Fuzzy-PSS and fuzzy neural network non-linear PI controller-based SSSC for damping inter-area oscillations

Abstract

In this paper, a new approach is recommended for damping out power system oscillations. In this approach, a new fuzzy neural proportional–integral (PI) controller (FNPIC) based static synchronous series compensator (SSSC) and a fuzzy-power system stabilizer (fuzzy-PSS) with a new structure are simultaneously employed. Adaptive learning rates based on the Lyapunov stability theory to enhance the convergence speed of the proposed controller are obtained. In the structure of fuzzy-PSS, a pattern search algorithm is used to adjust four gains. A two-area four-machine power system and a three-area, three-machine power system are employed to investigate the efficiency of the proposed method. Simulations results confirm the capabilities of proposed controller in suppressing the power system oscillations.

Keywords

Fuzzy neural network non-linear PI controller intelligent controller inter-area oscillations static synchronous series compensator (SSSC)

Introduction

Augmented demands on transmission, lack of long-term planning and the need to provide open access to generating companies and customers have created tendencies toward less security and reduced quality of supply. The flexible alternating current transmission system (FACTS) technology is indispensable for moderating some but not all of these difficulties. The FACTS technology creates new chances to control power and increase the usable capacity of the present, as well as new and upgraded lines. The opportunity that current and therefore power through a line can be controlled enables a large potential for enhancing the capacity of existing lines. These opportunities are achieved through the ability of FACTS controllers to control the interrelated parameters that govern the operation of transmission systems comprising series impedance, shunt impedance, current, voltage, phase angle and the damping of oscillations.

The static synchronous series compensator (SSSC), which is a member of the FACTS family, may have transiently rated energy storage or energy absorbing devices to improve the dynamic behaviour of the power system by additional temporary active power compensation, to enhance or reduce transitorily the overall active (resistive) voltage drop across the line (Hingorani and Gyugyi, 2000).

Because the inherent damping potential of an SSSC is low and under certain situations it may be insufficient (Farahani, 2012), to attain effective damping, an additional damping controller is added to the SSSC. Here, two main issues related to this additional damping controller may arise: the type of additional controller and type of input signal to the additional controller.

In general, controllers proposed as an additional damping controller for the SSSC can be divided into two groups. The first is controllers with fixed parameters, such as proportional–integral–derivative (PID) controllers (Acha et al., 2002) and lead-lag compensators (Abd-Elazim and Ali, 2016; Elazim and Ali, 2016; Farah et al., 2016; Kazemi et al., 2005; Khodabakhshian et al., 2016; Panda, 2009; Panda et al., 2008). These controllers benefit from a simple structure, whereas they require an accurate mathematical model to adjust their parameters. Furthermore, they are unable to provide adequate damping under diverse situations of a power system. The second is adaptive controllers, such as fuzzy controllers (Alizadeh and Tofighi, 2013; Radaideh et al., 2012; Reddy and Mohanta, 2008; Sadeghzadeh and Ansarian, 2006; Sadeghzadeh et al., 1998; Talaat et al., 2010), neural network controllers (Alizadeh et al., 2013; Badar and Khan, 2012; Farahani and Ganjefar, 2013) and self-tuning PID controllers (Farahani and Ganjefar, 2012), the parameters of which are updated during the control process. Although these controllers need hardware implementation and have more complicated structures than the first group, they can acquire knowledge from functions, are fault tolerant in the sense that they can handle noisy and faulty data, may overcome non-linear problems, and once trained may perform prediction and generalization at high speed.

The signals selected as the input of additional damping controllers can be divided to local and remote signals. The tie-line active power can be introduced as a local signal that has been used in some papers. The local signals are unable to provide high damping for the additional controller (Panda et al., 2008). The rotor speed of generators can be put into the remote signals category. These signals should be sent from the measuring location to the central control (Panda, 2009).

The contribution of this paper is to damp power system oscillations through an adaptive controller with a remote signal as an additional damping controller for enhancing the damping of an SSSC and a fuzzy-power system stabilizer (fuzzy-PSS). This adaptive controller is a fuzzy neural non-linear proportional–integral (PI) controller, which combines the abilities of non-linear PI controllers and fuzzy neural networks (FNNs) in controlling complicated systems. In order to accelerate the convergence speed of proposed controller, the adaptive learning rates are extracted from the Lyapunov stability theory. In addition, in the architecture of fuzzy-PSS, there are four gains that are tuned by a pattern search (PS) algorithm. The simulation results confirm the effectiveness of proposed approach in damping power system oscillations.

