Enhancement of small signal stability of power system using UPFC based damping controller with novel optimized fuzzy PID controller

Abstract

In this work a UPFC based novel optimal fuzzy PID controller is proposed whose gains are optimized by PSO-GWO algorithm considering integral of time weighted absolute error. The proposed controller has been employed to damp low frequency oscillations in power system considering wide range of operating conditions like change in input mechanical power, line loading, change in line reactance with single and multi machine power system. The fuzzy optimized PID controller has been compared with normal optimized PID controller with same optimization techniques to justify its effectiveness. Also the supremacy of PSO-GWO technique has been compared with PSO and GWO techniques by optimizing the gains of fuzzy PID controller. The detailed eigen value analysis and system response prove that proposed fuzzy PID controller is much superior than PID controller and PSO-GWO optimized controller is also superior as compared to PSO and GWO optimized fuzzy PID controllers to damp low frequency oscillations in power system and there by enhancing small signal stability of power system.

Keywords

FACTS UPFC PSO-GWO fuzzy PID small signal stability

1 Introduction

The recent power system networks interconnected via tie lines are putting continuously more challenges on security, control operation and stability of power system. The stability of an extended power system network can be addressed in terms of small and large signal analysis. The large signal analysis is related to transient stability and small signal analysis is related to dynamic stability. In this work the stability problem is confined to small signal stability analysis pertaining to damping of electro mechanical oscillations in power system, which is an all time issue of power system. These power system oscillations are instigated by disturbances in the system including malfunctioning of controllers, which may consequently integrate and finally lead to loss of synchronism if not damped efficiently [1]. Power system stabilizer (PSS) is being adopted for a long years to damp these oscillations, it suffers from some demerits like wide voltage profile change, leading power factor operation etc. [2]. During last couple of years, introduction of FACTS devices has changed the operating and control scenario of power system. PSS based on FACTS devices have several merit points in contrast to conventional PSS [3, 4]. Out of different FACTS devices UPFC is more versatile and can inject unconstrained voltage to power system [5, 6]. The steady state model of power system has been reported in the literature earlier [7]. For small signal analysis of power system, the linear Heffron-Phillips transfer function model has been reported in [8]. But, a systematic approach to design the UPFC damping controller is presented in [9], where conventional technique has been implemented to tune the parameters of controller. As per researches conventional methods to design controller are sluggish, put more pressure on computers [11]. Therefore a suitable optimization technique can provide fast computation, put less burden on computer and can enhance the efficacy of controller. On line tuning of UPFC based controller parameters is a prime decisive work for a modern power network and this can be handled by an efficient optimization technique. Recently swarm and evolutionary methods are gaining much popularity for efficient optimization of controller parameters. These techniques are inspired by animal’s nature, natural phenomenon and concept of evolution. PSO has been reported in [11, 12] to tune the controller parameters based on UPFC. DE and GSA techniques have been reported in [13, 14] to tune controller parameters based on SSSC. Hybridization of GA and GSA techniques to tune UPFC controller parameters have been reported in [15]. The GWO is a powerful swarm algorithm recently published in [16], which has been influenced by the hunting process of Grey Wolfs. This technique is simple and can easily be converted to computer programming. GWO has been implemented to design UPFC based controller in [17] and design of PSS for wide area in [18]. The hybridization of DE with GWO technique to design UPFC based controller has been reported recently in [19]. Due to ease of implementation, simplicity and robustness in performance, PSO has been very popular technique since last couple of years [20]. But, it may trap in local optima when handling a heavy constrained optimization problem. Whereas GWO can balance between exploitation and exploration very effectively and can avoid trapping in local optima. So hybrid PSO-GWO algorithm is proposed in this work here for optimal setting of UPFC based fuzzy PID controller parameters to damp oscillations in power system.

For controller design, conventional PID controller has been very much popular since so many years due to its simplicity, easy tuning, low maintenance, less cost and highly effectiveness. But, for higher order complex system and with time delay, conventional PID controllers do not provide effective results. In these cases fuzzy logic controller (FLC) can improve closed loop performance of PID controller by online updation of controller parameters [21 –23]. Another important aspect of FLC based PID controller, which must be addressed carefully is the lack of proper mathematical formulation for choosing of suitable fuzzy parameters like input and output scaling factor, rule base, membership function etc. Generally the fuzzy parameters are selected by some empirical rules and which may not be the optimal values and improper choice scaling factor at input and output may degrade the performance of controller drastically. Hence based upon above facts, the input and output scaling factors of fuzzy PID controller are optimized by the powerful PSO-GWO algorithm to improve the efficacy of controller.

