{H}) x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) - μ_{2} (x) (z_{3} + α_{2} + E_{q 0}^{'})

(31)

The time derivative of $V_{2}$ along the system trajectory is:

{\dot{V}}_{2} = - k_{1} z_{1}^{2} + z_{2} [z_{1} + (k_{1} - \frac{D}{H}) x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) - μ_{2} (x) (z_{3} + α_{2} + E_{q 0}^{'})]

(32)

Denote the error variable $z_{3} = x_{3} - α_{2}$ , where $x_{3}$ is the virtual control variable. In this step, to design the virtual control $α_{2}$ , we make $z_{3} = 0$ to ensure ${\dot{V}}_{2} \leq 0$ .

α_{2} = \frac{1}{μ_{2} (x)} [z_{1} + (k_{1} - \frac{D}{H}) x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) + k_{2} z_{2}] - E_{q 0}^{'}

(33)

where $k_{2} > 0$ is another design constant.

By substituting (33) in (29), the time derivate of $V_{2}$ can be written as:

{\dot{V}}_{2} = - k_{1} z_{1}^{2} - k_{2} z_{2}^{2} - μ_{2} (x) z_{2} z_{3}

(34)

Step 3: augment the Lyapunov function of Step 2 by:

V_{3} = V_{2} + \frac{1}{2} z_{3}^{2}

(35)

The time derivate of $z_{3}$ along the system trajectory is:

\begin{array}{l} {\dot{z}}_{3} = \frac{k_{c}}{T_{d 0}^{'}} u_{2} + μ_{3} (x) - x_{2} \frac{c o s (x_{1} + δ_{0})}{μ_{2} (x)} \\ [z_{1} + (k_{1} - \frac{D}{H}) x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) + k_{2} z_{2}] - \frac{1}{μ_{2} (x)} \{x_{2} + (k_{1} + k_{2} - \frac{D}{H}) [- \frac{D}{H} x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) - μ_{2} (x) (x_{3} + E_{q 0}^{'})] + μ_{1} (x) x_{2} c o s (x_{1} + δ_{0}) + k_{1} k_{2} x_{2}\} \end{array}

(36)

Thus the time derivate of $V_{3}$ is given by:

\begin{array}{l} {\dot{V}}_{3} = {\dot{V}}_{2} + z_{3} {\dot{z}}_{3} = - k_{1} z_{1}^{2} - k_{2} z_{2}^{2} + z_{3} - μ_{2} (x) z_{2} + \frac{k_{c}}{T_{d 0}^{'}} u_{2} + μ_{3} (x) - \\ x_{2} z_{3} \frac{c o s (x_{1} + δ_{0})}{μ_{2} (x)} [z_{1} + (k_{1} - \frac{D}{H}) x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) + k_{2} z_{2}] - \frac{1}{μ_{2} (x)} z_{3} x_{2} + (k_{1} + k_{2} - \frac{D}{H}) \cdot \\ [- \frac{D}{H} x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) - μ_{2} (x) (x_{3} + E_{q 0}^{'})] - \\ \frac{μ_{1} (x)}{μ_{2} (x)} z_{3} x_{2} c o s (g) (x_{1} + δ_{0}) + k_{1} k_{2} x_{2} \end{array}

(37)

The control input $u_{2}$ is chosen as:

μ_{2} = \frac{T_{d 0}^{'}}{k_{c}} \{- k_{3} z_{3} + μ_{2} (x) z_{2} - μ_{3} (x) + x_{2} \frac{c o s (x_{1} + δ_{0})}{μ_{2} (x)} [z_{1} + (k_{1} - \frac{D}{H}) x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) + k_{2} z_{2}] + \frac{1}{μ_{2} (x)} [x_{2} + (k_{1} + k_{2} - \frac{D}{H}) (- \frac{D}{H} x_{2} + \frac{ω_{s}}{H} P_{m} + μ_{1} (x) - μ_{2} (x) (x_{3} + E_{q 0}^{'})) + μ_{1} (x) x_{2} c o s (x_{1} + δ_{0}) + k_{1} k_{2} x_{2}]\}

(38)

where $k_{3} > 0$ is also a design constant.

