Design of a Proportional-Integral-Derivative Controller for an Automatic Generation Control of Multi-area Power Thermal Systems Using Firefly Algorithm

Ⅰ. INTRODUCTION

DUE to modern technologies escalation, industries are modernized and automated, which require power supplies of high quality. The power supply quality is deliberated by stability in the system frequency and voltage profile across the generator terminals. The single power system provides good quality of power, when load disturbance/demand occurs within the specified limit[1]. However, during large load demand condition, system stability, power quality and system performance are affected. In order to overcome these limitations, power generating units are interconnected through the tie-line.

During normal loading conditions each power generating unit carries its own load. Thus, when sudden load occurs in any one of the interconnected power systems, it shares the power between the control areas through tie-line to maintain system stability[2]. The interconnection of the power system increases the size and complexity. The primary control of the power system is achieved by speed governor, but the control action provided by the governor is not sufficient to match the generation with load demand[3]. The aforementioned issue with the power system is solved by implementing proper secondary/supplementary controller. From the literature, it is found that many supplementary controllers are successfully designed and implemented in a load-frequency control (LFC)/AGC crisis of single/multi area power system.

Kothari et al. discussed the AGC of two area thermal power system with generation rate constraint (GRC) and integral controller[4]. While, the AGC of two area power system with governor dead band and integral controller was conducted in[5]. With optimal controller, the AGC of two area hydrothermal power system was mentioned in[6]. Then, the AGC of two area reheat thermal power system using proportional integral (PI) controller was discussed in[7]. Das et al.[8] proposed the AGC of two area reheat thermal power system with variable structure controller (VSC). Tripathy et al. [9] suggested an AGC of two area reheat thermal power system by considering boiler dynamics and super magnetic energy storage (SMES) unit.

In 2010, Roy et al.[10] employed an evolutionary computation technique with the AGC of three area power system. In[11], the LFC of interconnected non reheat power system was analyzed with PI controller. In 2012, Gozde et al.[12] introduced the AGC of interconnected reheat thermal power system with PI and PID controller. In the same year, Daneshfar et al.[13] employed the genetic algorithm for multi objective design with the LFC problem. Fuzzy integral double derivative (FIDD) controller with the AGC of multi-area hydrothermal power system was presented in[14].

Debbarma et al.[15] investigated the AGC of multi-area thermal power system under deregulated environment by considering non-integer controller. In[16], the LFC of two area power system was discussed with 2-degree of freedom proportional-integral-derivative (2DOF-PID) controller for load frequency control of power system with governor dead-band (GDB) nonlinearity. Padhan and Majhi[17] presented the PID controller with the LFC of single/multi area power system. Shabani et al.[18] proposed a robust PID controller based on imperialist competitive algorithm for load frequency control of power systems. The AGC crisis in interconnected power system with PID controller was presented in[19]. Meanwhile, Taher et al. studied the LFC of the power system with fractional order PID controller (FOPID)[20].

From the preceding literatures, it is concluded that several controllers have been designed and implemented to solve the LFC/AGC concern of the power system. Since, proper selection of the controller gain is very crucial for better controlled performance. Recently by 2015, many bio-inspired algorithms are developed to optimize the controller gain values, such as 1) direct synthesis (DS) for tuning of PID controller parameters[21], 2) grey wolf optimizer algorithm for tuning of PI and PID controller gain values[22], 3) self adaptive modified bat algorithm (SMBA) for tuning of PI controller[23], 4) Cuckoo search (CS) algorithm for tuning of PI controller[24], 5) teaching learning based optimization (TLBO) for tuning of fuzzy PID controller gain values[25], 6) hybrid particle swarm optimization (PSO) and pattern search (PS) (hPSO-PS) optimization for tuning of fuzzy PI controller[26], 7) minority charge carrier inspired (MCI) algorithm was proposed for tuning of I and PI controller[27], and 8) modified harmony search algorithm (MHSA) for tuning PID controller parameters[28].

