Concordant controllers based on FACTS and FPSS for solving wide-area in multi-machine power system

Abstract

In this paper, a new controller based on combination of Fuzzy Power System Stabilizer (FPSS) and Flexible Ac Transmission Systems (FACTS) proposed in multi machine power system environment. In this model, the power system oscillation damping has been analyzed by considering the wide area control method. A fuzzy controller is a nonlinear controller and it is not so sensitive to system topology, parameter and operating condition changes as the conventional ones. According to wide area power system complexity and variation of the loads and network conditions, FPSS has been considered in this paper. Where, this structure is based on fewer fuzzy rules and less computational burden. Furthermore, the proposed FACTS controller has been improved by Vector Evaluated Honey Bee Mating Optimization (VEHBMO) as an optimization problem. Effectiveness of the proposed method has been applied over two case studies of single-machine infinite-bus (SMIB) and ten machine 39 bus New England power system. Obtained results demonstrate the superiority of proposed strategy.

Keywords

FACTS FPSS VEHBMO wide area control multi machine power system

1 Introduction

Power System Stabilizers (PSS) in synchronous generators are used to add damping to the rotor oscillations by modulating the excitation field voltage in order to produce an electrical torque in phase with the rotor oscillation. These generators exhibit three modes of rotor oscillations: Inter-machine, Plant-network and Inter-area [1]. Also, by increasing the complexity of power grid interconnections, power systems may become increasingly vulnerable to low frequency oscillations, especially inter-area oscillations. Actually, damping effectiveness of local measurement based controls is limited because local measurements have limited modal observability [1, 2]. In such situations, the use of wide-area signals in which the desired oscillatory modes may be readily observable could be more beneficial in damping inter-area oscillations of a large interconnected system. Recently, researchers interest to find technical methods to use the wide-area signals in power system stability enhancement. Where, according to significant investment in the U.S. in deploying synchrophasor measurement technology these techniques have been highlighted more and more.

These oscillations have also resulted into instability and blackouts in the power system. The traditional approach to damp out these oscillations is through Conventional Power System Stabilizer (CPSS), forming part of the generator excitation system. Besides PSS, FACTS devices are also applied to enhance system stability. Particularly, in multi-machine systems, using only conventional PSS may not provide sufficient damping for inter-area oscillations. In these cases, FACTS controllers are effective solutions. These controllers usually employ local signals as inputs and may not always be effective to damp out the interarea modes of oscillations. But CPSSs cannot satisfy the power system stability enhancement. For this purpose intelligent methods have been applied for tuning the parameters of PSSs [3 –5].

Recently, FPSS has been proposed for stability problem in multi-machine power systems. In this paper, Takagi Sugeno (T-S) based fuzzy controller has been proposed as FPSS controller. Actually, there are two types of methods to overcome the difficulty of classic T-S controller. One method is to exploit a good tradeoff between the conservatism and the computational burden by reducing unimportant decision variables [6]. However, the obtained controllers are still with a number of control rules, which may be unfavorable for implementation. The other method is that the original nonlinear model is first simplified as much as possible. Then, a fuzzy model with fewer fuzzy rules is constructed based on the simplified nonlinear model by using a fuzzy local approximation technique [7]. However, the designed control laws based on the fuzzy model may not guarantee the stability of the original nonlinear system. In this paper, T-S fuzzy controller based on local nonlinear models is exploited to describe the considered nonlinear systems. A new fuzzy control scheme with local nonlinear feedbacks is proposed, and the corresponding control synthesis conditions are developed in terms of solutions to a set of LMIs. In contrast to the existing methods for fuzzy control synthesis, the new proposed control design method is based on fewer fuzzy rules and less computational burden. Moreover, the local nonlinear feedback laws in the new fuzzy controllers are also helpful for achieving good control effects [8 –10].

The [11], investigates the problem of Hankel-norm output feedback controller design for a class of T–S fuzzy stochastic systems. The full-order output feedback controller design technique with the Hankel-norm performance is proposed by the fuzzy-basis-dependent Lyapunov function approach and the conversion on the Hankel-norm controller parameters.

Fuzzy controller for solving the nonlinear system based on mamdani model has been introduced in [12]. This model is based on simplified research procedure, according to the method of harmonic balance with simulation results.

In [13], focuses on analyzing a new model transformation of discrete-time T–S fuzzy systemswith time-varying delays is introduced and applying it to dynamic output feedback (DOF) controller design.