Then ${\dot{V}}_{3} = - k_{1} z_{1}^{2} - k_{2} z_{2}^{2} - k_{3} z_{3}^{2} \leq 0$ . Therefore, under the feedback control law (24), the zero dynamics closed-loop system is asymptotically stable and can be written as:

\{\begin{array}{l} {\dot{z}}_{1} = z_{2} - k_{1} z_{1} \\ {\dot{z}}_{2} = - k_{2} z_{2} - z_{1} - μ_{2} (x) z_{3} \\ {\dot{z}}_{3} = - k_{3} z_{3} + μ_{2} (x) z_{2} \end{array}

(39)

In fact, ${\dot{V}}_{3} \leq 0$ implies $V_{3} (t) \leq V_{3} (0)$ , i.e., $z_{1}$ , $z_{2}$ , $z_{3}$ are all bonded. Define $Ω = - {\dot{V}}_{3}$ , then $\int_{0}^{t} Ω (τ) d τ < V_{3} (t) - V_{3} (0)$ . Since $V_{3} (0)$ is bonded and $V_{3} (t)$ is non-increasingly bounded, $\underset{t \to \infty}{l i m} \int_{0}^{t} Ω (τ) d τ < \infty$ holds. In addition, since $\dot{Ω}$ is bonded, $\underset{t \to \infty}{l i m} Ω = 0$ holds due to Barbalat’s lemma. Therefore, $z_{1} \to 0$ , $z_{2} \to 0$ and $z_{3} \to 0$ as $t \to \infty$ . From the definitions of $x_{1}, x_{2}, x_{3}, α_{1}, α_{2}$ , it is clear that the system variables $x_{1}, x_{2}, x_{3}$ also converge to zero.

B. Design of Coordinated Passivation

In this sub-section, $u_{1}$ is deduced by the coordinated passivation control. Then, the whole system can be stabilized by the feedback passivation controller.

Let $W = V_{3}$ , the storage function $V = W (x) + y^{2} / 2$ is selected and the control law is designed as defined in the feedback passivation approach (21):

u_{1} = \frac{T_{s}}{K_{S}} \{- \frac{z_{2}}{y} [\frac{ω_{s}}{H} \frac{x_{d} - x_{d}^{'}}{2 X_{d Σ}^{'} X_{d Σ}} V_{s}^{2} s i n (2 (x_{1} + δ_{0})) - μ_{1} (x) - (\frac{ω_{s} V_{s} s i n (x_{1} + δ_{0})}{H X_{d Σ}^{'}} - μ_{2} (x)) (x_{3} + E_{q 0}^{'})] - \frac{z_{3}}{y} [- \frac{x_{d} - x_{d}^{'}}{T_{d 0}^{'} X_{d Σ}} \cdot V_{s} c o s (x_{1} + δ_{0}) - \frac{1}{T_{d 0}^{'}} \frac{X_{d Σ}^{'}}{X_{d Σ}} (x_{3} + E_{q 0}^{'}) - μ_{3} (x)] + v\}

(40)

The time derivate of $V$ along the system trajectory is:

\dot{V} = \dot{W} + y \dot{y} = {\frac{\partial W (x)}{\partial z} \dot{z}|}_{y = 0} - \frac{1}{T_{s}} y^{2} + v y \leq - \frac{1}{T_{s}} y^{2} + v y

(41)

In this way, the system output is strictly passive. The choice of $v = - β y (β > 0)$ yields $\dot{V} < 0$ if $y \to 0$ and $t \to \infty$ .

V. Simulation Results

The studied system is implemented in MATLAB and the simulation responses of the system have been carried out for the above control design strategies. The system parameters and the parameters of the proposed controller are provided in Appendix A.

To demonstrate its effectiveness, the proposed control strategy is compared with: ① robust excitation controller [

24]; ② separately designed linear AVR/PSS and SVC+PI controller.