Since, the PID controllers are widely used to control many kinds of systems, these controllers are used generally to improve the system dynamic response as well as to reduce/eliminate the steady-state error. Thus, for better performance, the optimal selection of the PID parameters is required. In this regard, the parameters selection can be considered an optimization problem, for which optimization algorithms can be used to determine the optimal controller gain/ parameters. Recently, the well-known firefly algorithm is adapted for solving different design problems[29]-[32]. Therefore, the main idea of the proposed system is to employ the cooperating fireflies optimization algorithm in order to optimize the PID controller parameters for a five area reheat thermal power systems. Additionally, the genetic algorithm (GA) and the PSO are well-established optimization algorithms. Thus, in this context, the proposed FFA-PID system performance is compared to both GA and PSO- based PID controllers.

The rest of the paper is organized as follows. Section Ⅱ describes the proposed system modeling, followed by Section Ⅲ that presents the design procedure of the PID controller. Section Ⅳ depicts the proposed bio-inspired algorithm. Then, Section Ⅴ provides the simulation result comparisons of the proposed algorithm to other optimization methods. Finally, the conclusion is conducted in Section Ⅵ.

Ⅱ. FIVE AREA INTERCONNECTED REHEAT THERMAL POWER SYSTEM

The transfer function model of the investigated multi-area interconnected reheat thermal power system was shown in Fig. 1. The multi-area power system incorporated five equal reheat thermal power generating units. Each unit consisted of suitable speed governing unit, turbine and generator unit with PID controller.

Figure  1.  Five area interconnected reheat thermal power system

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In Fig. 1, $R_{1}$-$R_{5}$ are the self regulation parameters for the governor in p.u. Hz; $Tg_{1}$-$Tg_{5}$ represent the speed governor time constants in second; $TR_{1}$-$TR_{5}$ are the reheat time constants in second; $K_{r1}$-$K_{r5}$ are the reheater gain; $Tt_{1}$-$Tt_{5}$ are the steam chest time constant in second; $Tp_{1}$-$Tp_{5}$ are the power system time constant in second. ($Tp = 2H/f\times D)$; $Kp_{1}$-$Kp_{5}$ are the power system gain ($Kp = 1/D)$; $B_{1}$-$B_{5}$ are the frequency bias parameters; $\Delta P_{\rm tie}$ is the incremental tie-line power change; $\Delta F_{1}$-$\Delta F_{5}$ are the incremental frequency deviations in Hz; $ACE_{1}$-$ACE_{5}$ stand for the area control error.

In current work, the PID controller is proposed as a secondary controller. The controller input and output are the area control error (ACE) and the control signal (u); respectively. The ACE is defined as the linear combination of the frequency deviation and the tie-line power flow deviation. The area control error for each area is given by

where $ACE_{1}$-$ACE_{5}$ are the area control error of areas 1-5; respectively. $\Delta F_1$-$\Delta F_5$ are the frequency deviation in areas 1-5; respectively, $\Delta P_{\rm tie}$ is the tie-line power deviation for the corresponding area.

Ⅲ. PROPORTIONAL-INTEGRAL-DERIVATIVE (PID) CONTROLLER

Nowadays, it is most common in all industries to use the significant features of the PID controller for simple construction and easy implementation. Thus, the transfer function of the PID controller is given by

where $K_{p }$ is the proportional gain, $K_{i }$ is the integral gain, $K_{d }$ is the derivative gain, $T_{i}$ is the Integral action time and $T_{d}$ is the derivative action time.