The major contribution of this work can be stated as: (i) UPFC based fuzzy PID controller is proposed to damp oscillations in power system (ii) PSO-GWO technique is proposed to optimize the controller parameters and scaling factors for controller (iii) Proposed controller has been compared with PSO and GWO optimized fuzzy controller to justify its supremacy. (iv) The controller is applied to SMIB system with different line loading and line reactance and applied to multimachine system with a distinct loading condition. (v) Detail eigen value analysis has been performed to validate the efficacy of proposed optimized fuzzy controller.

2 The single machine power system under study

In this case a single machine connected to infinite bus is considered as shown in Fig. 1. The initial condition of the system is given in appendix. The UPFC consists of two voltage source converters (VSC) is connected between generator and infinite bus. One VSC is series connected and another is shunt connected with the line. UPFC has four control actions which are m_B, δ_B, m_E and δ_E. Out of which m_B and m_E are modulation index for series and shunt VSC respectively. So on δ_B and δ_E are are phase angles of series and shunt VSC respectively.

Fig.1

The SMIB system under study.

2.1 Dynamic model of the system

2.1.1 Non linear model

By ignoring resistance of the line, non linear model of single machine power system can be represented by following equations [8]. $\dot{ω} = (\frac{P_{i} - P_{e} - D Δ ω}{M})$ (1) $\dot{δ} = ω_{0} (ω - 1)$ (2) ${\dot{E}}_{q} = (- E_{q} + E_{fd}) / T_{d 0}$ (3) ${\dot{E}}_{fd} = [- E_{fd} + K_{a} (V_{ref} - V_{t})] / T_{a}$ (4) $\begin{matrix} V_{dc} & = & \frac{3 m_{E}}{4 C_{dc}} (I_{Ed} sin δ_{E} + I_{Eq} cos δ_{E}) \\ + \frac{3 m_{B}}{4 C_{dc}} (I_{Ed} sin δ_{E} + I_{Eq} cos δ_{E}) \end{matrix}$ (5)

The active power between shunt VSC and series VSC can be balanced by Equation (6) as $Re (V_{B} I_{B}^{*} - V_{E} I_{E}^{*}) = 0$ (6)

2.1.2 Linear dynamic model

The linear model of power system can be obtained by linearizing the non linear model around the initial operating condition represented by following equations. $Δ \dot{δ} = ω_{0} Δ ω$ (7) $Δ \dot{ω} = (\frac{- Δ P_{e} - D Δ ω}{M})$ (8) $Δ {\dot{E}}_{q} = (- Δ E_{q} + Δ E_{fd}) / T_{d 0}$ (9) $Δ {\dot{E}}_{fd} = [- Δ E_{fd} + K_{a} (Δ V_{ref} - Δ V_{t})] / T_{a}$ (10) $\begin{matrix} Δ V d c & = & K_{7} Δ δ + K_{8} Δ E_{q}^{'} - K_{9} Δ V_{dc} + K_{cE} Δ m_{E} \\ + K_{c δ e} Δ δ_{E} + K_{cb} Δ m_{B} + K_{c δ b} Δ δ_{B} \end{matrix}$ (11)

Where

$\begin{array}{l} Δ P_{e} = & K_{1} Δ δ + K_{3} Δ E_{q}^{'} + K_{p d} Δ V_{d c} \\ + K_{p e} Δ m_{E} + K_{p δ E} Δ δ_{E} + K_{p i} Δ m_{B} \\ + K_{p δ b} Δ δ_{B} \end{array}$ (12)

$\begin{array}{l} Δ E_{d} = & k_{4} Δ δ + k_{3} Δ E_{q}^{'} + k_{q d} Δ V_{d c} + k_{q e} Δ m_{E} \\ + k_{q δ E} Δ δ_{E} + k_{q b} Δ m_{B} + k_{q δ B} Δ δ_{B} \end{array}$ (13) $\begin{array}{l} Δ V_{t} = & k_{5} Δ δ + k_{6} Δ E_{q}^{'} + k V_{d} Δ V_{d c} + k_{υ} Δ m_{E} \\ + k_{υ δ E} Δ δ_{E} + k_{υ b} Δ m_{B} + k_{υ δ B} Δ δ_{B} \end{array}$ (14)