By considering the coordination of FPSS and FACTS we will evaluate the effects of these devices in power system stability by considering the Wide-Area Coordinating (WAC). Accordingly, to enhance stability margins and control oscillatory modes by adding supplementary damping devices we will use the global remote signals have been suggested since the introduction of the Phasor Measurement Unit (PMU) technology. Remote signals transmit knowledge related to the overall network dynamics, in contrast with local signals, which often lack good observability of some significant inter-area modes [14]. Even though WAC controllers involve additional communication equipment, their implementation may turn out to be more cost effective than installing new control devices if the additional operating flexibility achieved in critical power systems compensates for the equipment cost [15].

The main contribution of this paper can be summarized as follows:

A new structure of fuzzy controller has been proposed to detect the wide-area control signal based on T-S model. Also, this controller has been improved by a meta-heuristic algorithm to partition the fuzzy space of the given input–output data. This optimization increased the robustness of fuzzy controller.

Also, the global signal of the centralized controller is employed in wide area control scheme to damp out the inter-area mode as well as local mode of oscillations. In this model, the parameters of FACTS controller have been evaluated by proposed optimization algorithm.

The remaining parts of the paper are organized as follows. In the second section, the structure power system has been presented. Section three presents the fuzzy and FACTS controllers. Section four introduce the proposed VEHBMO algorithm. Section five presents the simulation results and discussion. Section six concludes the paper.

2 Problem statement

2.1 Power system modeling

For testing the ability of proposed control strategy, two test cases have been considered in this paper. The first one is the single machine connected to infinite bus and the second one is the multi-machine 39 bus power system [16, 17]. Actually, the proposed power system consists of four generators and the electrical and mechanical part of ith generator is modeled as follow: ${\dot{δ}}_{i} (t) = w_{i} (t) - w_{0}$ (1) ${\dot{w}}_{i} (t) = \frac{w_{0}}{M_{i}} (P_{mi} - P_{ei} (t)) - \frac{D_{i}}{M_{i}} (w_{i} (t) - w_{0})$ (2) ${\dot{E}}_{qi} (t) = \frac{1}{T_{doi}^{'}} (E_{fi} (t) - E_{qi} (t))$ (3) $E_{qi} (t) = E_{qi}^{'} (t) + (x_{di} - x_{di}^{'}) . I_{di} (t)$ (4) $V_{ti} (t) = \frac{1}{x_{dsi}} [x_{si}^{2} E_{qi}^{2} + V_{si}^{2} x_{qi}^{2} + 2 x_{si} x_{di} x_{dsi} P_{ei} cot δ_{i}]^{1 / 2}$ (5) $P_{ei} (t) = \sum_{j = 1}^{n} E_{qi}^{'} (t) E_{qj}^{'} (t) (B_{ij} sin (δ_{ij} (t))$ (6) $\begin{matrix} + G_{ij} cos (δ_{ij} (t))) \\ Q_{ei} (t) = \sum_{j = 1}^{n} E_{qi}^{'} (t) E_{qj}^{'} (t) (G_{ij} sin (δ_{ij} (t)) \end{matrix}$ (7) $\begin{matrix} + B_{ij} cos (δ_{ij} (t))) \\ I_{di} (t) = \sum_{j = 1}^{n} E_{qj}^{'} (t) (G_{ij} sin (δ_{ij} (t)) - B_{ij} cos (δ_{ij} (t))) \end{matrix}$ (8) $\begin{matrix} = \frac{Q_{ei} (t)}{E_{qi}^{'} (t)} \\ I_{qi} (t) = \sum_{j = 1}^{n} E_{qj}^{'} (t) (B_{ij} sin (δ_{ij} (t)) - G_{ij} cos (δ_{ij} (t))) \end{matrix}$ (9) $\begin{matrix} = \frac{P_{ei} (t)}{E_{qi}^{'} (t)} \\ E_{qi} (t) = x_{adi} I_{fi} (t) \end{matrix}$ (10)