The PID controller consists of three modes, namely the proportional, integral and derivative modes. Based on the input of the PID controller, it generates appropriate control output signal to keep the power system response within the specified limit. The expression for the generated control signal is given by

In the current study, the parameters of the PID controller's gain values are optimized by using three different bio-inspired optimization algorithms, namely the genetic algorithm (GA), the particle swarm optimization (PSO) and the firefly algorithm (FFA). The controller's proper design is based on suitable selection of its objective function. The objective function is defined based on the required specification and constraints as well as the closed loop response of entire system output with time domain specification. The output of the time domain specification analysis is the peak overshoot, undershoot, settling time and steady state error. The different cost functions that can be involved are the integral square error (ISE), integral time square error (ITSE), integral absolute error (ITAE) and integral time absolute error (ITAE). Based on the literature, it is evident that the ITAE based objective function provides more superior performance compared to other objective functions[26],[33]. Thus, the ITAE objective function is used in the proposed system using the following expression.

This objective function is to be considered with the optimization algorithms.

Ⅳ. FIREFLY ALGORITHM

The firefly algorithm (FFA)[33]-[37] is a bio-inspired meta-heuristic search algorithm inspired by the behavior of fireflies. Meta-heuristics do not assure that the global optimal solution can ever be achieved, though such global optimality can be found in many cases in practice. Almost all the meta-heuristics use some form of stochastic components. Their power comes from their attempt to emulate the best features in nature, biological systems that have specially evolved by natural selection, over millions of years. There are significant attributes of meta-heuristic algorithms, though the selection of the best fit and environmental adaptability are very important. From the algorithm behavior point of view, intensification and diversification are the two key components. Intensification tends to explore local regions around the region of the existing best solutions, selecting the best solutions or candidates. In contrast, miscellany tries to explore the search space more competently by generating solutions with higher diversity. In traditional gradient-based methods, gradient of the function to be optimized has vital information for rapid finding and optimization of the solutions for a specific problem. Though, in the case of dealing contrary to the necessary conditions to the relevance of these methods (highly nonlinear, not differentiable, non-smooth, non-convex problems) face complexities on convergence and often getting trapped in local optima.

The firefly optimization algorithm can remarkably improve the technique of the global search and local optimization ability. The FFA (non-gradient based) is a simple objective function based evolutionary technique that can produce an effective result when dealing with highly nonlinear dynamic optimization problems having quite a few limitations. Like all other well tested meta-heuristic algorithms for optimization, FFA can also find an optimal solution to a problem by iteratively making an effort to enhance a candidate solution considering a specified measure of the solution quality. A nature stimulated meta-heuristic algorithm was developed by Yang et al. in 2007, namely firefly algorithm (FFA), which is based on the fashing patterns and behavior of fireflies[34],[37]-[40]. This modern meta-heuristic algorithm uses three idealized rules as stated below:

1) Fireflies are unisexual. They move towards more appealing and brighter fireflies irrespective of their sex.

2) Attractiveness is proportional to brightness. Brightness is inversely proportional to the distance amongst fireflies. For any two fashing fireflies, the less bright moves towards the brighter one. A firefly moves randomly, if there is no brighter firefly than the particular one.

3) The landscape of the objective function determines the brightness of a firefly. In most of the problem domain, the objective function value is proportional to the brightness.

The two most significant issues in the firefly algorithm are:

1) Light intensity variation; and

2) Formulation of attractiveness.

The firefly's attractiveness is proportional to the light intensity seen by adjacent fireflies. A monotonically decreasing function, namely the attractiveness function $\beta (r)$ with the distance $r$ ($r$ is the distance between two adjacent fireflies) can be represented as the following generalized form:

where $\beta _0$ denotes the maximum attractiveness at $r = 0$. $\gamma$ is a fixed light absorption coefficient, which controls the decrease in the light intensity. Although, $\gamma \in \left[ {0, \infty } \right]$ but still in practice the value of $\gamma$ is determined by the characteristic length of the system to be optimized which normally ranges within 0.1 to 10[40].

In addition, the characteristic distance $\Gamma$ is the distance over which the attractiveness changes significantly. For a given characteristic length scale in an optimization problem, the parameter $\gamma$ can be typically initialized by

For fixed$\gamma$, the characteristic distance will be: $\Gamma = \gamma ^{-1}\to 1$ as $m\to \alpha$. The distance between any two fireflies $i$ and $j$ at $x_i$ and $x_j$ can be computed by

where $x_{i, k}$ is the $k$th component of the spatial coordinate $x_i$ of $i$th firefly and $d$ denotes the number of dimensions.