3 Small signal model of single machine system

The Heffron Philips transfer function model of single machine power system is shown in Fig. 2. The ‘K’ constants of this model are calculated with reference to initial operating condition and system parameters [9] given in appendix. This model has been developed by using Equations (7 –11) and modification of basic Heffron Philips model with UPFC. In this model [ΔU] is the control vector in column form and [K_pu], [K_vu], [K_qu], [Kcu] vectors are in row form given by following expressions [ΔU] = [Δm_E Δδ_E Δm_B Δδ_B] ^T, [K_pu] = [K_pe K_pδe K_pb K_pδb], [K_vu] = [K_ve K_vδe K_vb K_vδb], [K_qu] = [K_qe K_qδe K_qb K_qδb], [K_cu] = [K_ce K_cδeK_cb K_cδb].

Fig.2

Modified Heffron-Phillips model with UPFC.

4 Structure of controller and objective function

4.1 Damping controller

The fuzzy PID controller structure is depicted in Fig. 3 [23]. Here two fuzzy PI and fuzzy PD controllers are combined together. In this controller there are four scaling factors out of which ks1, ks2 are at the input side and ks3, ks4 are at the output side. The input to fuzzy logic controller (FLC) are speed deviation Δω and derivative of speed deviation $Δ \overset{´}{ω}$ . The triangular membership function has been taken here, which has five linguistic variables denoted by bigger negative (Bn), smaller negative (Sn), zero (Z), smaller positive (Sp) and bigger positive (Bp) for both input and output. The triangular membership function provides very good and fast response with simple line segment putting less computational pressure. Mamdani fuzzy inference method is used here, as it is widely adopted and very much popular. Membership functions for speed deviation, derivative of speed deviation and FLC output is shown in Fig. 4. The two dimension rule base for speed deviation, derivative of speed deviation and FLC output is presented in Table 1 and rule surface is shown in Fig. 5. The centre of gravity method is used here for defuzzyfication to get FLC output.

Fig.3

Structure of fuzzy PID controller.

Fig.4

Membership functions for I/P and O/P of FLC.

Fig.5

The rules surface for controller.

Table 1

Rule base for speed deviation, derivative of speed deviation and FLC output

Δω	$Δ \overset{´}{ω}$
	Bn	Sn	Z	Sp	Bp
Bn	Bn	Bn	Sn	Sn	Z
Sn	Bn	Sn	Sn	Z	Sp
Z	Sn	Sn	Z	Sp	Sp
Sp	Sn	Z	Sp	Sp	Bp
Bp	Z	Sp	Sp	Bp	Bp

It has been reported earlier [6] that modulation index of series and phase angle of shunt converters are best control actions for UPFC based damping controller design to damp oscillations in a power network. So in this work m_B and δ_E control actions are selected to design damping controller.

4.2 Objective function

The stability function to damp oscillations is formulated to an objective function of ITAE type [10]. The main objective here is to minimize the error e(t), which is speed deviation resulting from disturbance in the power system. This function is given by J and for objective function 10 % rise in mechanical input power to generator is taken here. $J = \int_{0}^{tsim} t | e (t) | dt$ (15)

5 Overview of optimization techniques

5.1 PSO optimization technique

It is a simple and popular optimization technique. Here a number of particles are allowed to move in a searched space in multi dimensions [12]. During the course of searching, the velocity of each particle is updated by

$\begin{matrix} v_{i}^{k + 1} & = & {wv}_{i}^{k} + c_{1} {rand}_{1} (P_{i, pbest}^{k} - x_{i}^{k}) \\ + c_{2} {rand}_{2} (P_{i, gbest}^{k} - x_{i}^{k}) \end{matrix}$ (16)

Where, c₁ and c₂ are the acceleration coefficients, w is the inertial weight varying between 0.9 to 0.4, rand₁ and rand₂ are the two random variables in the range of [0,1].

