Application of Bat Algorithm to Optimize Scaling Factors of Fuzzy Logic-Based Power System Stabilizer for Multimachine Power System

2.1 Test system model

The general representation of a power system using nonlinear differential equations can be given by

where X and U represent the vector of state variables and the vector of input variables. As in Refs [15, 30, 31], the power system stabilizers can be designed by use of the linearized incremental models to a power system around an operating point. The system representation based on algebraic equations is given in Appendix. The state equations to a power system can be written as

Analysis of the IEEE ten-machine thirty-nine-bus test power system as in Figure 1 can be carried out by simultaneous solution of equations consisting of prime movers, synchronous machines with excitation systems, transmission line network, dynamic and static loads, and other devices like static VAR and HVDC converters-based compensators. The dynamics of generator rotors, excitation, prime movers, and other related devices are being represented by differential equations. Thus, the complete multimachine model consists of large numbers of ordinary differential equations (ODE) and algebraic equations [19, 32]. These are linearized about an operating point (nominal) to derive a linear model for the small signal oscillatory behaviour of power systems. The range of variation in operating point can generate a set of linear models corresponding to each operating point/condition.

Figure 1:

Line diagram of New England 39 bus test power system.

Figure 2:

Heffron-Philips model for multimachine power system.

The state equations of a power system, consisting “N” number of generators and N_pss number of power system stabilizers, can be written as in eq. (2) where A is the system matrix of an order as 4N×4N (40×40) and is given by ∂f/∂X, while B is the input matrix with order 4N×N_pss (40×10) and is given by ∂f/∂U. The order of state vector ΔX is 4N×1 (40×1), and the order of ΔU is N_pss×1 (10×1). Here, the well-known Heffron-Phillip linearized model is used to represent the large multimachine power system as in Figure 2 [2, 15, 30].

2.2 Objective function

To optimize the PD parameters (Kp and Kd) an objective function is formulated, wherein the damping is maximized in terms of reduced overshoots and settling time in system oscillations. As in integral square error (ISE), the error is heavily reduced/penalized in the beginning (during large errors) and low for light errors. Since, the speed deviation Δω of the generator is sensed from the shaft of the generator. As an objective function, the ISE-based cost function considered in Ref. [26] is represented as

subjected to

Typical ranges of the optimized parameters are selected as in Ref. [26]. The considered system is New England ten-machine system wherein each excitation system of generator is equipped with automatic voltage regulator and fuzzy logic controller in decentralized manner. Considering one of the above objectives, the proposed approach employs BA algorithm to solve this optimization problem for an optimal set of scaling factors for FPSS. The scaling factors of the FPSS are the factors by which the input signals such as error (speed deviation) and derivative of error (acceleration) are being controlled. As per analogy developed in Sambariya, 2013, the scaling factors of FPSS are analogous to PD controller parameters [26]. The same analogy has been considered to optimize scaling factors. Therefore, these scaling factors are optimized using bat algorithm to have appropriate degree of input signals to FPSS and resulting with better performance as compared to FPSS without optimal set of FPSS. The process of optimization of scaling factors (in turn the PD parameters) is carried out according to the arrangement as in Figure 3, wherein generators of the test system is equipped with Kp and Kd scaling factors of the FPSS. The used bat algorithm is introduced in Section 3.

Figure 3:

Scheme of tuning for PD controller parameters/scaling of fuzzy logic power system stabilizer using bat algorithm.

2.3 Fuzzy logic power system stabilizer

The fuzzy logic controller is considered as the fuzzy logic-based power system stabilizer (FPSS). The main parts of FPSS are fuzzification, knowledge base, inference, and defuzzification process [3]. The knowledge base consists of the storage of the linguistic variables (LVs), membership functions (MFs), and the fuzzy rules defined by the user. The number of MFs is equal to the number of LVs. The increased number of LVs improves the performance of the FPSS but simultaneously increases the computational time and computational burden. Therefore, a compromise is taken in the selection of the number of LVs and MFs. The selected number of LVs is five and named as LN (large negative), MN (medium negative), Z (zero), MP (medium positive), LP (large positive), and P (positive) while the selected MFs is the triangular type. The number of rules in the knowledge base is square of the number of LVs, i. e. twenty-five in this case [3]. The main objective of FPSS is to map crisp input signal to the crisp output control signal, which is applied to the excitation system to modulate the excitation control and to increase the damping of the system.