A firefly $i$ gets attracted to another more appealing (brighter) firefly$j$ where the relation between the new and old position of firefly $i$ is determined by

where the 2nd term in (16) is due to attraction. The 3rd term is randomization with $\alpha$ being the randomization parameter, and $\varepsilon _i^t$ is a vector of random numbers drawn from a Gaussian distribution or uniform distribution at time $t$.

A Firefly Algorithm

The main objective of this paper is to use the FFA to optimize the PID control parameters of five reheat thermal power systems. The basic steps of the FFA[34]-[36] can be summarized by the following pseudo code:

Algorithm 1. FFA algorithm
Begin
Define the objective function
$f\left(x \right): x=\left({kp_n, ki_n, kd_n } \right)^T, n=1, 2, {\ldots}, 5$.
Generate initial population of fireflies $y_{i}$, $i = 1, 2, {\ldots}, s$.
Light intensity ${I}pidval_{i}$ at $x_{i}$ is determined by $f(x_i)$.
Define light absorption coefficient $\gamma$
while ($t <$ MaxGeneration)
for $i = 1$ to $s$ all $s$ fireflies
for $j = 1$ to $i$ all $s$ fireflies
if ${I}pidval_{i} < {I}pidval_{j}$
Move firefly $i$ towards $j$
end if
Attractiveness varies with distance $r_{ij}$ via exp$\left[{-\gamma \times r_{ij} } \right]$
Evaluate new solutions and update light intensity
end for $j$
end for $i$
Rank the fireflies and find current best
end while
Post process on the best so far results and visualization
End

 | Show Table

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Ⅴ. SIMULATION RESULTS AND DISCUSSIONS

Extensive efforts were directed to the optimization problems as well as PI controllers and load frequency control[41]-[52]. It is known that, during normal operation of the power system, it carries its own load and keeps the system parameters within the limit. However, when sudden disturbance occurs in the power system, it affects the system parameters. In order to overcome this problem, it is required to use controllers. In power system, the primary control loop is considered speed regulator and the secondary controller is considered supplementary controller.

In the current work, the PID controller is equipped as a secondary controller to generate appropriate control signal based on the error signal developed by the power system. The control signal generated by the controller is used by the power system as a reference signal. The error signal is linear combination of the system frequency error and tie-line power flow error between connected areas.

Consequently, this work investigated multi-area reheat thermal power system which has been developed and implemented using MATLAB/Simulink environment. The designed power system is simulated by considering one percent step load perturbation (1 % SLP) in area 1 and PID controller to measure the proposed algorithm superiority. The optimization algorithms for tuning controller parameters are written as a separate MATLAB file and stored. The simulation results are carried out for two different scenarios: 1) Simulation performed for 100 iterations, 2) Simulation performed for 150 iterations. In these scenarios the gain values of the controller parameters are optimized by considering the GA, PSO and FFA algorithms.

A Scenario (a) With 100 Iterations

The simulation is performed by considering 100 iterations and responses of the under investigation power system is shown in the Figs. 2-16. In these mentioned figures, the solid lines indicate the response of the GA based PID controller equipped power system; the dash-dotted line show the response of the PSO based PID controller equipped power system and the solid bold line illustrates the response of the FFA based PID controller power system.

Figure  2.  The frequency deviation of area 1 for 100 iterations.

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Figure  3.  The frequency deviation of area 2 for 100 iterations.

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Figure  4.  The frequency deviation of area 3 for 100 iterations.

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Figure  5.  The frequency deviation of area 4 for 100 iterations.

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Figure  6.  The frequency deviation of area 5 for 100 iterations.

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Figure  7.  Tie-line power flow from area 1 for 100 iterations.