The swarm position is updated by $x_{i}^{new} = x_{i} + v_{i}$ (17)

The best solution to further iteration is given by $x_{i}^{k + 1} = {\frac{x_{i, new} iff (x_{i, new}) ⩽ f (x_{i})}{x_{i} otherwise}}$ (18)

5.2 Grey wolf optimizer technique

This is a swarm intelligence type recent metaheuristic technique inspired by the attitude of Grey-Wolf as they hunt a prey [16]. Grey wolves remain in a pack and are put in a rank to implement the hunting process. For mathematical modeling of the hunting process, the most fittest solution is given by the position of α group of wolf followed by β, γ and δ groups. For initiating the process to hunt for prey, the wolves form a circle around the victim, which can be formulated as $\vec{D} = | \vec{C} . {\vec{X}}_{P} (t) - \vec{X} (t) |$ (19) $\vec{X} (t + 1) = | {\vec{X}}_{P} (t) - \vec{A} . \vec{D} |$ (20)

Where, ‘t’ shows the current iteration, A and C are coefficient vectors. Xp and X vectors locate victim position and Grey wolf positions respectively. A and C vectors are represented by Equations (21 and 22). $\vec{A} = 2 . \vec{a} . {\vec{r}}_{1} - \vec{a}$ (21) $\vec{C} = 2 . {\vec{r}}_{2}$ (22)

Where, r₁ and r₂ are random vectors in the range of [0 1]. During iterations the component ‘a’ decreases from 2 to 0.

The process of hunting can be formulated as $\begin{matrix} {\vec{D}}_{α} & = & | {\vec{C}}_{1} . {\vec{X}}_{α} - \vec{X} |, {\vec{D}}_{β} = | {\vec{C}}_{2} . {\vec{X}}_{β} - \vec{X} |, \\ {\vec{D}}_{δ} & = & | {\vec{C}}_{3} . {\vec{X}}_{δ} - \vec{X} | \end{matrix}$ (23) $\begin{matrix} {\vec{X}}_{1} & = & {\vec{X}}_{α} - {\vec{A}}_{1} . ({\vec{D}}_{α}), {\vec{X}}_{2} = {\vec{X}}_{β} - {\vec{A}}_{2} . ({\vec{D}}_{β}), \\ {\vec{X}}_{3} & = & {\vec{X}}_{δ} - {\vec{A}}_{3} . ({\vec{D}}_{δ}) \end{matrix}$ (24)

The best position of prey can be found out by taking average value of positions of α, β and δ wolves as $\vec{X} (t + 1) = \frac{{\vec{X}}_{1} + {\vec{X}}_{2} + {\vec{X}}_{3}}{3}$ (25)

6 Hybrid PSO-GWO technique

Despite of several merits of PSO technique like its simplicity, robustness to parameter variations, easy implementation, there are still chances of being trapped in local minima when subjected to heavy constrained problem. GWO has very good balance between exploration and exploitation and it avoids trapping in local optima. Therefore both these special features of PSO and GWO are combined in hybrid PSO-GWO algorithm. The flow chart of PSO-GWO technique is given in Fig. 6. The following steps de-scribe the implementation of PSO-GWO technique.

6.1. PSO operation

Fig.6

Flow chart for PSO-GWO algorithm to find optimal controller parameters.

6.1.1. With the objective function in Equation 15, the fitness function of all particles are evaluated.

6.1.2. Individual P best and global G best are computed.

6.1.3. The velocity for each swarm is updated by Equation 16.

6.1.4. Swarm position is updated by Equation 17.

6.1.5. Fitness values of each particle based on the objective function is computed.

6.1.6. By comparing the fitness values the best solution is selected for next iteration by Equation 18.

6.2. GWO operation

6.2.1. The final population with PSO is taken as the initial population for GWO.

6.2.2. Parameters A, C and a are updated by Equations 21, 22.

6.2.3. Random position generated for each search agent.

6.2.4. Fitness values are computed for Grey wolves by objective function.

6.2.5. Grey wolves position are updated and also for parameters A, C and a.

6.2.6. By comparing the fitness functions, best solution is choosed for further iteration.

6.2.7. X_α, X_β and X_δ are updated.

6.2.8. Steps 6.1.2 to 6.2.7 are repeated till stopping criterion is met.

6.3. Final optimal controller parameters are obtained.

7 Verification of PSO-GWO algorithm

For verification of the proposed PSO-GWO technique, it has been tested with some standard benchmark functions given in Table 2. The table also shows the dimension (n), the range of variables and the optimum value of the functions f_opt. The proposed method was applied to the minimization of these standard functions then, results obtained were put into comparison with other two algorithms PSO and GWO. In all cases, population size is set to 50 (N = 50). The dimension is 30 (n = 30) and the maximum iteration (k_max) is 500 for functions in Table 3.