The FPSS is a two input and one output device as speed deviation (Δω), acceleration (Δω˙=Δa), and voltage (ΔVpss) respectively. The selected input signal Δω is sensed from the rotor shaft of the generator while the acceleration is computed as Δω˙=Δa=Δω(kT)−Δω(k−1)T/T, where T is the sampling time.

The sensed crisp signals are applied as input to the fuzzification system and is mapped to the MFs to fuzzify it. The triangular MF is shown in Refs [3, 26]. The output of fuzzification process is applied to the inference process to get a fuzzy control output signal according to the fuzzy rules. In defuzzification process, the fuzzy control signal is applied to defuzzification process to get crisp control signal, and the method of defuzzification is considered as centroid. The crisp control signal is added to modulate the excitation signal to increase the damping of the system. The considered rule base is shown in Table 1.

3.1 BA procedural steps

In an optimization problem, the objective function is represented by minimization of F(x) which is subjected to xk∈Xk, k = 1, 2, 3…N.

Step 1: Initialization

1. As an initial step, the bat population is initiated as position xk and velocity as vk with k = 1, 2, 3, … n.

2. Initial pulse frequency is defined as fk∈[fmin,fmax]

3. The pulse rates rk and the loudness Ak are also set as above.

4. Check number of iterations or t<T_max

Step 2: Generation of new solutions

1. New solutions may be generated by adjusting the pulse frequency and keeping wavelength as constant.

2. For each bat (k), its position xk and velocity vk in a d-dimensional search space should be defined. xk and vk should be subsequently updated during the iterations. The new solutions xkt and velocities vkt at time step “t” can be calculated by:

   (6)fk=fmin+(fmax−fmin)β

   (7)vkt=vkt−1+(xkt−1−x′)fk

   (8)xkt=xkt−1+vkt

where β is defined for uniform distribution as a vector and selected as β∈[0,1]. In this application, the value of β is considered as 0.4 after several simulations and found as fit on the trial and method. The x′ stands as the best location in search space after comparing solutions of all the n bats. The product of fk and νk represents the velocity increment. The velocity increment can be adjusted by changing one and keeping fixed another according to a problem. The generally used range of frequency is 0≤f≤100 and each bat at initialization step is selected from f=[fmin,fmax].

Step 3: Local search

Once the best current solution is selected among the available solutions, then a new solution is generated by using local random walk and assigned to each bat as in eq. (9). If ε∈[−1,1] represents a random number range and At=<Akt> stands for average value of loudness of all initiated n bats at time t,

As the number of iteration increases, the loudness Ak and the rate rk of pulse emission have to be updated. As a micro bat reaches to its target/prey the rate of pulse emission increases while the loudness decreases. The loudness is generally selected from [A⁰, A_min] = [0, 1]. The A⁰ = 1 represents the maximum loudness of emitted pulse by micro bat in search of prey, while A_min = 0 indicates that the micro bat got the target/prey and was not emitting any loudness. Thus, the loudness and the rate of pulse emission is updated as

where α and γ represent the constant values. Here, α is similar to the cooling factor of a cooling schedule in the simulated annealing [33, 34] and the range of these constants is as 0<α<1 and 0<γ<1.

To make optimization simpler, the value of α and γ should be selected as same; therefore, in this study α=γ=0.9 after carrying out several simulation studies. However, the variation of α=γ can be considered as linearly varying but to minimize computational time and computation burden, a constant value is suggested. As in eq. (11), the initial loudness and emission rate may be represented by Ak0 and rk0, respectively. The value of emission rate at time t can be selected from rk0∈[0,1].

Step 5: Checking the stopping criterion

If the maximum count of iterations is reached and the stopping criterion is satisfied, then the process of computation is terminated. Otherwise, go to Steps 3 and 4 to repeat the process. Moreover the used pseudo code of bat algorithm is included in Table 2.

