Modified PID controller for automatic generation control of multi-source interconnected power system using fitness dependent optimizer algorithm

In this paper, a modified form of the Proportional Integral Derivative (PID) controller known as the Integral- Proportional Derivative (I-PD) controller is developed for Automatic Generation Control (AGC) of the two-area multi-source Interconnected Power System (IPS). Fitness Dependent Optimizer (FDO) algorithm is employed for the optimization of proposed controller with various performance criteria including Integral of Absolute Error (IAE), Integral of Time multiplied Absolute Error (ITAE), Integral of Time multiplied Square Error (ITSE), and Integral Square Error (ISE). The effectiveness of the proposed approach has been assessed on a two-area network with individual source including gas, hydro and reheat thermal unit and then collectively with all three sources. Further, to validate the efficacy of the proposed FDO based PID and I-PD controllers, comprehensive comparative performance is carried and compared with other controllers including Differential Evolution based PID (DE-PID) controller and Teaching Learning Based Optimization (TLBO) hybridized with Local Unimodal Sampling (LUS-PID) controller. The comparison of outcomes reveal that the proposed FDO based I-PD (FDO-I-PD) controller provides a significant improvement in respect of Overshoot (Osh), Settling time (Ts), and Undershoot (Ush). The robustness of an I-PD controller is also verified by varying parameter of the system and load variation.


Introduction
The electrical power system consist of highly complex structures having various network of varied loads are interconnected. The basic purpose of AGC is to provide desired amount of power within satisfactory quality to entire users. The system will be stable when there is an equilibrium between generated power and consumers load. Since, the consumers load normally changes, the active power drawn from the generator increases which reduce the speed of generator or turbine due to variation in frequency. The modern power system comprises of numerous network areas which are connected to transmission lines via tie-lines. AGC plays a key role to sustain the exchange of power TLBO-I-PD.
The respite of the research work is structured as follows: Section 2 represents the material and methods followed by controller structure and optimization technique, While in section 3 implementation and results are demonstrated. The presented research work is concluded in section 4.

Controller Structure
In early AGC method, the integral controller had been used to control the system frequency and tie-lines power. However, due to its slower time response, the researchers used Proportional Integral (PI) controller which have the advantages of its low cost, simple structure and faster time response. The poor dynamic performance of PI controller has been improved by PID controller and their modified form I-PD controller which are nowadays very commonly used in practical [33]. In I-PD controller, the proportional parameter and the derivative parameter are put in feedback form while, integrator parameter are put in feed forward direction, as depicted in  The input of PID /I-PD controller for area 1 and 2 is specified by Area Control Error (ACE).
where β 1 and β 2 represents biased parameters of frequency for area-1 and 2 respectively. ∆F 1 and ∆F 2 represents the frequency variation for area-1 and 2 respectively. Similarly, ∆P tie12 shows the tie line variation from area-1 to area-2 and ∆P tie12 represent the tie line variation from area-2 to area-1.
To optimize the parameters of a controller, one of the essential steps is to determine the objective function. In this paper four different performance criteria, i.e., ITSE, ISE, ITAE and IAE are applied to verify the system performance and are given in below equations.

Fitness Dependent Optimizer
Several optimization methods are introduced by researchers in the field of power system to optimize the gains of the controllers by determination of fitness function. A nature-inspired metaheuristic computational technique known as Fitness Dependent Optimizer (FDO) is deployed in this work to tune the various gains of the proposed controller. FDO starts from the initialization population of scout randomly in search space K m ; m = 1, 2, 3, . . . . . . , n. The number of the scout bees were equal to population size and each scout contains three parameters known as K p , K i and K d denote the gains of PID/I-PD Controller. Each scout bee denotes a new exposed hive (solution). Scouts bee are randomly searching more position to find best solutions. A previously discovered hive is ignored when new search space is find out. So each time the algorithm finds a better new solution the previous one is discarded. Moreover, if the scout bee does not find the best solutions by moving forward, then it comes back to its previous direction hoping for optimal solutions. However, if the prior solution is unable to provide a better new solution then the existing solution will be considered as the best solution that has been discovered yet. During random movement of the scout bees, each time adding pace to the present position, the scout hopes to determine the best solution. The results are compared with global best and hence it is repeated until the optimal solution not achieved or generation is stopped The movement of scouts is represented as below: where m denotes current search agent, t denotes current iteration, K indicate scout bees and pace p is the movement rate and direction. Generally, pace p depends on Fitness Weight w f and can be articulated as follows: where k * m,t,f represents the value of the best global solution, k m,t,f is the value of current solution and α is a weight factor which is used for the controlling of w f and its value is either 1 or 0. If the value of α is 1 then, it represent high level of convergence while if the value is 0 then there is no effect on equation (4) but often it provides more stable search. The value of w f should be in the range of [1,0]. But in some cases it may be equal to 1, for instance, when the current and global best solution are equal. The value of w f will be equal to 0 when k * m,t,f is equal to 0. Finally, the case k m,t,f = 0 should be ignored. Hence, the rules given below must be considered: where R is the random number in the range of [1, −1]. The elementary steps of the FDO are shown in the algorithm 1.