The robustness of the proposed controller is tested according to the short-circuit periods. Two cases are studied in this paper.

1) Case 1: the short-circuit period is set to be $100 m s$ .

The following fault sequences are simulated as follows.

Stage 1: the system is in a pre-fault steady-state.

Stage 2: a three-phase short circuit occurs in one of the three transmission lines at $t = 0.5$ s.

Stage 3: the fault is removed by opening the breaker of the faulty line at $t = 0.6$ s.

Stage 4: the system is in a post-fault state.

Figures 4 and 5 show the responses of relative speed and terminal voltage generator in case 1, respectively. Figures 6 and 7 show the responses of SVC terminal voltage and field voltage, respectively. Figures 8-10 show the responses of rotor angle, electrical power, and TCR susceptance in case 1, respectively.

Fig. 4 Response of relative speed in case 1.

Fig. 5 Response of terminal voltage generator in case 1.

Fig. 6 Response of SVC terminal voltage in case 1.

Fig. 7 Response of field voltage in case 1.

Fig. 8 Response of rotor angle in case 1.

Fig. 9 Response of electrical power in case 1.

Fig. 10 Response of TCR susceptance in case 1.

Table I shows the values of the main variables constituting the SMIB power system with SVC in steady state, and Table II shows the stabilization time and steady-state errors of different variables for each controller type. Overall, the proposed controller has the best performance (in transient and permanent phase) compared with all other controllers.

Table I Main Variables Constituting SMIB-SVC Power System in Steady State

Controller	$δ$ (rad)		$ω$ $(r a d / s)$		$P_{e}$ (p.u.)		$V_{t}$ (p.u.)		$V_{m}$ (p.u.)
Controller	Pre- fault	Post- fault	Pre- fault	Post- fault	Pre- fault	Post- fault	Pre- fault	Post- fault	Pre- fault	Post- fault
AVR/ PSS and SVC PI controller	$1.246$	$1.326$	$0$	$0$	$0.9$	$0.9$	$0.9947$	$0.9947$	$0.9638$	$0.9638$
Robust excitation controller	$1.246$	$1.430$	0	0	$0.9$	$0.9$	$1$	$1$	$0.9638$	$0.6946$
Coordinated excitation and SVC controller	$1.246$	$1.246$	$0$	$0$	$0.9$	$0.9$	$1$	$1$	$0.9638$	$0.9638$

Table II Stabilization Time and Steady-state Errors of Different Variables for Each Controller Type

Controller	$T_{s}$ (s)					$E_{S S}$ $(%)$
Controller	$δ$	$ω$	$P_{e}$	$V_{t}$	$V_{m}$	$δ$	$ω$	$P_{e}$	$V_{t}$	$V_{m}$
AVR/PSS and SVC+PI controller	$3.658$	$3.09$ 0	$2.241$	$0.519$	$2.339$	6	$0$	$0$	$0$	0
Robust excitation controller	$0.874$	$0.78$ 0	$0.664$	$1.127$	$0.708$	14	$0$	$0$	0	30
Coordinated excitation and SVC controller	$0.748$	$0.507$	$1.497$	$0.519$	$0.426$	0	$0$	$0$	0	0

The power angle in Fig. 8 reaches its initial value after the disappearance of the defect under a transient condition lasting $0.748$ s compared with other controllers. Figure 4 shows the evolution of the relative speed. It is clearly shown that with the proposed controller, the speed stabilizes faster than other controllers with oscillation damping $T_{s} = 0.507$ s.

The response of relative speed is shown in Fig. 4, i.e., the absence of the static error by the proposed controller linearization. Figure 6 shows the SVC terminal voltage. With the proposed controller, the SVC terminal voltage returns to its initial value of $0.9638$ p.u.. However, with only the excitation controller and without SVC, the voltage drop in the middle of the transmission line is $0.6946$ p.u. as shown in Table I, which justifies the contribution of SVC installed in the line for voltage regulation.

2) Case 2: the short-circuit period is set to be $155 m s$ .