The rest of this paper is organized as follows: in the next section, the proposed power system is studied. Then we describe the structure of the SSSC. The proposed control approach is presented and simulation results are given. Finally, some concluding remarks and discussion are presented.

Overview of SSSC and power system under study

Figure 1(a) shows the power system comprising two fully symmetrical areas. As seen in this figure, the two areas are linked together through two 230-kV lines of 220 km length (Kundur et al., 1994). In this power system, there are two 20-kV/900-MVA synchronous generators. The load is represented as constant impedances and split between the areas. As the surge impedance loading of a single line is about 140 MW (Kundur et al., 1994), the system is somewhat stressed, even in steady state. Meanwhile, to damp the power system oscillations, a 100-MVA SSSC is installed in series with line-2. A dynamical model of a synchronous machine is given in Appendix A. A modal analysis of acceleration powers of the four machines shows three dominant modes:

An inter-area mode (Δω_IA) (f_n=0.64 Hz, damping rate=−0.026) involving the entire area 1 against area 2.

A local mode of area 1 (f_n=1.12 Hz, damping rate=0.08) involving this area’s machines against each other.

A local mode of area 2 (f_n=1.16 Hz, damping rate=0.08) involving this area’s machines against each other.

This modal analysis demonstrates that the inter area-mode is not stable, because it has a negative damping. In addition, a three-area power system shown in Figure 1(b) is used to confirm the effectiveness of the proposed approach. A 100-MVA SSSC is installed in series with line-1.

The SSSC as a member of FACTS family uses power electronics to control power flow and improve power oscillation damping (Hingorani and Gyugyi, 2000). The SSSC is able to present the virtual compensation of the transmission line by injecting the controllable voltage (V_q) in series with the transmission line. Because of operating an SSSC in capacitive as well as inductive mode, it can be employed as an effective tool to control the power flow of the system. Furthermore, to improve the stability of a power system, an auxiliary stabilizing signal can also be attached to the power flow control function of the SSSC (Hingorani and Gyugyi, 2000). The virtual reactance caused by injecting V_q has effect on electric power flow in the transmission lines (Hingorani and Gyugyi, 2000). The injected voltage value is varied by a voltage-sourced converter (VSC) connected on the secondary side of a transformer. To control the compensation level dynamically, the magnitude and polarity of V_q is variable. In this case, the SSSC can work in inductive and capacitive modes.

The proposed approach

The strategy of PID control has been one of more superior and most regularly used methods in the industry. This is because that the structure of PID controllers is simple and they have strong robustness in broad operating conditions. However, the requirement of control precision becomes higher and higher in conformity with the complexity of systems. A conventional PID controller with fixed parameters may ordinarily diminish the performance of the controller. Different types of modified PID controllers have been introduced, such as the intelligent PID controller (Åström et al., 1992), self-tuning discrete PID controller (Dey and Mudi, 2009) and self-tuning predictive PID controller (Ganjefar and Farahani, 2012; Vega et al., 1991).

However, if severe non-linearity is involved in the controller process, a non-linear control approach will be more useful, especially in the case of a power system. Currently, neural networks or wavelet neural networks have proved a promising approach to solve complex non-linear control problems. Hence, it motivates us to combine an FNN with a non-linear PID controller. It is expected that the combination will make the most of the simplicity of non-linear PID controller and the FNN’s powerful capability to learn, adapt and tackle non-linearity.

Strategy of control

The proposed strategy of control includes two parts: fuzzy-PSS tuned by a PS algorithm and a fuzzy neural PI controller (FNPIC)-based SSSC.

FNPIC-based SSSC

Figure 2 shows a block diagram of the FNPIC to modulate the injected voltage V_q, in which the FNPIC contains the FNN, online training mechanism and adaptive learning rates. As seen in this figure, the error e and e(1−z⁻¹) are selected as the inputs of the FNPIC. The variable e is defined by the error between the inter-area oscillations $Δ ω_{I A}$ and its desired $Δ ω_{d}$ , in which $Δ ω_{I A}$ are the inter-area oscillations in the power systems. Moreover, the output of intelligent control system is supplemented to the reference voltage to modulate the injected voltage of the SSSC. In this case, the output of SSSC is so varied that the inter-area oscillations are effectively damped out.