Algorithm 1 : Fitness Dependent Optimizer
Cost function J(.); Generate scout bee population k m,t ; m = 1, 2, · · · , n While Boundary not reached do for all For each scout bee k m,t do  [15,29], ISE [24,25] and ITAE [11,13] are mostly used for AGC. For the comparison among various performance criteria Eq (2a)-Eq (2d) are executed in Matlab and identified that ITSE provides minimum error as compared to others criteria which are show in Table 2. Hence, ITSE criteria is preferred as an objective function to tune the gains of controller. Further, for the sake of comparison three other methods such as PSO, TLBO and FA are used for tuning of I-PD/ PID controller. The convergence rate for different methods using ITSE criteria is depicted in  Table 11. For simulation a population of 30 and generation of 60 numbers are considered. The optimization was performed by 30 times for each algorithm and the best optimal gains are picked during optimization which are specified in the Table 1, 4, 6, and 8 for reheat thermal unit, hydro, gas and multi-source unit respectively.     Table 3.    Table 3 reveal that FDO-PID controller completely eliminates overshoot O sh as compared to PSO, FA and TLBO based PID controller which is dire need of a controller for the system stability.  Table 3.

Two Area Hydro Power System
The transfer function(TF) diagram for hydro power unit with two area is shown in Fig  11 which consist of the Transfer Function (TF) of hydro governor   The TF model of two-area hydro power system have been assessed with FDO-PID and FDO-I-PD. The results are compared with PID and I-PD with other tuning techniques i.e.PSO, TLBO and FA. The results obtained from two-area hydro power system with PID and I-PD controllers are given in Fig (12-17).    Table 5. The representation of bold values indicates the best results.

Two-Area Gas Power System
The Transfer Function (TF) diagram of two area gas power generation is show in Fig 18. The two-area are interconnected by tie lines whereas, each area indicate the TF of valve position, speed governor, compressor discharge and fuel with combustion reaction. K g , R g1 and R g2 denote the gas constant, and gas droop constant for area 1 and 2 respectively. The TF model of two-area gas power system are established in Matlab/Simulink using Appendix Table 10. The system is assessed with FDO base optimized PID/I-PD controllers to evaluate the achievement of the proposed techniques and their outcomes are compared with some recent optimization algorithms like TLBO, FA and PSO. The results obtained from two-area gas power system for area 1 and 2 with PID and I-PD controllers are given in Fig (19-24). Table 6. Optimum gains of PID /I-PD controllers optimized with different methods for gas generation unit.
Controller with Techniques Gas Power system

Two-Area Multi-Source Interconnected Power System
The Transfer Function (TF) diagram for multi-source with two-area IPS are depicted in Fig 25. Various power generation sources like reheat thermal, gas power and hydro     Fig (26-31).

Sensitivity Analysis
The robustness of an I-PD controller is verified by varying the system parameters of two areas multi-generation system within a range of ± 25%. Fitness dependent algorithm (FDO) is employed to check the performance of the proposed controller by varying some parameters of the system from their nominal values such as turbine constant (T t ), droop constant R and governor constant (T g ). The results yielded from sensitivity analysis for ∆F 1 , ∆F 2 , and ∆P tie with variation in T t , R and T g is depicted in Figure 33-41. It can be observed from results that our proposed controller provides robustness by changing system parameters T t , R and T g for ∆F 1 , ∆F 2 , and ∆P tie respectively within a range of ± 25%. Further, it can be also observed that optimal values of controller gains need not be re-tuned for a wide change of system parameters and load conditions.