Also, it can be presented after mathematical transformers where, ΔPe, Δw and ΔVt are quantities; $\begin{matrix} [\begin{matrix} Δ \dot{P_{ei}} \\ Δ \dot{w_{i}} \\ Δ \dot{V_{ti}} \end{matrix}] & = [\begin{matrix} \frac{S_{Ei} - S_{Vi}}{T_{di}^{'} S_{Vi}} & S_{Ei}^{'} & - \frac{R_{Vi} S_{Ei}}{T_{di}^{'} S_{Vi}} \\ - \frac{w_{0 i}}{H_{i}} & - \frac{D_{i}}{H_{i}} & 0 \\ \frac{S_{Ei} - S_{Vi}}{T_{di}^{'} R_{Vi} S_{Vi}} & \frac{S_{Ei}^{'} - S_{Vi}}{R_{Vi}} & - \frac{S_{Ei}}{T_{di}^{'} S_{Vi}} \end{matrix}] [\begin{matrix} Δ P_{ei} \\ Δ w_{i} \\ Δ V_{ti} \end{matrix}] \\ + [\begin{matrix} \frac{R_{Ei}^{'}}{T_{d 0 i}} \\ 0 \\ \frac{R_{Ei}^{'}}{T_{d 0 i} R_{Vi}} \end{matrix}] Δ E_{fdi} \end{matrix}$ (11)

Where,

ΔP_ei is the state deviation in generator electromagnetic power for the ith subsystem

Δw_i is the state deviation in rotor angular velocity for the ith subsystem,

ΔV_ti is the state deviation in the terminal voltage of the generator for the ith subsystem $\begin{matrix} S_{Ei} = \frac{E_{qi} U_{si}}{X_{d \sum i}} cos δ_{i} + U_{si}^{2} \frac{X_{d \sum i} - X_{q \sum i}}{X_{d \sum i} X_{q \sum i}} cos 2 δ_{i} \\ S_{Ei}^{'} = \frac{E_{qi}^{'} U_{si}}{X_{d \sum}^{i^{'}}} cos δ_{i} + U_{si}^{2} \frac{X_{d \sum}^{i^{'}} - X_{q \sum i}}{X_{d \sum}^{i^{'}} X_{q \sum i}} cos 2 δ_{i} \end{matrix}$ (12) $\begin{matrix} R_{Ei} = \frac{U_{si}}{X_{d \sum i}} sin δ_{i} \\ R_{Ei}^{'} = \frac{U_{si}}{X_{d \sum}^{i^{'}}} sin δ_{i} \\ S_{Vi} = S_{Ei} - R_{Vi} \frac{\partial U_{ti}}{\partial δ_{i}}, R_{Vi} = S_{Ei} / \frac{\partial V_{ti}}{\partial E_{qi}} \end{matrix}$ (13)

The linearized rotor motion equation for synchronous generator can be described as: $T_{J} \frac{d Δ w}{dt} = Δ T_{m} - Δ T_{e} - Δ T_{D}$ (14)

Where,

ΔT_m is the mechanical input torque

ΔT_e is the electromagnetic torque and ΔTe = K1Δδ+ K2Δδ,

By neglecting the K2ΔE’q the formulation can be described as; ΔTe = K1Δδ+ ΔTD

D is the natural damping constant

Accordingly the above equation after Laplace transformer and Δw = sΔδ/w0 can be described as; $T_{J} \frac{s^{2} Δ δ}{w_{0}} = - K_{1} Δ δ - D \frac{s Δ δ}{w_{0}}$ (15)

Which can be described as; $T_{J} s^{2} + Ds + w_{0} K_{1} = 0$ (16)

OR $s^{2} + 2 ζ_{n} w_{n} s + w_{n}^{2} = 0$ (17)

Consequently, we can achieve the following equation from above equations; $\begin{matrix} ζ_{n} = D / 2 \sqrt{w_{0} T_{J} K_{1}} \\ w_{n} = \sqrt{w_{0} K_{1} / T_{J}} \end{matrix}$ (18)

Where,

ξ_n is the damping factor

w_n is the un-damped mechanical oscillation frequency

Because of the complexity and multi-variable conditions of the power system, conventional control methods may not give satisfactory solutions. Also, Fig. 1, presents the power system modelling for ith generator. Also, the parameters definition is presented in APPENDIX A.