Table 2
Benchmark functions

Function name Test function N S fopt

Sphere $F_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$ 30 [-100, 100] ⁿ 0

Schwefel $F_{2} (x) = \sum_{i = 1}^{n} | x_{i} | + \prod_{i = 1}^{n} | x_{i} |$ 30 [-10, 10] ⁿ 0

Rotated hyper ellipsoid $F_{3} (x) = \sum_{i = 1}^{n} {(\sum_{j = 1}^{i} x_{j})}^{2}$ 30 [-100, 100] ⁿ 0

Schwefel F₄ (x) = max { |x_i|, 1 ⩽ i ⩽ x} 30 [-100, 100] ⁿ 0

Step $F_{5} (x) = \sum_{i = 1}^{n} ([x_{i} + 0.5])^{2}$ 30 [-100, 100] ⁿ 0

Noisy Quadric $F_{6} (x) = \sum_{i = 1}^{n} {ix}_{i}^{4} + random [0, 1)$ 30 [-1.28, 1.28] ⁿ 0

Ackley $F_{7} (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - exp (\frac{1}{n} \sum_{i = 1}^{n} cos 2 π x_{i}) + 20 + e$ 30 [-32, 32] ⁿ 0

Griewank function $F_{8} = \sum_{i = 1}^{n} - x_{i} sin (\sqrt{| x_{i} |}$ 30 [-500, 500] ⁿ 0

Penalised $F_{9} (x) = \frac{π}{n} {10 {sin}^{2} (π y_{i}) + \sum_{i = 1}^{n} (y_{i} - 1)^{2} [1 + 10 {sin}^{2} (3 π y_{i + 1})]}$ $+ \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4)$ $y_{i} = 1 + \frac{x_{i} + 1}{4}$ 30 [-50, 50] ⁿ 0

Function name	Test function	N	S
Sphere	$F_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	30	[-100, 100] ⁿ
Schwefel	$F_{2} (x) = \sum_{i = 1}^{n} \| x_{i} \| + \prod_{i = 1}^{n} \| x_{i} \|$	30	[-10, 10] ⁿ
Rotated hyper ellipsoid	$F_{3} (x) = \sum_{i = 1}^{n} {(\sum_{j = 1}^{i} x_{j})}^{2}$	30	[-100, 100] ⁿ
Schwefel	F₄ (x) = max { \|x_i\|, 1 ⩽ i ⩽ x}	30	[-100, 100] ⁿ
Step	$F_{5} (x) = \sum_{i = 1}^{n} ([x_{i} + 0.5])^{2}$	30	[-100, 100] ⁿ
Noisy Quadric	$F_{6} (x) = \sum_{i = 1}^{n} {ix}_{i}^{4} + random [0, 1)$	30	[-1.28, 1.28] ⁿ
Ackley	$F_{7} (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - exp (\frac{1}{n} \sum_{i = 1}^{n} cos 2 π x_{i}) + 20 + e$	30	[-32, 32] ⁿ
Griewank function	$F_{8} = \sum_{i = 1}^{n} - x_{i} sin (\sqrt{\| x_{i} \|}$	30	[-500, 500] ⁿ
Penalised	$F_{9} (x) = \frac{π}{n} {10 {sin}^{2} (π y_{i}) + \sum_{i = 1}^{n} (y_{i} - 1)^{2} [1 + 10 {sin}^{2} (3 π y_{i + 1})]}$ $+ \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4)$ $y_{i} = 1 + \frac{x_{i} + 1}{4}$	30	[-50, 50] ⁿ