Fuzzy neural PI controller

The architecture of a four-layer FNN is shown in Figure 3. In this figure, the structure of the FNPIC is thoroughly illuminated. In the first layer, the output of each node is computed as follows:

μ_{i j} = \exp (- {(\frac{x_{i} - c_{i j}}{σ_{i j}})}^{2})

(1)

where x_i, $μ_{i j}$ , $c_{i j}$ and $σ_{i j}$ denote the inputs of the FNN, membership function, centre and standard deviation of fuzzy membership function corresponding to the input x_i in the jth rule, respectively. In the third layer, the number of nodes is equal to the number of fuzzy rules. Actually, each node matches the antecedent part of the fuzzy rule. The output of these nodes is given by:

z_{j} = Π_{i = 1}^{n_{i}} μ_{i} = Π_{i = 1}^{n_{i}} \exp (- {(\frac{x_{i} - c_{i j}}{σ_{i j}})}^{2})

(2)

where z_j represents the firing strength of the jth rule; n_i is the number of input variables x_i. The third layer corresponds to the consequent part of the fuzzy rules, and the number of nodes in this layer is equal to the number of rules. To improve the capability of non-linear mapping in Takagi–Sugeno–Kang (TSK)-type fuzzy systems, the output of the jth node in this layer is described as a non-linear PI-like neural network:

\begin{matrix} y_{j} = \sum_{j = 1}^{n_{w}} ψ_{j} \\ ψ_{j} = \tanh (\sum_{i = 1}^{2} t_{i} w_{i j}) \end{matrix}

(3)

where t_i denotes the inputs of non-linear PI-like neural network; w_ij represents the connection weights. In the fourth layer, the defuzzification phase is carried out and the output of the overall FNPIC is represented by

\begin{matrix} u = a / a b \\ a = \sum_{j = 1}^{n_{w}} z_{j} y_{j}, \\ b = \sum_{j = 1}^{n_{w}} z_{j} \end{matrix}

(4)

where n_w is the number of trained rules in the FNPIC. The learning method of the FNPIC is explained in the following section.

Online training mechanism

In the training algorithm, the purpose is to compute a gradient vector wherein each element in the training algorithm is expressed as the derivative of an objective function in terms of parameters of the FNPIC such that the objective function is minimized. For this purpose, the chain derivative rule is benefited and the overall method is referred as the back error propagation learning rule, for the reason that the gradient vector is calculated in the direction opposite to the flow of the output of each node. First, take into consideration the following objective function:

J = \frac{1}{2} e^{2} = \frac{1}{2} {[Δ ω_{d} - Δ ω_{I A}]}^{2}

(5)

From the output of the FNN, the following error term should be propagated:

δ_{o} = - \frac{\partial J}{\partial u} = [- \frac{\partial J}{\partial e} \frac{\partial e}{\partial u}] = [- \frac{\partial J}{\partial e} \frac{\partial e}{\partial Δ ω_{I A}} \frac{\partial Δ ω_{I A}}{\partial u}]

(6)

The parameters c_ij and σij can be updated by using the following equations:

\begin{matrix} c_{i j} (k + 1) = c_{i j} (k) - η_{c} \frac{\partial J}{\partial c_{i j}} = c_{i j} (k) - η_{c} \frac{\partial J}{\partial u} \frac{\partial u}{\partial z_{j}} \frac{\partial z_{j}}{\partial c_{i j}} \\ = c_{i j} (k) + η_{c} δ_{o} \frac{ψ_{j} - u}{b} z_{j} \frac{2 (x - c_{i j})}{σ_{i j}^{2}} \end{matrix}

(7)

\begin{matrix} σ_{i j} (k + 1) = σ_{i j} (k) - η_{σ} \frac{\partial J}{\partial σ_{i j}} = σ_{i j} (k) - η_{σ} \frac{\partial J}{\partial u} \frac{\partial u}{\partial z} \frac{\partial z}{\partial σ_{i j}} \\ = σ_{i j} (k) + η_{σ} δ_{o} \frac{ψ_{j} - u}{b} z_{j} \frac{2 {(x - c_{i j})}^{2}}{σ_{i j}^{3}} \end{matrix}