3 Structure of proposed controller

3.1 Fuzzy power system stabilizer

Regarding to some complexity in wide area power system and variation of the loads and network conditions, it can be considered as multi-input and multi-output System. The T-S fuzzy controller is the best choice of this application. So, we can obtain the following T-S fuzzy model as: $\begin{matrix} IF v_{1} (t) is Γ_{i 1} and v_{2} (t) is Γ_{i 2}, \dots, v_{p} (t) is Γ_{ip} \\ THEN \dot{x} (t) = A_{i} x (t) + B_{1 i} ω (t) + B_{2 i} u (t) + G_{i} φ (t) \\ z (t) = C_{1 i} x (t) + D_{1 i} ω (t) + D_{2 i} u (t) + G_{zi} φ (t) \end{matrix}$ where i = 1, …, r. r is the number of IF–THEN rules, v (t) = [v₁ (t) v₂ (t) … v_p (t)] T ∈ Rp × 1 are the premise variables, and Γ_ij are the fuzzy sets [18]. By using the fuzzy inference method with a singleton fuzzifier and product inference and center average defuzzifiers, the final T-S fuzzy model is obtained as follows: $\dot{x} (t) = \frac{\sum_{i = 1}^{r} W_{i} (v (t)) (A_{i} x (t) + B_{1 i} ω (t) + B_{2 i} u (t) + G_{i} φ (t))}{\sum_{i = 1}^{r} W_{i} (v (t))}$ $z (t) = \frac{\sum_{i = 1}^{r} W_{i} (v (t)) (C_{1 i} x (t) + D_{1 i} ω (t) + D_{2 i} u (t) + G_{zi} φ (t))}{\sum_{i = 1}^{r} W_{i} (v (t))}$

Where $W_{i} (v (t)) = Π_{j = 1}^{p} η_{ij} (v_{j} (t)) . η_{ij} (v_{j} (t))$ is the grade of membership of v_j (t) in Γ_ij, where it is assumed that $\sum_{i = 1}^{r} W_{i} (v (t)) > 0, W_{i} (v (t)) ⩾ 0,$ i = 1, 2, …, r. Denote $\begin{matrix} α_{i} (v (t)) = (w_{i} (v (t))) / \end{matrix}$ $\begin{matrix} ((\sum_{i = 1}^{r} W_{i} (v (t)))); \end{matrix}$ then 0 ≤ α_i (v (t)) ≤1 and $\sum_{i = 1}^{r} α_{i} (v (t)) = 1$ .

Where α_i (v (t)) is said to be normalized membership functions.

Let α (v (t)) = [α₁ (v (t)) , α₂ (v (t)) , …, α_r (v (t))] ^T, and denote

α (v (t)) as α for a brief description. So, we can write; $\dot{x} (t) = A (α) x (t) + B_{1} (α) ω (t) + B_{2} (α) u (t) + G (α) φ (t)$ $z (t) = C_{1} (α) x (t) + D_{1} (α) ω (t) + D_{2} (α) u (t) + G_{z} (α) φ (t)$

With $\begin{matrix} A (α) = \sum_{i = 1}^{r} α_{i} (t) A_{i} B_{1} (α) = \sum_{i = 1}^{r} α_{i} (t) B_{1 i} \\ B_{2} (α) = \sum_{i = 1}^{r} α_{i} (t) B_{2 i} G (α) = \sum_{i = 1}^{r} α_{i} (t) G_{i} \\ C_{1} (α) = \sum_{i = 1}^{r} α_{i} (t) C_{1 i} D_{1} (α) = \sum_{i = 1}^{r} α_{i} (t) D_{1 i} \\ D_{2} (α) = \sum_{i = 1}^{r} α_{i} (t) D_{2 i} G_{z} (α) = \sum_{i = 1}^{r} α_{i} (t) G_{zi} \end{matrix}$

In fact, A (α) x (t) , B₁ (α) , B₂ (α) , G (α) , C₁ (α) x (t) , D₁ (α) , D₂ (α), and G_z (α) in fuzzy model are the new descriptions of $\bar{f} a (x (t))$ , h (x (t)), g (x (t)), fb (x (t)), $\bar{f} za (x (t))$ , hz (x (t)), gz (x (t)), and fzb (x (t)) by fuzzy membership functions, respectively. The following lemma will be useful in the sequel.