Figures 11-13 show the robustness of the coordinated excitation and SVC controller. Even if we increase the short-circuit time, the system can support it and converge quickly without stability loss.

Fig. 11 Response of power angle in case 2.

Fig. 12 Response of terminal voltage generator in case 2.

Fig. 13 Response of SVC terminal voltage in case 2.

Linear excitation control of AVR/PSS and the SVC+PI controller can dampen the oscillation in case the duration of short circuit does not exceed $100 m s$ (case 1). However, the performance gets worse when the duration of short circuit reaches $155 m s$ (case 2). Moreover, even in case 1, system performance with the control designed separately is not so good as that of a coordinated control scheme, which shows the effectiveness of the proposed coordinated control.

The proposed coordinated excitation and SVC controller can improve transient stability and effectively dampen oscillations, regardless of short-circuit duration, and simultaneously achieve good post-fault voltage regulation.

VI. Conclusion

We aim to design and implement a nonlinear coordinated control of SMIB power system equipped with an SVC installed in the middle of the transmission line in order to improve the power angle stability, terminal voltage generator regulation and voltage regulation at SVC.

For controller design, applying coordinated passivation approach divides the system into two parts. On one hand, the excitation controller is designed via back-stepping controller. On the other hand, the passivation theory is applied to design the SVC controller, which eventually brings back to the whole system stability. In order to investigate the effectiveness of the proposed controller, a three-phase short circuit occurs. Simulations results verify the effectiveness of the proposed controller, especially the elimination of the static error of the voltage at the SVC terminals after the elimination of the defect by comparing with the results given by the control of the excitation and without SVC.

Appendix

APPENDIX A

The power system parameters are shown in Table AI.

Table AI Power System Parameters

Parameter	Value
Synchronous speed	$ω_{0} = 314.159 r a d / s$
Damping constant	$D = 5$
Inertia constant	$H = 4$
d-axis transient open-circuit time constant	$T_{d 0}^{'} = 6.9 s$
Generator d-axis reactance	$x_{d} = 1.863 p . u .$
Generator d-axis transient reactance	$x_{d}^{'} = 0.257 p . u .$
Transmission line reactance	$x_{L} = \frac{x_{L 1} x_{2}}{x_{L 1} + x_{L 2}}, x_{L 1} = 0.4853 p . u .$ , $x_{L 2} = 0.4853 p . u .$
Transformer reactance	$x_{T} = 0.127 p . u .$
Initial TCR susceptance	$B_{l 0} = 1 p . u .$
Capacitor bank susceptance	$B_{c} = 1 p . u .$
Constant time of SVC	$T_{S} = 0.5 s$
SVC gain	$K_{S} = 150$

AVR parameters include: $G_{A V R} = K_{A} / (1 + s T_{A})$ , $K_{a} = 200, T_{a} = 0.15$ . PSS parameters include: $K_{s t a b} = 9.5$ , $T_{w}$ = $1.41$ , $G_{P S S} = K_{s t a b} [s T_{w} / (1 + s T_{w})] [(1 + s T_{1}) / (1 + s T_{2})]$ , $T_{1} = 0.154, T_{2} = 0.154 .$ PI SVC controller parameters include: $K_{p} = 9.5$ , $K_{I} = 0.005 .$ The proposed controller parameters include: $K_{1} = 10$ , $K_{2} = 500$ , $K_{3} = 300$ , $a n d$ $β = 0.5$ .

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Nonlinear Coordinated Passivation Control of Single Machine Infinite Bus Power System with Static Var Compensator PDF

Abstract

Keywords

I. Introduction

II. Modelling of Power System

A. Mechanical Equations

B. Generator Electrical Dynamics

C. SVC Model

D. Electrical Equations

III. Overview on Coordinated Passivation Approach

IV. Design of Coordinated Controller for SMIBPower System with SVC

A. Design of Backstepping Controller for x-subsystem

B. Design of Coordinated Passivation

V. Simulation Results

VI. Conclusion

Appendix

APPENDIX A

REFERENCES