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Figure  8.  Tie-line power flow from area 2 for 100 iterations.

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Figure  9.  Tie-line power flow from area 3 for 100 iterations.

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Figure  10.  Tie-line power flow from area 4 for 100 iterations.

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Figure  11.  Tie-line power flow from area 5 for 100 iterations.

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Figure  12.  Area control error of area 1 for 100 iterations.

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Figure  13.  Area control error of area 2 for 100 iterations.

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Figure  14.  Area control error of area 3 for 100 iterations.

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Figure  15.  Area control error of area 4 for 100 iterations.

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Figure  16.  Area control error of area 5 for 100 iterations.

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From the previous figures responses, it is clearly shown that the FFA algorithm optimized PID controller equipped power system achieved the least damping oscillations with the fastest settled response, in addition to the minimal overshoot peak. Thus, it provided a superior controlled performance response compared to using the GA and PSO based PID controller equipped power system. It obvious that the frequency deviation of area 1 has the maximum undershoot peak compared to the other areas. While, it has the minimal overshoot peak compared to the other areas. With respect to the tie-line power flow, it is illustrated that area 1 has negative undershoot peak, while the other areas have positive values.

The numerical value comparisons of settling time, peak overshoot and undershoot are given in the Table Ⅰ along with a statistical bar chart comparisons of the peak overshoot, undershoot and settling times using 100 iterations.

Table  Ⅰ.  COMPARISON VALUES OF SETTLING TIME, PEAK OVER SHOOT AND PEAK UNDER SHOOT FOR 100 ITERATIONS

	Settling Time			Peak overshoot			Peak undershoot
	GA	PSO	FFA	GA	PSO	FFA	GA	PSO	FFA
$\Delta F_{1}$	19.12	16.97	14.55	0.00146	0.0015	0.0014	0.0107	0.0103	0.011
$\Delta F_{2}$	21.09	16.3	12.94	0.00068	0.0003	0.00029	0.0048	0.0049	0.0059
$\Delta F_{3}$	22.17	20.01	16.12	0.0001	0.00001	0.00014	0.0053	0.0052	0.0052
$\Delta F_{4}$	21.09	18.92	16.12	0.0005	0.00045	0.0005	0.006	0.0055	0.0053
$\Delta F_{5}$	23.86	20.86	15.67	0.00056	0.00048	0.00012	0.0052	0.0051	0.0056
$\Delta P_{\rm tie1}$	25.83	24.83	24.53	0.0004	0.0003	0.0004	0.0076	0.0072	0.0078
$\Delta P_{\rm tie2}$	19.2	16.75	14.52	0.0005	0.00042	0.00038	0.005	0.0047	0.0059
$\Delta P_{\rm tie3}$	24.76	24.59	24.05	0.0018	0.0018	0.0019	0.00007	0.00005	0.00001
$\Delta P_{\rm tie4}$	17.42	16.36	13.94	0.0006	0.00044	0.00038	0.0048	0.0047	0.0059
$\Delta P_{\rm tie5}$	23.25	21.70	18.92	0.0018	0.0017	0.0019	0.00013	0.00008	0.00017
$ACE_{1}$	17.84	16.33	13.53	0.00071	0.00042	0.00038	0.0048	0.0047	0.0056
$ACE_{2}$	26.63	22.7	21.8	0.00063	0.00065	0.00075	0.00092	0.00099	0.0015
$ACE_{3}$	17.19	16.54	14.55	0.0006	0.00042	0.00038	0.0048	0.0049	0.0059
$ACE_{4}$	24.33	23.23	22.77	0.00089	0.00078	0.00085	0.0019	0.0015	0.0017
$ACE_{5}$	17.19	16.33	13.79	0.0006	0.00042	0.00038	0.0048	0.0047	0.0056

 | Show Table

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B Scenario (b) With 150 Iterations

In this scenario, the gain values of the controller parameters are optimized by considering GA, PSO and FFA algorithms with 150 iterations. The response of the investigated power system is shown in the Figs. 17-31.