Table 3

Minimization comparison with different algorithms

Function name	Method	Best	Worst	Mean	Median	STD	Mode
F1	PSO	94.3732	262.4175	163.0999	164.3581	34.2104	94.3732
	GWO	94.27	169.162	139.9708	141.4834	17.339	94.27
	PSO-GWO	– 9.98	– 7.8809	– 8.8697	– 9.99	0.5181	– 9.98
F2	PSO	53.7589	72.2576	62.4696	63.1159	5.0132	53.7589
	GWO	10.965	28.10	19.7388	18.878	2.4493	10.965
	PSO-GWO	0	1.15×10^–5	7.61×10^–7	9.99×10^–9	2.48×10^–6	0
F3	PSO	7.74×10³	2.13×10⁴	1.28×10⁴	1.34×10⁴	3.12×10³	7.74×10³
	GWO	153.9494	1.32×10⁴	2.24×10³	1.18×10³	3.47×10³	153.9494
	PSO-GWO	0	2.89×10^–26	1.60×10^–27	7.19×10^–34	6.15×10^–27	0.00
F4	PSO	19.695	41.2969	27.7502	27.0653	5.2951	19.695
	GWO	2.3239	3.2637	2.8789	2.941	0.2407	2.3239
	PSO-GWO	0.0017	0.003	0.0027	0.0027	6.39×10^–4	0.0017
F5	PSO	3.88×10⁴	6.38×10⁵	2.40×10⁵	2.29×10⁵	1.48×10⁵	3.88×10⁴
	GWO	4.29×10⁴	0.05×10⁶	2.95×10⁵	2.54×10⁵	1.0×10⁵	4.29×10⁴
	PSO-GWO	3.81×10^–6	0.2721	0.0137	5.47×10^–4	0.0499	3.81×10^–6
F6	PSO	544.7005	3.01×10³	1.72×10³	1.67×10³	505.8671	544.7005
	GWO	96.6406	178.0129	147.3656	149.6641	13.8933	96.6406
	PSO-GWO	0.3×10^–7	0.82×10^–5	3.04×10^–6	1.03×10^–6	4.34×10^–6	0.30×10^–7
Multi modal functions
F7	PSO	– 182.775	– 120.6168	– 155.4972	– 156.609	13.2993	– 182.775
	GWO	– 5.06×10³	– 1.07×10³	– 2.70×10³	– 2.82×10³	0.04×10³	– 5.06×10³
	PSO-GWO	– 26.027	– 5.89	– 6.56	– 7.8905	1.6765	– 26.027
F8	PSO	17.4788	4.40×10⁴	2.60×10³	47.1633	9.62×10³	17.4788
	GWO	1.008	14.5567	5.7757	5.0361	2.43	1.008
	PSO-GWO	1.2×10^–12	0.122×10^–5	0.42×10^–6	2.64×10^–7	1.87×10^–6	1.2×10^–12
F9	PSO	7.3724	9.3164	8.7138	8.7964	0.4071	7.3724
	GWO	6.8	11.7	8.44	8.3	0.1414	6.8
	PSO-GWO	– 3.68	– 3.67	– 3.670	– 3.6708	7.40×10^–6	– 3.6708

The algorithms were run for 30 times with the results being noted down. From the simulation results obtained, statistical analyses were performed. Table 3 shows the summary of these results. For each method, the worst, mean, median, best, and standard deviation from the 30 independent runs were calculated and compared.

Table 3 lists down the data for best value, worst value, mean value, median, standard deviation and mode for the given nine benchmark functions as mentioned in Table 2 for PSO, GWO and PSO-GWO optimization techniques. The results of Table 3 indicate that the mean fitness assessed by PSO-GWO for all the 500 iterations were lower for all functions than those values computed by PSO and GWO.

8 Simulation and results with single machine system

The initial operating condition of the system is given in appendix. The prime objective here is optimal parameter setting for UPFC based fuzzy PID controller, which has been performed here considering PSO,GWO and proposed PSO-GWO algorithms. For this purpose ITAE of Equation (15) is taken to minimize its fitness value. As per research on damping of oscillations m_B and δ_E are the best controllers [6], so these two control actions are taken in this work. The optimum values of parameters of damping controller are given in the Table 4. The effectiveness of the proposed controller has been justified by taking different operating conditions as mentioned below.