Lemma 2: If the following conditions hold: $\begin{matrix} M_{ii} < 0, 1 \leq i \leq r \\ \frac{1}{r - 1} M_{ii} + \frac{1}{2} (M_{ij} + M_{ji}) < 0, 1 \leq i \neq j \leq r \end{matrix}$

Then we can write; $\begin{matrix} M_{ii} < 0, 1 \leq i \leq r \\ \sum_{i = 1}^{r} \sum_{j = 1}^{r} α_{i} α_{j} M_{ii} < 0 \\ where, α_{i}, 1 \leq i \leq r, satisfy 0 \leq α_{i} \leq 1, \sum_{= 1}^{r} α_{i} = 1 \end{matrix}$

3.2 Fuzzy PSS control strategy

In this paper, the H∞ control strategy and guaranteed cost control have been considered. Furthermore, the local nonlinear feedbacks control and the corresponding control synthesis conditions is solved by Linear Matrix Inequalities (LMIs) method. As this section is not the main focus of this paper, its details is presented in [19].

3.3 Flexible Ac transmission systems controller

The structure of proposed FACTS controller has been presented in Fig. 2. Also, local signals of FACTS devices are applied for the damping control. In this paper, the active power flow through the series FACTS device line P is employed [20, 21].

The parameters of proposed FACTS controller have been optimized by IHBMO in this paper. Many input signals have been proposed for the FACTS to damp the system oscillations. Signals, which carry invaluable information about the inter-area mode, can be considered as the input signals. Since FACTS controllers are located in the transmission systems, local input signals are always preferred. In this paper, the suitable WAC signals are selected for FACTS controllers. Then, through the designed centralized controller embedded in the WAC center, the control output signals are sent to FACTS device and FPSS simultaneously. Also, for implementation of the local control strategy using wide area information, the local controller, the WAC output can be adopted by the related FACTS local controllers to provide stabilizing control for the stable and secure operation of power systems. Beside of this process we proposed a new algorithm for optimizing the parameters of FACTS devise. In this process, the parameters of T1, T2 T3 and T4 are selected for finding the good abilities of FACTS devise. Obtained results of this controller is presented in Table 1. The upper limit of these parameters are considered 3 and the lower bound is 0.01. For this purpose we introduce this algorithm in the nest section.

4 Vector evaluated honey bee mating optimization

The collective behavior of decentralized, self-organized systems, natural or artificial is Swarm intelligence. Honey bee mating optimization (HBMO) is a new powerful evolutionary algorithm for solving single or multi-objective optimization problems with discrete parameters [22]. Recently, HBMO is employed for solving multi-objective optimization problems also this technique achieved desired results in many single objective optimization problems [23].

4.1 Review classic HBMO

The honey bee is a social insect that can survive only as a member of a community, or colony. This means that they tend to live in colonies where all the individuals are of the same family, sometimes the offspring of a queen [23, 24]. A colony of honey bees concludes a queen, several hundred drones, 30,000 to 80,000 workers and broods. Each bee undertakes sequences of actions which unfold according to genetic, ecological and social condition of the colony [24]. The queen is the most important member of the hive because she is the one that keeps the hive going by producing new queen and worker bees. Drones’ role is the mating with queen. Workers utilize some heuristic mechanisms such as crossover. Although the honey bee queen thought that she is the most important member of her colony, the workers sometimes determine that their colony needs a new queen. This happen due to space constrictions, weak performance associated with age and the unexpected death of the queen [22]. A mating flight starts with a dance performed by the queen who then starts a mating flight during which the drones follow the queen and mate with her in the air. When a virgin queen flies to a site where thousands of male honey bees may be waiting, it mates with several males in flight. A male drone will mount the queen and the mating flight may be considered as a set of transitions in a state-space (the environment). The queen’s size of spermatheca number equals to the maximum number of mating of the queen in a single mating flight is determined. A drone mates with a queen probabilistically using an annealing function as [25]: $prob (Q, D) = e^{\frac{- Δ (f)}{S (t)}}$

Where, Prob (Q, D) is the probability of adding the sperm of drone D to the spermatheca of queen Q (that is, the probability of a successful mating); Δ (f) is the absolute difference between the fitness of D (i.e., f (D)) and the fitness of Q (i.e., f (Q)); and S (t) is the speed of the queen at time t. It seems that this function acts as an annealing objective function, where the probability of mating is high when both the queen is still in the start of her mating–flight and therefore her speed is high, or when the fitness of the drone is as good as the queen’s. After each transition in space, the speed of queen, S (t), and it’s energy, E (t), decay using the following equations: $S (t + 1) = α \times S (t)$ $E (t + 1) = E (t) - γ$

Where, α (t) is speed reduction factor and γ is the amount of energy reduction after each transition (α, γ ∈ [0, 1]). In general, the whole process of the HBMO algorithm as shown in Fig. 3 can be summarized at the five main steps as follows [23]:

The algorithm starts with the mating–flight, where a queen (best solution) selects drones probabilistically to form the spermatheca (list of drones). A drone is then selected from the list at random for the creation of broods.