Figure  17.  Frequency deviation of area 1 for 150 iterations.

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Figure  18.  Frequency deviation of area 2 for 150 iterations.

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Figure  19.  Frequency deviation of area 3 for 150 iterations.

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Figure  20.  Frequency deviation of area 4 for 150 iterations.

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Figure  21.  Frequency deviation of area 5 for 150 iterations.

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Figure  22.  Tie-line power flow from area 1 for 150 iterations.

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Figure  23.  Tie-line power flow from area 2 for 150 iterations.

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Figure  24.  Tie-line power flow from area 3 for 150 iterations.

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Figure  25.  Tie-line power flow from area 4 for 150 iterations.

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Figure  26.  Tie-line power flow from area 5 for 150 iterations.

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Figure  27.  Area control error of area 1 for 150 iterations.

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Figure  28.  Area control error of area 2 for 150 iterations.

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Figure  29.  Area control error of area 3 for 150 iterations.

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Figure  30.  Area control error of area 4 for 150 iterations.

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Figure  31.  Area control error of area 5 for 150 iterations.

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Figs. 17-31 established that the power system response with FFA based PID controller achieved superior performance compared to PSO and GA based PID controller performance in terms of minimal settling time with lesser damping oscillations and peak over and under shoot values. The numerical value comparisons of the settling time, peak overshoot and undershoot are given in the Table Ⅱ along with statistical bar chart comparisons of the peak overshoot, undershoot and settling times using 150 iterations.

Table  Ⅱ.  COMPARISON VALUES OF SETTLING TIME, PEAK OVER SHOOT AND PEAK UNDER SHOOT FOR 150 ITERATIONS

	Settling Time			Peak overshoot			Peak undershoot
	GA	PSO	FFA	GA	PSO	FFA	GA	PSO	FFA
$\Delta F_{1}$	15.48	14.62	14.12	0.0017	0.0014	0.0011	0.0117	0.0114	-0.009
$\Delta F_{2}$	14.82	14.2	14.12	0.00051	0.00074	0.00048	0.0061	0.0063	0.0052
$\Delta F_{3}$	23.62	11.16	18.16	0.00032	0.00079	0.00023	0.0062	0.0062	0.0044
$\Delta F_{4}$	14.62	13.83	13.6	0.0012	0.00068	0.00049	0.0067	0.0064	0.0049
$\Delta F_{5}$	20.86	16.37	16.44	0.00069	0.00082	0.00047	0.0071	0.0058	0.0051
$\Delta P_{\rm tie1}$	24.32	23.2	23.1	0.00014	0.000016	0.00011	0.0089	0.0085	0.0069
$\Delta P_{\rm tie2}$	18.3	17.23	20.8	0.002	0.0021	0.0017	0.00009	0.000032	0.00007
$\Delta P_{\rm tie3}$	27.55	26.01	25.81	0.0021	0.0021	0.0015	0.000045	0.000056	0.0000056
$\Delta P_{\rm tie4}$	22.36	20.23	21.03	0.0023	0.0021	0.0017	0.000079	0.000047	0.000016
$\Delta P_{\rm tie5}$	19.78	25.81	24.32	0.0024	0.002	0.0017	0.0000016	0.00002	0.00002
${ACE}_{1}$	21.84	21.74	20.8	0.0008	0.00047	0.0001	0.011	0.0111	0.0091
${ACE}_{2}$	21.53	20.38	19.38	0.0007	0.00084	0.00075	0.0012	0.0018	0.0014
${ACE}_{3}$	26.9	22.82	21.3	0.00073	0.0008	0.00057	0.0012	0.0015	0.00071
${ACE}_{4}$	25.81	25.01	24.12	0.00088	0.00083	0.0007	0.0019	0.0017	0.0012
${ACE}_{5}$	18.87	23.89	18.16	0.00095	0.00071	0.00073	0.0022	0.0011	0.0014

 | Show Table

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Statistical calculations are carried out to check the supremacy of proposed FFA optimization technique. A comparison of PID controllers performance designed on the basis of three different optimization techniques, namely FFA-PID, GA-PID and PSO-PID with respect to the average values of the settling time is illustrated in Figs. 32-39. Fig. 32 illustrates the bar chart of the peak overshoots using 100 iterations based PID controller performance with GA, PSO and FFA optimization algorithms.