(8)

where η_c and η_σ denote the learning rates of the centre and standard deviation of fuzzy membership function, respectively. In the non-linear PI-like neural network, the adaption law for w_ij is represented by:

\begin{matrix} w_{i j} (k + 1) = w_{i j} (k) - η_{w} \frac{\partial J}{\partial σ_{i j}} = w_{i j} (k) - η_{w} \frac{\partial J}{\partial u} \frac{\partial u}{\partial w_{i j}} \\ = w_{i j} (k) + η_{w} δ_{o} \frac{z_{j}}{b} t_{i} (1 - \tanh^{2} (t_{i} w_{i j})) \end{matrix}

(9)

where η_w is the learning rate of connection weights.

In (6), $\frac{\partial Δ ω_{I A}}{\partial u}$ should be computed using the power system dynamics. The gradient of complex non-linear dynamic systems cannot be found obviously. In this study, instead of the system gradient, the delta adaptation rule proposed in Alizadeh et al. (2013) is utilized

δ_{o} ≅ e + A \overset{\cdot}{e}

(10)

where A is a positive number.

Convergence analyses of FNPIC through ALRs

Selecting the appropriate learning rates affects the convergence of the FNPIC. For the reason that the learning rate is a key factor to determine the convergence of the FNPIC trained by the gradient descent method, it is important to acquire the ideal learning rate at each iteration of the training algorithm (Alizadeh et al., 2013; Farahani and Ganjefar, 2013). In most cases, a learning rate is often selected as a fixed parameter by using trial and error. In this paper, the adaptive learning rates, which can be adapted rapidly, are obtained.

Consider the following discrete Lyapunov function:

V (k) = J (k) = \frac{1}{2} {(e (k))}^{2}

(11)

The change of Lyapunov function is given by

Δ V (k) = V (k + 1) - V (k) = \frac{1}{2} [e^{2} (k + 1) - e^{2} (k)]

(12)

The error difference can be represented by

Δ e (k) = e (k + 1) - e (k) ≅ {[\frac{\partial e (k)}{\partial W_{i}}]}^{T} Δ W_{i}

(13)

where W_i is an arbitrary component of the vector W of the FNPIC:

W = {[c, σ]}^{T}

(14)

and the corresponding change of W_i is symbolized by $Δ$ W_i and is obtained by

Δ W_{i} = η_{W} δ_{o} \frac{\partial u}{\partial W_{i}}

(15)

where η_W is the learning rate corresponding to the vector component W_i.

Theorem 1. Let η_W=diag[η₁,η₂]=diag[η_c,η_σ] be the learning rates of the parameters of the FNPIC and define C_max as

C_{\max} = {[C_{\max}^{1}, C_{\max}^{2}]}^{T} = {[\max ‖ \frac{\partial u}{\partial c} ‖, \max ‖ \frac{\partial u}{\partial σ} ‖]}^{T}

(16)

where ||.|| represents the Euclidean norm. Then, the convergence is guaranteed if η_l are chosen to satisfy η_W=λ/(C^l_max)², l=1,2 and 3, where λ is a positive value.

Proof. See Appendix B.

In order to prove Theorem 2, the following lemmas are used (Farahani, 2013):

Lemma 1.Let $f (t) = t \exp (- t^{2})$ . Then, $| f (t) | < 1$ $\forall t \in ℜ$ .

Lemma 2.Let $g (t) = t^{2} \exp (- t^{2})$ . Then, $| g (t) | < 1$ $\forall t \in ℜ$ .

Theorem 2. Let η_c and η_σ be the learning rates of the translation and dilation for the FNN, respectively. The convergence is guaranteed if the learning rates satisfy

η_{c} = \frac{λ}{n_{w}} {[\frac{{| σ_{i j} |}_{\min}}{2 \exp (- 0.5) ({| \frac{ψ_{j} - u}{b} |}_{\max})}]}^{2}

(17)

η_{σ} = \frac{λ}{n_{w}} {[\frac{{| σ_{i j} |}_{\min}}{2 \exp (- 0.5) ({| \frac{ψ_{j} - u}{b} |}_{\max})}]}^{2}

(18)

where | $ψ$ _j |_max and |σ_ij|_min are the maximum of the absolute value of weights vector and the minimum of the absolute value of dilations vector, respectively; $λ$ is a positive constant value.