4.1 Experimental plant creation

The line diagram of ten-machine thirty-nine-bus power system (New England thirty-nine-bus test system) is as in Figure 1, while the system data are as in Ref. [35]. A linear representation of the system without PSSs is formed around a nominal point, and the power system models (plants) are generated by applying fault at different bus numbers as in Table 3. The plants generated are subjected to operate without PSS (i. e. under open loop) and the speed responses of generators (ten in number) in each plant are recorded. The speed response of all ten generators of plant 1–4 is shown in Figures 4(a), 4(b), 5(a), and 5(b), respectively. Clearly, none of the generators of any plant are showing stable operation. The proposed BAFPSS and FPSS [26] would be implemented on these plants, and the performance analysis is carried out in next section.

Figure 4:

Speed variation under open condition for (a) plant-1 and (b) plant-2.

Figure 5:

Speed variation under open condition for (a) plant-3 and (b) plant-4.

Figure 6:

Plot of fitness function variation in the tuning of PD controller parameters using bat algorithm for plant-1 of considered test power system.

4.2 Optimization of scaling factors of FPSS

The above-generated plants equipped with the PD controller in turn scaling factors of FPSS as in Figure 3 and optimized by bat algorithm are subjected to the objective function as defined in eq. (3) with the parametric bounds such as 0.00 ≤ K_p ≤ 50 and 0.00 ≤ K_p ≤ 50 as in Ref. [26]. The optimized parameter values for all four plants are enlisted in the Table 4 and the performance of bat algorithm in tuning of scaling factors for plant-1 in terms of fitness function with iteration count is shown in Figure 6. It is better to declare that the concept and relationship in between scaling factors of FPSS and gain factors of PD controller are considered as in Ref. [26] and, therefore, not included in this paper.

4.3 Simulation results

4.3.1 Speed response analysis

The objective of the paper is to optimize the scaling factors by tuning the PD controller parameters using bat algorithm as in Figure 3. Therefore, the system responses without PSS, with FPSS [26], and with BAFPSS (proposed) are compared. The speed response of generators for plant-1 under open condition, with FPSS and with BAFPSS, is shown in Figures 7–11. It is very clear that the response of each generator with BAFPSS is settling to steady-state position earlier than the response with FPSS. The responses for plant conditions 2–4 are not shown because of space constraint. However, the speed response in terms of settling time, for each plant with FPSS and with BAFPSS, is provided in Table 5. The oscillations occurring in the response for each generator of the system are damped much faster with the proposed BAFPSS as compared to the response of the system with FPSS in [26]. This illustrates the potential and superiority of the proposed design approach to obtain an optimal set of parameters of FPSS using BA.

Table 4:

BA-optimized PD controller parameters or scaling factors of FPSS [26].

Generators of power system	Scaling factors of FPSS
	Kp	Kd
Gen-1	35.3269	32.2919
Gen-2	27.6603	10.9836
Gen-3	38.6411	11.4786
Gen-4	18.6061	45.5573
Gen-5	42.8332	20.1814
Gen-6	15.9692	30.4709
Gen-7	45.5187	45.4640
Gen-8	29.6206	16.6953
Gen-9	42.6679	22.1757
Gen-10	45.2273	1.7557

Table 5:

Settling time for speed response of generators in the plants 1–4 with FPSS [26] and BAFPSS (proposed).

Genrs.	Controller	Plant-1	Plant-2	Plant-3	Plant-4
Gen-1	FPSS	11.56	13.87	10.37	12.96
	BAFPSS	2.8	6.21	3.07	6.29
Gen-2	FPSS	11.73	12.88	11.56	14.96
	BAFPSS	2.7	5.0	3.11	5.77
Gen-3	FPSS	11.53	13.73	13.11	14.43
	BAFPSS	3.73	5.47	5.55	9.25
Gen-4	FPSS	13.88	14.21	15.51	16.40
	BAFPSS	4.18	6.78	11.88	10.35
Gen-5	FPSS	12.64	14	14.74	17.29
	BAFPSS	4.15	4.42	5.51	10.59
Gen-6	FPSS	12.82	13.68	13.37	19.29
	BAFPSS	2.46	6.70	3.81	15.47
Gen-7	FPSS	13.69	13.96	11.71	14.51
	BAFPSS	2.55	6.89	3.70	4.81
Gen-8	FPSS	12.71	12.01	12.45	16.37
	BAFPSS	2.96	5.53	5.45	10.61
Gen-9	FPSS	12.98	15.85	16.78	19.14
	BAFPSS	2.11	8.1	10.56	15.31
Gen-10	FPSS	12.6	13.83	11.86	16.66
	BAFPSS	3.62	4.7	3.65	7.68