Table 4
Optimized parameter for single machine system

Control action Optimization technique ks1 ks2 ks3 ks4

m_B Control action PSO 2.1519 0.1899 3.9160 2.6106

GWO 2.2173 0.0748 5.1480 2.6253

PSO-GWO 3.4501 0.0348 5.8202 2.1495

δ_E Control action PSO 0.6565 0.0318 3.2627 1.6909

GWO 2.9620 0.0348 1.0691 0.3702

PSO-GWO 1.9159 0.1939 4.4500 0.7026

Control action	Optimization technique	ks1	ks2	ks3	ks4
m_B Control action	PSO	2.1519	0.1899	3.9160	2.6106
	GWO	2.2173	0.0748	5.1480	2.6253
	PSO-GWO	3.4501	0.0348	5.8202	2.1495
δ_E Control action	PSO	0.6565	0.0318	3.2627	1.6909
	GWO	2.9620	0.0348	1.0691	0.3702
	PSO-GWO	1.9159	0.1939	4.4500	0.7026

8.1 Condition-I(Base condition)

Here Pe = 0.8, Qe = 0.17 and line reactance Xe = 0.5. In this case the response of the system is considered for a 10% change in mechanical input power. The change in speed deviation and line power deviations are shown in Figs. 7 and 8 respectively for m_B based controller and Figs. 9 and 10 respectively for δ_E based controller. In these figures the fuzzy PID controller has also been compared with simple PID controller optimized by same techniques. The Table 5 shows the eigen value of the system response for different controllers. For this condition it was observed that the results obtained with proposed technique are much better as compared to standard PSO and GWO techniques. The line power deviation has been illustrated here and for further conditions, the speed deviations are presented. Also the performance of proposed optimized controller is compared in terms of eigen values with PSO and GWO optimized controller for both m_B and δ_E control actions. The system eigen values obtained for both control actions with proposed controller shift to left half of complex plane by a large extent, which proves the superiority of this technique.

Fig.7

Δω with m_B action for condition-I.

Fig.8

ΔPe with m_B action for condition-I.

Fig.9

Δω with δ_E action for condition-I.

Fig.10

ΔPe with δ_E action for condition-I.

Table 5

System eigen values with single machine system

Condition-I				Condition-II			Condition-III
Eigen values	PSO	GWO	PSO-GWO	PSO	GWO	PSO-GWO	PSO	GWO	PSO-GWO
m_B action	0	0	0	0	0	0	0	0	0
	– 99.818	– 99.8328	– 99.838	– 99.8118	– 99.828	– 99.848	– 99.808	– 99.8338	– 99.8338
	– 0.0044±4.0959i	– 0.53±2.0959i	– 0.79±0.40957i	– 0.0040±4.0959i	– 0.043±2.0959i	– 0.53±0.40959i	– 0.0046±4.091i	– 0.057±2.0961i	– 0.69±0.40961i
	– 0.4820	– 0.4820	– 0.4820	– 0.4820	– 0.4820	– 0.4820	– 0.4819	– 0.4819	– 0.4819
	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018
	0	0	0	0	0	0	0	0	0
δ_E action	0	0	0	0	0	0	0	0	0
	– 99.8008	– 99.8238	– 99.8338	– 99.8338	– 99.8338	– 99.8418	– 99.808	– 99.8338	– 99.8338
	– 0.0038±4.0960i	– 0.47±2.0960i	– 0.93±0.40960i	– 0.0033±4.0960i	– 0.039±2.0959i	– 0.56±0. 40958i	– 0.0043±4.0961i	– 0.050±2.0960i	– 0.58±0.4096i
	– 0.4819	– 0.4819	– 0.4819	– 0.4820	– 0.4820	– 0.4820	– 0.4819	– 0.4819	– 0.489
	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018	– 0.0018
	0	0	0	0	0	0	0	0

8.2 Condition-II (Light loading case)

In this case Pe = 0.6, Qe = 0.02 and the line reactance is decreased by 40% from the base value. Under this condition results for change in speed deviation are shown in Figs. 11 and 12 for m_B and δ_E based controllers respectively. The system eigen values are shown in Table 5. It has been found that system eigen values are shifted to left half of compex plane with much damped oscillations. Hence performance of proposed controller is much better as compared to PSO and GWO optimized controller.

Fig.11

Δω with m_B action for condition-II.

Fig.12

Δω with δ_E action for condition-II.

8.3 Condition-III (Heavy loading condition)

In this situation Pe = 1.1, Qe = 0.38 and the line reactance is increased by 30% from the base value. With this condition, the changes in speed deviations are shown in Figs. 13 and 14 for m_B based controller and δ_E based controller respectively. The system eigen values are shown in Table 5, where it has been shown that eigen values are much shifted to left half of complex plane with reduced oscillations for proposed controller. Hence the system responses show that the performance of proposed controller is much better than PSO and GWO optimized controller.