Figure  32.  Bar chart comparisons of peak overshoots using 100 iterations.

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Figure  33.  Bar chart comparisons of peak undershoots using 100 iterations.

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Figure  34.  Bar chart comparisons of settling time for 100 iterations.

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Figure  35.  Bar chart comparisons of peak overshoot for 150 iterations.

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Figure  36.  Bar chart comparisons of peak undershoot for 150 iterations.

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Figure  37.  Bar chart comparisons of settling time for 150 iterations.

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Figure  38.  Bar chart comparisons of the improvement percentage ($y$-axis) for proposed FFA over GA and PSO for 100 iterations.

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Figure  39.  Bar chart comparisons of the improvement percentage ($y$-axis) for proposed FFA over GA and PSO for 150 iterations.

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Fig. 32 established that in terms of the $\Delta F_1$, $\Delta F_2$, $\Delta F_5$, $\Delta P_{\rm tie2}$, $\Delta P_{\rm tie4}$, $ACE_1$, $ACE_3$ and $ACE_5$ values, the proposed FFA based controller has effectively reduced the peak overshoot compared to the other optimization algorithms.

Fig. 33 illustrates the bar chart comparison with respect to the peak undershoots using 100 iterations based PID controller performance with GA, PSO and FFA optimization algorithms. Fig. 33 depicted that in terms of the $\Delta F_3$, $\Delta F_4$ and $\Delta P_{\rm tie3}$, the proposed FFA based controller has effectively reduced the peak undershoot compared to the other GA-PID and PSO-PID approaches. Fig. 34 illustrates the bar chart comparison of the settling time using 100 iterations based PID controller performance with GA, PSO and FFA optimization techniques.

Fig. 34 indicated that in terms of the $\Delta F_1$, $\Delta F_2$, $\Delta F_3$, $\Delta F_4$, $\Delta F_5$, $\Delta P_{\rm tie1}$, $\Delta P_{\rm tie2}$, $\Delta P_{\rm tie3}$, $\Delta P_{\rm tie4}$, $\Delta P_{\rm tie5}$, $ACE_1$, $ACE_2$, $ACE_3$, $ACE_4$ and $ACE_5$, the proposed FFA based controller provided the fastest settled response compared to other optimization techniques based controller for the same investigated power system.

Meanwhile, the comparison values in Table Ⅱ are statistically illustrated for 150 iterations in Figs. 35-37. Fig. 35 illustrates the bar chart comparisons of peak overshoots for 150 iterations based PID controller performance with GA, PSO and FFA algorithm techniques.

Fig. 35 depicts that with respect to the $\Delta F_1$, $\Delta F_2$, $\Delta F_3$, $\Delta F_4$, $\Delta F_5$, $\Delta P_{\rm tie2}$, $\Delta P_{\rm tie3}$, $\Delta P_{\rm tie4}$, $\Delta P_{\rm tie5}$, $ACE_1$, $ACE_3$ and $ACE_4$, the proposed FFA based controller has effectively reduced the peak overshoot. Fig. 36 demonstrates the bar chart comparison of the peak undershoots using 150 iterations based PID controller performance with GA, PSO and FFA algorithm techniques.