Figure 7:

Speed response for plant-1 without PSS, with FPSS [26] and with BAFPSS (proposed) of (a) Generator-1 and (b) Generator-2.

Figure 8:

Speed response for plant-1 without PSS, with FPSS [26] and with BAFPSS (proposed) of (a) Generator-3 and (b) Generator-4.

Figure 9:

Speed response for plant-1 without PSS, with FPSS [26] and with BAFPSS (proposed) of (a) Generator-5 and (b) Generator-6.

Figure 10:

Speed response for plant-1 without PSS, with FPSS [26] and with BAFPSS (proposed) of (a) Generator-7 and (b) Generator-8.

Figure 11:

Speed response for plant-1 without PSS, with FPSS [26] and with BAFPSS (proposed) of (a) Generator-9 and (b) Generator-10.

4.3.2 Performance index-based analysis

For completeness and clear perceptiveness about the system response for all the system conditions, three performance indices that reflect the settling time and overshoot are introduced and evaluated. These indices are defined as in Refs [26, 29] and represented by eqs (12)–(14).

ITAE: Integral of the Time-weighted Absolute Error

(12)ITAE=∫t=0Tsimt|Δω(t)|dt

ISE: Integral Square Error

(13)ISE=∫t=0Tsim|Δω(t)|2dt

IAE: Integral of the Absolute Error

(14)IAE=∫t=0Tsim|Δω(t)|dt

where t_sim is the simulation time of the system.

The performance indices as ITAE, IAE, and ISE calculated for a speed response of plants 1–4, with FPSS [26] and with BAFPSS (proposed), are mentioned in Tables 6, 7, and 8, respectively. Here, it should be cleared that the smallest value of performance index represents the best performance of the system. The ITAE of system generators with BAFPSS (proposed) is slightest as compared to the performance with FPSS [26]. To make clearer in the continuous time domain, the plots of ITAE, IAE, and ISE for plant-1 with FPSS and with BAFPSS are shown in Figures 12, 13, and 14, respectively. Clearly, the associated ITAE, IAE, and ISE values are least with BAFPSS.

Figure 12:

ITAE response for all generators of plant-1 (a) with FPSS and (b) with bat algorithm optimized FPSS.

Figure 13:

IAE response for all generators of plant-1 (a) with FPSS and (b) with bat algorithm optimized FPSS.

Figure 14:

ISE response for all generators of plant-1 (a) with FPSS and (b) with bat algorithm optimized FPSS.

Table 6:

ITAE performance index for speed response of each generator of plants (1–4) with FPSS [26] and BAFPSS (proposed).

Genrs.	Controller	Plant-1	Plant-2	Plant-3	Plant-4
Gen-1	FPSS	0.0156	0.0122	0.0045	0.0133
	BAFPSS	0.0094	0.0068	0.0005	0.0021
Gen-2	FPSS	0.0098	0.0101	0.0039	0.0071
	BAFPSS	0.0012	0.0014	0.0007	0.0029
Gen-3	FPSS	0.0175	0.0142	0.0053	0.0151
	BAFPSS	0.0043	0.003	0.0026	0.007
Gen-4	FPSS	0.0254	0.0254	0.0098	0.0252
	BAFPSS	0.0082	0.0058	0.0035	0.0039
Gen-5	FPSS	0.0228	0.0228	0.009	0.0207
	BAFPSS	0.0033	0.0035	0.0019	0.005
Gen-6	FPSS	0.0222	0.0224	0.0085	0.0193
	BAFPSS	0.0034	0.0043	0.0013	0.0084
Gen-7	FPSS	0.0221	0.0224	0.0087	0.0192
	BAFPSS	0.0044	0.0059	0.0014	0.0054
Gen-8	FPSS	0.0133	0.0127	0.0051	0.0208
	BAFPSS	0.0025	0.0033	0.0028	0.006
Gen-9	FPSS	0.0251	0.0244	0.0099	0.0642
	BAFPSS	0.0095	0.0096	0.004	0.0306
Gen-10	FPSS	0.0102	0.0107	0.0039	0.0155
	BAFPSS	0.0014	0.0021	0.0011	0.0043

Table 7:

IAE performance index for speed response of each generator of plants (1–4) with FPSS and BAFPSS.