Creation of new broods by crossoverring the drones’ genotypes with the queen’s.

Use of workers (heuristics) to conduct local search on broods (trial solutions). It can transfer the genes of drones and the queen to the jth individual based on Equation (4).

child = Drone + β (Queen - Drone)

Where β is the decreasing factor (β ∈ [0, 1]).

Adaptation of workers’ fitness based on the amount of improvement achieved on broods. The population of broods is improved by applying the mutation operators as follows:

\begin{matrix} {Brood}_{i}^{k} = {Brood}_{i}^{k} \pm (δ + ɛ) {Brood}_{i}^{k} \\ δ \in [0, 1], 0 < ɛ < 1 \end{matrix}

Replacement of weaker queens by fitter broods.

If the new brood is better than the current queen, it will take the position of the queen. If the brood fails to replace the queen, for improving the algorithm in the next mating flight of the queen this brood will be one of the drones. Furthermore, if the termination criterion is satisfied then finish the algorithm; else discard the all previous trial solutions (brood set). Then generate new drones set and go to stage 2.

4.2 VEHBMO

Honey bees live in a colony and they forage and store honey in their constructed colony. HBMO is a population based algorithm which it exploits a population of individuals to probe promising regions of the search space. On the other hand, in this method the objectives are optimized separately by sub-populations that communicate among themselves. This concept is used in Vector Evaluated PSO (VEPSO) [24]. In VEPSO, which is inspired by VEGA, each swarm is exclusively evaluated with one of the objective functions; however information coming from other swarm(s) is used to influence a swarm’s motion in the search space. Information communicated to each of the other swarms contains the global best particle found by each of the swarms. The authors argue that the exchange of this information among the swarms can lead to Pareto-optimal solutions. In this technique authors assume that the search behavior of a bee is affected by neighboring drones. The procedure of exchanging information among drones can be clearly viewed as a migration scheme in a parallel computation framework. This selecting the scheme is according to the “ring” migration topology [22]. The Vector Evaluated Honey Bee Mating Optimization assumes that D drones, (S₁, S₂, …, S_D) of size N aim to optimize simultaneously M-objective functions. Each colony is evaluated according to one of the objective functions. The basic concept of VEHBMO algorithm is illustrated in Fig. 4. As an example, for the case of two objective functions, D₁ and D₂ are population of drone 1 (colony 1) and drone 2, respectively, while queen1 and queen2 are the best answer (queen) for colony 1 and colony 2, respectively. The objective functions f₁ and f₂ calculates with D₁ and D₂, respectively. Each drone is exclusively calculated according to the respective objective function. The queen of the second drone (D₂) is used for the calculation of the new queen of the first drone’s (D₁) colony and accordingly, queen of the first drone (X₁) is used for the calculation of the new queen of the second drone (D₂).

4.3 Strength Pareto Optimal (SPO) solutions

If we plot the objective values f₁ and f₂ of these optimal solutions against each other in one plot. A multi-objective optimization algorithm tries to approximate these solutions but uses a different approach to obtain these solutions. The supposed algorithm sorts the population based on non-dominated fronts. The first front found is ranked the best/highest and the last one the lowest. This ranking is used in the mating flight selection process.

In addition to, for assure diversity in a population (honey bee) employed crowding distance measure. The main steps of the SPO algorithm are explained in more detail as follows:

4.4 Non-Dominated sort

This set of non-dominated solutions is called a Pareto front. A multi-objective algorithm selects and improves solutions based on this domination principle to approximate the real trade-off curve. Non-domination occurs when a candidate cannot be improved any further in one objective while degrading in another objective. Figure 5 explains this strategy in more detail. The figure shows that crowding distance for solution y (x₄) is calculated relative to the solutions from the same front which are all colored blue. The distance is the sum of the length and width of a cubical that can be drawn through these two closest neighbors.

4.5 Crowding distance

The population density around a particular solution i is estimated by the average distance of the two solutions at either side of i with each of the objectives. Crowding distance is assigned front wise and comparing the crowding distance between two individuals in different front is meaningless.