Fig.13

Δω with m_B action for condition-III.

Fig.14

Δω with δ_E action for condition-III.

9 Simulation and results with Multi machine system

The power system shown in Fig. 15 has been taken here for study with three machines [8]. IEEE ST1A excitation is taken for all machines. The machine-1 taken here has same data considered for single machine system earlier. The data for other two machines are given in appendix. UPFC is installed between machine 1 and 3 because this is the longest line with highest power flow. A special load with negative reactive power is connected at Bus-3, which is given in appendix as L3 and the input power to generator – 1 is being raised by 10%. With this disturbance the fuzzy PID UPFC controller is employed to damp oscillations, and the parameters of controller are optimized by PSO, GWO, and PSO-GWO techniques. The inter area speed deviation among machines 1 and 3 is taken as input to controller and m_B and δ_E control actions are considered for UPFC. The speed deviation between machines 1 and 3 is represented as w13 and in between machines 1 and 2 is is represented as w12. The speed deviation w13 with m_B and δ_E control actions are given in Figs. 16 and 17 respectively. The speed deviation w12 with m_B and δ_E control actions are given in Figs. 18 and 19 respectively. The Table 6 represents optimized parameters for multi machine system and the prominent electromechanical oscillatory mode eigen value with m_B and δ_E control actions. The response of the system subjected to disturbance indicates that with proposed controller, the system oscillations are much damped and eigen value of oscillatory mode being much shifted to left half of S plane promising much better results with respected to other optimized controllers.

Fig.15

Multi machine test system with damping controller.

Fig.16

Δω₁₃ with m_B based controller.

Fig.17

Δω₁₃ with δ_E based controller.

Fig.18

Δω₁₂ with m_B based controller.

Fig.19

Δω₁₂ with δ_E based controller.

Table 6

Optimized parameters and eigen value of inter area swing for multi machine system

	Algorithms	ks1	ks2	ks3	ks4	Inter area swing
m_B controller (multi)	PSO	1.7627	0.1156	2.1394	2.7325	0.0224±j4.108
	GWO	2.2361	0.0747	4.1216	0.8484	0.058±j2.1533
	PSO-GWO	2.9269	0.1577	4.7361	2.3268	0.2116±j1.7618
δ_E controller (multi)	PSO	3.8556	0.0559	0.5123	1.9202	0.0217±j4.1101
	GWO	1.0794	0.0780	2.7205	0.5863	0.0567±j2.1413
	PSO-GWO	1.4545	0.1809	3.2691	0.2458	0.2236±j1.7421

10 Conclusion

It was observed from experimentations with UPFC based optimized PID and optimized fuzzy PID controller that the PID with fuzzy optimal controller can damp system oscillations much effectively in comparison to other. Also PSO-GWO technique has been proven to be much more efficient as compared to PSO and GWO techniques to tune the controller parameters, since it does not trap in local optima. The system eigen values are much shifted to negative half of complex plane and peaks of oscillations, settling time are also reduced to a large extent with proposed controller in contrast to others.

Footnotes

Appendix

(All the datas are in per unit other than constants)

A1. Single machine infinite bus test system data: C_dc = 1, H = 4MJ/MVA, Ka = 100, Ta = 0.01, T_d0 = 5.044 sec, D = 0, δ₀ = 47.13⁰, V_b = 1, V_dc = 2, V_t = 1, X_B = X_E = 0.1, X_BV = 0.3, X_d = 1, X_E = 0.1, X_d = 0.3, X_q = 0.6, Xe = 0.5.

A2. Multi machine system data: H₂ = 20, H₃ = 11.8, D₂ = D₃ = 0, T_d02 = 7.5 sec, T_d03 = 4.7 sec, T_dc = 0.01, K_dc = 5, X_q2 = 0.16, X_q3 = 0.33, X_d2 = 0.19, X_d3 = 0.41, X_d2 = 0.076, T_A2 = 0.01, K_A2 = 100, K_A3 = 20, T_A2 = 0.01, Z₁₃ = j0.6 (double lines), Z₂₃ = j0.1, L₃ = 0.8-j1.253, V₃ = 1 < 0⁰, V₂ = 1 < 5⁰.

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