Genrs.	Controller	Plant-1	Plant-2	Plant-3	Plant-4
Gen-1	FPSS	0.0052	0.0036	0.0017	0.0037
	BAFPSS	0.0027	0.0014	0.0003	0.0007
Gen-2	FPSS	0.0026	0.0028	0.0013	0.0019
	BAFPSS	0.0005	0.0006	0.0002	0.0007
Gen-3	FPSS	0.0054	0.0041	0.0019	0.0042
	BAFPSS	0.0018	0.001	0.0005	0.0009
Gen-4	FPSS	0.0069	0.007	0.0033	0.0065
	BAFPSS	0.0013	0.0014	0.0006	0.001
Gen-5	FPSS	0.0061	0.0062	0.003	0.0053
	BAFPSS	0.0008	0.001	0.0005	0.001
Gen-6	FPSS	0.0061	0.0063	0.0029	0.0052
	BAFPSS	0.0011	0.0015	0.0005	0.0024
Gen-7	FPSS	0.006	0.0062	0.003	0.0051
	BAFPSS	0.0012	0.0018	0.0005	0.0017
Gen-8	FPSS	0.0038	0.0036	0.0018	0.0058
	BAFPSS	0.0008	0.001	0.0007	0.0017
Gen-9	FPSS	0.0064	0.0065	0.0032	0.017
	BAFPSS	0.0013	0.002	0.0007	0.0073
Gen-10	FPSS	0.003	0.0033	0.0015	0.0044
	BAFPSS	0.0008	0.0011	0.0006	0.0017

Table 8:

ISE performance index for speed response of each generator of plants (1–4) with FPSS and BAFPSS.

Genrs.	Controller	Plant-1	Plant-2	Plant-3	Plant-4
Gen-1	FPSS	7.28E-06	2.61E-06	6.75E-07	2.72E-06
	BAFPSS	4E-06	6.06E-07	7.31E-08	1.32E-07
Gen-2	FPSS	1.16E-06	1.38E-06	3.71E-07	5.79E-07
	BAFPSS	1.36E-07	1.67E-07	2.09E-08	1.41E-07
Gen-3	FPSS	7.15E-06	3.28E-06	8.32E-07	3.36E-06
	BAFPSS	2.23E-06	4.77E-07	5.76E-08	1.18E-07
Gen-4	FPSS	8.68E-06	8.84E-06	2.38E-06	7.46E-06
	BAFPSS	4E-07	6.83E-07	5.07E-08	2.46E-07
Gen-5	FPSS	6.68E-06	7.24E-06	1.95E-06	5.04E-06
	BAFPSS	2.27E-07	5.26E-07	6.55E-08	1.88E-07
Gen-6	FPSS	6.64E-06	7.26E-06	1.91E-06	5.11E-06
	BAFPSS	5.38E-07	8.59E-07	1.14E-07	1.25E-06
Gen-7	FPSS	6.48E-06	7.09E-06	1.87E-06	4.89E-06
	BAFPSS	5.46E-07	1.18E-06	1.16E-07	8.08E-07
Gen-8	FPSS	2.60E-06	2.53E-06	7.18E-07	7.13E-06
	BAFPSS	2.17E-07	3.92E-07	1.43E-07	9.61E-07
Gen-9	FPSS	7.38E-06	7.26E-06	1.97E-06	5.14E-05
	BAFPSS	2.78E-07	9.47E-07	1.89E-07	2.04E-05
Gen-10	FPSS	1.62E-06	2.28E-06	6.14E-07	3.88E-06
	BAFPSS	3.98E-07	6.42E-07	1.78E-07	1.32E-06