Proof. See Appendix C.

Fuzzy-PSS

The structure of Fuzzy-PSS

A fuzzy inference system (FIS) maps given inputs to outputs using fuzzy logic. The fuzzy PID controller uses a parallel structure (Xu et al., 2000), as shown in Figure 4. It is a combination of fuzzy PI control and fuzzy proportional–derivative (PD) control. The change of measurement −(y(k)−y(k−1)) is used, instead of change of error e(k)−e(k−1), as the second input signal to FIS to prevent the step change in reference signal from directly triggering the derivative action.

FIS settings such as its style, membership functions and rule base to obtain a desired non-linear control surface are adjusted. Gaussian curve membership function is selected as the membership function. Each input set has two terms (Positive and Negative). The following rules are defined:

If E is Negative and CE is Negative then u is −20.

If E is Negative and CE is Positive then u is 0.

If E is Positive and CE is Negative then u is 0.

If E is Positive and CE is Positive then u is 20.

The 3-D non-linear control surface is plotted in Figure 5. It has a higher gain near the centre of the E and CE plane than the linear surface has, which helps reduce the error more quickly when the error is small. When the error is large, controller becomes less aggressive so that control action is limited to avoid possible saturation.

Design of fuzzy-PSS

To achieve the best performance of fuzzy-PSS in damping the power system oscillations, the difference between input mechanical power to generator P_m and output electrical power from generator P_e and its reference value, which is zero, are selected as parameters y and r in structure shown in Figure 4, respectively. The next phase is to determine the parameters of fuzzy-PSS. The optimal solutions of the fuzzy-PSS are taken into account, as an optimization problem and PS algorithm will be utilized to solve them.

Optimization problem

It should be that the design purpose of the fuzzy-PSS is to damp the power system oscillations. Thus, the purpose is formulated as the minimization of the objective function F represented by

F = f_{1} + f_{2}

(19)

where

f_{1} = \int_{t_{1} = 0}^{t_{2} = t_{s i m}} t | Δ ω_{I A} (t) | d t

(20)

f_{2} = \int_{t_{1} = 0}^{t_{2} = t_{s i m}} t | Δ ω_{L A} (t) | d t

(21)

where t_sim is the simulation time period and $Δ ω_{L A}$ is local mode of power system oscillations. To calculate the objective function, the time-domain simulation of the system model considering all limits of control signals is carried out for the simulation period. The design problem can be formulated as the following constrained optimization problem, where the constraint is the parameter bounds:

\begin{matrix} G C E^{\min} ⩽ G C E ⩽ G C E^{\max} \\ G E^{\min} ⩽ G E ⩽ G E^{\max} \\ G C U^{\min} ⩽ G C U ⩽ G C U^{\max} \\ G U^{\min} ⩽ G U ⩽ G U^{\max} \end{matrix}

(22)

Pattern search algorithm

The benefits of being a trouble-freeing concept, easy to implement and efficient computationally algorithm can be stated for the PS algorithm. Unlike other heuristic algorithms such as the genetic algorithm (GA), the PS has a flexible and well-balanced operator to enhance and adjust the global and fine tune local search (Al-Othman and El-Nagger, 2007).

The start point of the PS algorithm is to calculate a sequence of points that may or may not be closed to the optimal point. The algorithm begins by generating a set of points called mesh, around the given point. This current point could be the initial point that is entered by the user or it could be calculated from the previous run of the algorithm. The mesh is formed by adding the current point to a scalar multiple of a set of vectors called a pattern. If the mesh includes a point that improves the objective function at the current point, in the next iteration, the current point is considered the new point. This maybe better explained by the following (Al-Othman and El-Nagger, 2007).