In this study, in order to obtain good performance, number of queens, drones, broods, workers, the queen’s spermatheca size, α, β, γ and ɛ is chosen as number of dimension 40, 25, 15, 35, 0.94, 0.98, 0.97 and 0.8, respectively.

5 Numerical results and discussion

5.1 One machine infinite bus

For stability assessment of power system adequate mathematical models describing the system are needed. The system behaviour following such a disturbance is critically dependent upon the magnitude, nature and the location of fault and to a certain extent on the system operating conditions. A schematic diagram for the proposed first test case has been presented in Fig. 6.

The disturbances are given at t = 1 sec. System responses in the form of slip (S_m) are plotted. The following types of disturbances have been considered [26].

Scenario 1: A step change of 0.1 pu in the input mechanical torque.

Scenario 2: A three phase-to-ground fault for 100 ms at the generator terminal.

The convergence trend of proposed algorithm is presented in Fig. 7. Also, the achieved results for FACTS controller has been presented in Table 1.

In the following, Fig. 8 presents the system response at the lagging power factor operating conditions with weak transmission system for scenario 1. It is clear that the proposed method could provide better stability condition.

In this test case, the proposed model has been compared with other three algorithm which have been resented in [26]. The mentioned algorithms are Particle Swarm Optimization (PSO) and two other version from this algorithm which named PSO-TVAC and PSO-TVIW [27]. Also, Fig. 9 presents the system response in scenario 2 with inertia H’ = H/4.

Furthermore, to demonstrate the robustness performance of the proposed method, in the some operating condition for scenario 1, the Eigen values of the system with comparison of these methods has been presented in Table 2. All of the presented numerical results have been quoted directly from [26]. Where, by considering to this table it can be claim that the proposed method could evaluate better results in all of the Min, Mean and Max values. Also, the iteration number of proposed method is lower than other methods which demonstrate the high speed of this algorithm with better standard deviation value. It is clear to see that the eignvalues of the system with proposed model are farther than the imaginary axis and the system stability margin is more than other methods.

For more information about the proposed algorithm the computational results which are used in this paper through several runs of proposed technique. The computational results are shown in Table 3.

5.2 10 machine- 39 bus new england power system

A 10-machine 39-bus power system is considered as a test case for this paper which is presented in Fig. 10. To assess the effectiveness and robustness of the proposed method over a wide range of loading conditions, different operation conditions are considered. Details of the system data and operating condition are given in Ref. [27, 28]. The proposed power system is divided to four area as shown in this figure. The G₅, G₇ and G₉ are indicated for the place of proposed FPSS and FACTS controllers. In this test the critical mode of system has been presented in Table 4.

Also, damping contribution of FPSS and FACTS devices using the available line current as the control input for inter-area oscillation modes has been presented in Fig. 11. In this figure, (a) presents the changing the scenario 1 and (b) presents the changing in scenario 2. These scenarios have been presented in the following for TAFM power system.

Figure 11 shows the damping contribution of the FPSS and FACTS devices, which use different line current as the control input for each concerned inter-area mode. From Fig. 11(a) and (b), it can be found that when the FPSS wide-area controller selects the specified WAC signals, it can achieve better damping performance for mode 1 and 3. Thus, the current in Line 82 and Line 76 is selected for the FPSS wide-area controller to damp mode 1 and 3. Similarly, from Fig. 11(c) and (d), it can be found that the FACTS WAC, which selects the specified WAC signals, can achieve better damping performance for mode 2 and 4 than the FPSS. Thus, the current in Line 86 and Line 48 is selected as the WAC input of the FACTS wide-area controller to damp modes 2 and 4.

It is worth noting that although the FACTS is the candidate for the WAC network, from Fig. 11, it can be found that compared to the FACTS and the FPSS, the FPSS cannot provide the effective damping for the concerned inter-area modes. In addition, from the aforementioned analysis, it can be seen that the FACTS and the FPSS are sufficient to damp these four inter-area modes. Therefore, combination of these two devices could satisfy the stability problem in power system. Also, in meta-heuristics algorithms the mature convergence can be occurred. To tackle of the mentioned problem, this algorithm has been run 10 times over proposed problem. Consequently, the best value of the consequence parameters of 10 runs are presented in Fig. 12.