Fig. 36 illustrates the superiority of the proposed FFA-PID controller system with the $\Delta F_1$, $\Delta F_2$, $\Delta F_3$, $\Delta F_4$, $\Delta F_5$, $\Delta P_{\rm tie1}$, $\Delta P_{\rm tie3}$, $\Delta P_{\rm tie4}$, $\Delta P_{\rm tie5}$, $ACE_1$, $ACE_3$ and $ACE_4$, where it effectively reduces the peak undershoot. Fig. 37 demonstrates the bar chart comparison of the settling time using 150 iterations based PID controller performance with GA, PSO and FFA algorithm techniques.

Fig. 37 clearly depicts that in $\Delta F_1$, $\Delta F_2$, $\Delta F_4$, $\Delta P_{\rm tie1}$, $\Delta P_{\rm tie3}$, $ACE_1$, $ACE_2$, $ACE_3$, $ACE_4$ and $ACE_5$, the proposed FFA based controller provided fast settled response compared to other optimization techniques based controller performance in the same investigated power system.

Generally, a comparison chart of the percentage improvement of the firefly algorithm based PID controller performance over GA and PSO optimized PID controllers' response for the same investigated power system for 100 iterations is shown in Fig. 38.

It is clearly shown that the overall performance of the proposed system for all areas is improved compared to GA and PSO optimization techniques. Additionally, Fig. 39 demonstrates comparison chart of the percentage improvement of firefly algorithm based PID controller performance over GA and PSO optimized PID controllers' response for the same investigated power system for 150 iterations.

Fig. 39 establishes that in all areas the performance of the proposed controlled system is improved compared to GA and PSO optimization techniques except the responses of $\Delta F_3$, $\Delta P_{\rm tie2}$, $\Delta P_{\rm tie4}$, $\Delta P_{\rm tie5}$ for the investigated power system.

Furthermore, a computational time comparison is done between the proposed FFA-PID controller and the GA-PID and PSO-PID controllers as shown in Table Ⅲ.

Table  Ⅲ.  COMPARISON VALUES OF THE COMPUTATIONAL TIME OF THE THREE ALGORITHMS

	Execution time (min)
Iteration	GA	PSO	FA
100	12.4235	42.3184	4.1594
150	15.2182	61.4776	6.2194

 | Show Table

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Table Ⅲ proves that the computational time using the proposed FFA-PID is superior to the ones obtained with the GA-PID and the PSO-PID as shown in the above table when using 100 or 150 iterations.

It is established from the above results that with increasing number of iterations, the settling time using the FFA is slightly increased, while it is decreasing both the peak overshoot and undershoot efficiently. With 150 iterations, the FFA settling time, peak overshoot and undershoot values are superior to those obtained using GA and PSO.

Consequently, the preceding results depicted the superiority of the PID-FFA over the PID-GA and the PID-PSO. Since, the differential evolution (DE) is a popular optimization method, thus it is recommended to compare the proposed system performance using FFA-PID to the DE-PID controller performance.

Ⅵ. CONCLUSION

The proposed work studied and developed an automatic generation control (AGC) of five area interconnected equal reheat thermal power system by considering proportional-integral-derivative (PID) controller. The controller parameters, namely the proportional gain ($K_{p})$, integral gain ($K_{i})$ and Derivative gain ($K_{d})$ are tuned by using nature bio-inspired firefly algorithm (FFA) considering one percent step load perturbation (1\, ${%}$ SLP) in area 1. The performance of proposed algorithm was compared to both the genetic algorithm (GA) and the particle swarm optimization (PSO) technique based PID controller response. The superiority of the proposed algorithms was tested with changing the number of iterations of each algorithm. Finally, the simulation result obviously demonstrated that the FF algorithm tuned PID controller gained superiority (lesser damping oscillations, minimal settling time with less peak overshoot/undershoot values) compared to using the GA and PSO tuned controller performance. It is established that the FFA performance is improved with increasing the number of iterations. However, the proposed approach required to solve the tradeoff between increasing the number of iterations which lead to the increase in the settling time with decreasing the peak overshoot and undershoot.

Appendix

Nominal parameters of the five area interconnected reheat thermal power system are