First, the PS starts at the initial point X₀ that is entered as a starting point by the user. At the first iteration, with a scalar=1 called mesh size, the pattern vectors are built as [01], [10], [−1 0] and [0 −1], they may be called direction vectors. The direction vectors are then added to the initial point X₀ by the PS to compute the following mesh points:

\begin{matrix} X_{0} + [\begin{matrix} 0 & 1 \end{matrix}] \\ X_{0} + [\begin{matrix} 1 & 0 \end{matrix}] \\ X_{0} + [\begin{matrix} - 1 & 0 \end{matrix}] \\ X_{0} + [\begin{matrix} 0 & - 1 \end{matrix}] \end{matrix}

Figure 6 shows how to form the mesh and pattern vectors. The algorithm calculates the objective function at the current mesh points in the stated order. The algorithm polls the mesh points by computing their objective function values to find one with a value smaller than the objective function value of X₀. If there is such a point, then the poll is successful and the algorithm sets this point as new point X₁.

After a successful poll, the algorithm moves to iteration 2 and multiplies the current mesh size by 2 (this is called the expansion factor and has a default value of 2). At iteration 2, the mesh has the following points: 2*[1 0] +X₁, 2*[0 1] +X₁, 2*[−1 0] +X₁, 2*[0 −1] +X₁,

The algorithm again polls the mesh points to find one whose value is smaller than objective function value of X₁. If there is such a point, it will be called X₂ and the poll is successful. Because of being a successful poll, the algorithm multiplies the current mesh size by 2 to obtain a mesh size of 4 at the third iteration because the expansion factor is 2.

Second, if iteration 3 (mesh size=4), ends up with an unsuccessful poll, in other words, none of the mesh points has a smaller objective function value than the value at X₂, in this case the algorithm transfers the current point without any change to the next iteration, i.e. X₃=X₂. At the next iteration, the algorithm multiplies the current mesh size by 0.5 (a contraction factor) so that the mesh size at the next iteration is smaller. The algorithm then polls with a smaller mesh size.

The above steps will be iterated by the PS optimization algorithm to obtain the optimal solution to minimize the objective function.

Simulation results and discussions

Here, the performance of the proposed approach is evaluated under several operating conditions and with different disturbances. In this regard, the different loading conditions given in Table 1 are considered. Simulations are carried out for the example power system subjected to various disturbances. Table 2 provides the architecture of the FNPIC used in the simulations.

Table 1.

Operating conditions used for the simulations (in p.u.).

		Operating condition
		Case 1	Case 2
G₁	P ₁	0.78	0.78
	Q ₁	0.10	0.21
G₂	P ₂	0.78	0.56
	Q ₂	0.13	0.26
G₃	P ₃	0.80	0.80
	Q ₃	0.10	0.07
G₄	P ₄	0.78	0.89
	Q ₄	0.10	0.22

Table 2.

The structure of fuzzy neural proportional–integral (PI) controller (FNPIC) for the simulations.

	Number of existing neurons
	Input layer (n_i)	Hidden layer (n_w)	Output layer
FNPIC	2	6	1

Before presenting the simulation results, the PS algorithm is used to determine the optimal parameters of Fuzzy-PSS. It should be noted that the optimization process are run several times and the best parameters are then selected from the obtained results. In addition, the optimal parameters of fuzzy-PSS are obtained in the presence of the FNPIC-based SSSC. Figure 7 demonstrates changes of objective function and mesh size during the optimization process. The optimization process was completed at iteration of 342. The final parameters obtained from the PS are presented in Table 3. It should be noted that fuzzy-PSS is installed only on two generators G₁ and G₃.

Table 3.

Final parameters of fuzzy-power system stabilizer (fuzzy-PSS) obtained from the pattern search (PS).

Generator	Fuzzy-PSS				f ₁	f ₂
Generator	GCE	GCU	GE	GU	f ₁	f ₂
G₁	3.83	8.34	6.38	4.16	0.011775	0.023446
G₂	3.85	8.42	6.41	4.16	0.012552	0.023354

Case 1

In this case, the power system shown in Figure 1(a) is used to demonstrate the performance of proposed approach.

As the first simulation, a 220-ms two-phase to ground fault is applied to the middle of line-1. To validate the effective performance of the proposed controller in suppressing power system oscillations, the results obtained from the proposed approach are compared with the results obtained from a lead-lag compensator (or CPSS), which has been proposed in Kundur et al. (1994). Lead-lag compensators are broadly utilized in power systems for various goals because of their simple structure and acceptable performance.