To demonstrate the effectiveness of the proposed strategy, this method has been tested over three case studies through the comparison of without wide area controller and [8]. The simulation results have been tested in different load condition and faults where two scenarios have been considered to find the capability of proposed wide area control strategy. The achieved results demonstrate the validity of proposed method in comparison of other methods. For this purpose, at first we describe different scenarios in literature as;

Scenario One: For the second case study, same scenarios have been applied for this power system. Accordingly, for the first scenario, performance of the proposed strategy under transient conditions is verified by applying a 3-cycle three-phase fault at t = 1 sec, on bus 25 at the end of line 25-26. The fault is cleared by permanent tripping the faulted line. For obtained results from simulation, Fig. 13, is presented to show the speed deviations of the generators G5, G7, G9 and G11 under heavy load condition are shown in Fig. 13. The information of load for this power system is presented in [29]. By considering of the presented figure we can say that the proposed strategy of wide-area control provides good stability for this power system. Where, CPSS has swinging behavior however, the oscillation of proposed strategy is a little better than WATSFDC with good convergence and stability characteristics.

Scenario Two: in this scenario same as the previous case the loading power factor is considered through the decreasing of load with a 0.2 p.u. step in mechanical torque at t = 1.0. The mentioned scenario is applied at bus-4 where by decreasing load the proposed strategy of wide-area control has been challenged through this condition.

Accordingly, Fig. 14 presents the variation of three controllers by proofing this fact that the proposed strategy is superior. Also, to demonstrate performance robustness of the proposed method, the integral of the time multiplied absolute value of the error (ITAE) and figure of demerit (FD) based on the system performance characteristics are being used as objective functions where, overshoot (OS), undershoot (US) and settling time of frequency deviation is considered for evaluation of this criteria. The mentioned objectives are presented in Equations 30 and 31, respectively.

These Tables of 5 and 6, presents the FD and ITAE numerical results, respectively. $ITAE = \int_{0}^{t_{sim}} | \sum_{i = 1}^{N_{g}} ω_{i} | . dt$ (19) $\begin{matrix} FD & = \sum_{i = 1}^{N_{g}} ((| max (ω_{i}) \times 5000 |)^{2} \\ + (| min (ω_{i}) \times 10000 |)^{2} + 0.1 \times T_{sGi}^{2}) \end{matrix}$

For the last part, tables of numerical results are presented through three load condition of heavy, light and nominal. By considering to this information the superiority of proposed strategy is obvious. The behavior of proposed strategy is better in all of generators in comparison with [29]. However, this method could achieve better result in G₇.

However, by considering to the numerical results analysis in Tables 5 and 6, it can be said that the proposed strategy is better in overall.

By considering the proposed numerical results and generators output in Figs. 8, 9, 13 and 15, it can be said that the proposed method could provide good results in comparison with other methods. Where it can be claimed that the overshoot, undershoot and settling time of proposed method is smaller than other compared methods. Moreover, the proposed method could provide better results in nominal, heavy and light load conditions in different scenarios through the comparison with other method.

Furthermore, according to the small oscillation in generators output and good numerical results based on well-known criteria in power system stability, the superiority of proposed method has been proofed.

6 Conclusion

In this paper coordination of FPSS and FACTS have been considered in multi-machine power system. This strategy has been categorized in two main stages. At first stage the proposed new fuzzy control is presented to detect the wide-area control signal. And the second stage an intelligent algorithm is applied over fuzzy controller to partition the fuzzy space of the given input–output data. Through introducing the suitable WAC signals, the WAC network can be constructed with the advanced control ability for enhancing the overall stability of large-scale interconnected systems. Where, after determining the damping contribution of the FPSS and FACTS devices, which use different line current as the control input for each concerned inter-area mode, the stability of proposed power system satisfied. By the mentioned strategy, the application and robustness of fuzzy controller has been increased. Also, the global signal of the centralized controller is employed in wide area control scheme to damp out the inter-area mode as well as local mode of oscillations. Moreover, the parameters of FACTS controller have been evaluated by VEHBMO. Effectiveness of the proposed method has been applied over two case studies of single-machine infinite-bus (SMIB) and ten machine New England power system. Obtained results demonstrate the superiority of proposed strategy.

Also, by implementing a new version of TS controller based on Hankel-norm performance, discrete time model and DOF design, we can propose a new and strong controller for solving the mentioned problem. This section can be proposed as a future work.

Footnotes

Appendix

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