In addition, the following performance index, which is based on the system performance characteristics, is defined.

\begin{matrix} ITAE = 1000 \\ \int_{0}^{t_{s i m}} t (| Δ ω_{1, 3} | + | Δ ω_{1, 4} | + | Δ ω_{2, 3} | + | Δ ω_{2, 4} | + | Δ ω_{1, 2} | + | Δ ω_{3, 4} |) d t \end{matrix}

(23)

where t_sim is the time period of simulations. A smaller value of the integral of time-weighted absolute error (ITAE) index shows better performance and dynamical response. The inter-area and local modes of oscillations are shown in Figures 8(a) and (b), respectively. As seen in this figure, the performance of proposed controller is better than CPSS. Figure 8(c) shows numerical results of the ITAE performance index for the two controllers. Clearly, the minimum value belongs to the proposed approach, which means a better dynamical response and faster damping of oscillations.

For the next simulation, it is assumed that a three-phase to ground fault occurs at the beginning of line-1 at t=1 s and cleared 220 ms later. Figures 9(a) and (b) show the inter-area and local modes of oscillations for this fault. It is clear from this figure that the best performance belongs to the proposed controller. Numerical results of the ITAE performance index for two controllers are presented in Figure 9(c). As expected, the ITAE index in the presence of proposed approach shows a smaller value.

Case 2

In order to verify the effectiveness of proposed approach, the performance of proposed controller is evaluated on case 2 given in Table 1. Figures 10(a) and (b) illustrate the inter-area and local modes of oscillations for when line-1 is disconnected at t=1 s (F3). Clearly, the proposed approach can suppress the entire oscillations satisfactorily. Numerical results of the ITAE performance index for two controllers can be seen from Figure 10(c).

For completeness, a single-phase to ground fault is applied to the end of line-2 during 170 ms. The response of example power system to this fault is shown in Figures 11(a) and (b). As seen in this figure, the performance of the proposed controller is still satisfactory. Numerical results of the ITAE performance index for two controllers are drawn in Figure 11(c).

Case 3

Now, the second power system displayed in Figure 1(b) is employed to show the effectiveness of the proposed approach. It is assumed that a 100-ms three-phase to ground is occurred at the end of line-2. Figures 12(a) and (b) show the inter-area oscillations and ITAE index, respectively. It is clear that the proposed approach is still effective in damping oscillations.

A comparative study

In order to demonstrate the successful performance of the proposed controller in damping oscillations, the results obtained from the proposed approach are compared with the results obtained from a neural network that was proposed in Farahani (2013). In Farahani (2013), a wavelet neural network (WNN) has been proposed to control the power system oscillations. Figures 13(a) and (b) display the inter-area and local modes of oscillations for a 200-ms two-phase to ground fault applied to the end of line-1. Moreover, numerical results of ITAE index are given in Figure 13(c). As can be seen from these results, the proposed approach provides a better dynamical situation for the power system such that both inter-area and local modes of oscillations are damped out faster. The advantages of the proposed method over some other ones are as follows:

The proposed approach combines abilities of FNNs and PIDs to control complicated non-linear systems.

In the proposed approach, a fuzzy PID controller with new structure is used as a supplementary controller.

Conclusion

This paper has successfully shown the application of a new intelligent controller to damp the power system oscillations in power systems. In this controller, a FNN-based non-linear PID controller with a simple structure is trained in the online mode. In order to accelerate the convergence speed of the proposed controller, the adaptive learning rates obtained from the Lyapunov method are employed. This controller does not need any identifier to identify the dynamic of controller systems, owing to the learning ability of the proposed controller. To enhance the power system oscillations, a fuzzy-PSS is used in excitation system of the generator. The performance of proposed controller is verified by a two-area, four-machine power system. A PS optimization algorithm is used to tune the fuzzy-PSS. All the simulation results show the satisfactory performance of proposed approach in damping the power system oscillations. Features such as adaptive performance, online tuning of parameters, lack of identifier, adaptive learning rates and simple structure can be stated as benefits of proposed approach, whereas the hardware implementation is the main weakness of the proposed controller.

Footnotes

Appendixes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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