Robust Load Frequency Control of Hybrid Solar Power Systems Using Optimization Techniques

Abstract

It is necessary to predict solar photovoltaic (PV) output and load profile to guarantee the security, stability, and reliability of hybrid solar power systems. Severe frequency fluctuations in hybrid solar systems are expected due to the intermittent nature of the solar photovoltaic (PV) output and the unexpected variation in load. This paper proposes designing a PID controller along with the integration of a battery energy storage system (BESS) and plug-in hybrid electric vehicle (PHEV) for frequency damping in the hybrid solar power system. The solar PV output is predicted with high accuracy using artificial neural networks (ANN) given that solar irradiance and cell temperature are inputs to the model. The variation in load is also forecasted considering the factors affecting the load using ANN. Optimum values of the PID controller have been found using genetic algorithm, particle swarm optimization, artificial bee colony, and firefly algorithm considering integral absolute error (IAE), integral square error (ISE), and integral time absolute error (ITAE) objective functions. IAE, ISE, ITAE, Rise time, settling time, peak overshoot and maximum frequency deviation have been measured for comparison and effectiveness. The transient behavior has been further improved by utilizing the power from BESS/PHEV to the power system. The results demonstrate the efficacy of the suggested design for frequency control using the genetic algorithm method along with ISE objective function compared with those obtained from the conventional, particle swarm optimization, artificial bee colony, and firefly algorithm techniques.

Introduction

Solar photovoltaic (PV) systems are intermittent in nature. The variation in generated power and load demand is common in solar PV connected power systems. Under normal operating situations, microgrids are powered by the integration of both solar PV systems and the power grid. However, when the output from the solar PV system is lower than expected, and the load demand is high, the power grid may not be able to support enough power because of its slow dynamic response. This will cause the power system to be unstable which leads to severe frequency fluctuations in the system (Sadat, 2012).

Hybrid solar power systems are a source for generating electrical power that use inverters coupled with batteries to store energy later use. In this way, hybrid solar systems will be able to use the stored energy during blackouts and compensate for renewable energy sources output fluctuation (Newkirk, 2015). Integrating microgrids with renewable energy systems is considered an adequate solution to meet the increasing demand of electric power. Such microgrids can utilize plug-in hybrid electric vehicles and battery energy storage systems to maintain the balance between generated power and the load demand. Due to the intermittent nature of renewable energy sources, the change in generation power and load demand would lead the system to instability causing severe frequency fluctuations. Mathematical models were developed to predict the behavior of microgrids in order to dampen frequency fluctuations. Power systems with a high penetration of renewable energy sources were integrated with battery energy storage system (BESS) and plug-in hybrid electric vehicles to maintain the frequency deviations within limits (Akula, 2019).

Mellit (Mellit and Pavan, 2010) uses artificial neural network methods for forecasting the electric power produced by a 20 grid-connected PV system placed on a rooftop in the municipality of Trieste, Italy. According to Huang (Huang et al., 2016), improved the solar photovoltaic (PV) output by using the solar irradiance and temperature along solar zenith angle and solar azimuth angle with the help of artificial neural network methods. A data-driven ensemble approach was used by Al-Dahidi in (Al-Dahidi et al., 2019) to predict the day ahead solar photovoltaic (PV) output (230 capacity) placed on the top of the Applied Science Private University, Amman, Jordan to overcome the intermittent nature of solar energy system.

Hote (2018) studied the PID controller calibrating approaches for load frequency control of Power systems. The prime challenge in LFC was to design a PID controller that could maintain the frequency fluctuations rigorously within the defined limits. Fractional Order PID controller was tuned by Taher (2014) for load frequency control for three area power system with the help of imperialist competitive algorithm (ICA). The response of the interconnected power system towards disruption was smooth and less vibratory as a result of using the proposed controller. Mosaad (Mosaad and Salem., 2014) presented a methodological design for an adaptive PID load frequency using artificial neural networks (ANN) and adaptive Neuro-Fuzzy Inference systems (ANFIS). The PID Controller was able to maintain the effective performance at any load point in the power system with the superiority of ANFIS over ANN with respect to integral absolute error (IAE), integral square of errors. According to Otani (2017), there are possible chances of frequency fluctuations after losing balance between power generated and load demand. Battery storage was incorporated because of its faster response to overcome load frequency deviations using recurrent neural networks.

The desired frequency of the power system deviates due to disturbances in the electric power generation and load fluctuations. Load frequency control has been considered the main issue for power system operation and control for effective and reliable power supply (Sundaram and Jayabarathi, 2011). PI, PID and fuzzy controllers have been used to stabilize the power system when it is integrated with wind power. Kumari et al. (2016) introduce the ANN-based PID controller to sufficiently damp frequency fluctuations produced as a result of load changes. An artificial intelligent controller has been designed using the backpropagation algorithm to reduce the area control error of a two-area hydrothermal power system for automatic generation control purposes in (Rao, 2012).

Meta-heuristic optimization techniques have been employed to get the optimal gains of PID controller for LFC problem. According to (Das et al., 2010), genetic algorithm based PID controller was tuned for autonomous hybrid generation system. Fractional order PID controller was tuned using genetic algorithm for hybrid power system integrated with renewable energy sources (Regad et al., 2019). Multi-objective artificial bee colony technique was adopted to tune the PID controller parameters with the help of integral time multiplied absolute error (ITAE) and integral of time weighted squared error (ITSE) for two area thermal power system (Naidu et al., 2014). Particle swarm optimization has been utilized to tune the fuzzy logic controller for the frequency restoration of multi-area power system (Jaber et al., 2013). PSO is used for obtaining the optimal PID parameters for two area system with integral square error as cost function and showing best convergence profile (Nagarjuna and Shankar, 2015). PID optimal parameters obtained for two and three area power system using firefly algorithm considering the generation rate constraint and governor dead band (Padhan et al., 2014).

Energy storage systems are also used in previous studies for frequency regulation. The role of battery energy storage system has been presented for regulating the frequency in interconnected power system. The results are compared to the conventional PID controller that describe BESS as efficient for LFC purpose (Kalyani et al., 2012). Peak frequency deviation and settling time of two area power system is reduced using the battery energy storage system and choosing the proper integral gain (Aditya and Das, 2001). Plug-in electric vehicles power is utilized for frequency stability of the islanded power system with the help of controllable power rate strategy (Qi et al., 2018). The energy from electric vehicles that are charged during daytime can be utilized in the night-time for small residential power system (Takagi et al., 2009).

The purpose of this study to address the limitations in the previous studies and fulfill the gap. Many load frequency control techniques have been applied to power system for its stability purposes, i.e., robust control, decentralized aspect, linear quadratic, pole shifting, and variable structure. But there have been some drawbacks of these techniques which decrease their execution efficiency. To overcome such problem, artificial intelligence techniques, fuzzy logic (FL) and neural network (NN) have been adopted and applied to solve the non-linear models of the power system. These techniques have proved their effectiveness and efficiency to solve the problem. But they still have some disadvantages. For example, it is hard to train the neural network again and again to set the number of neurons and other parameters to get the best output response. Similarly, fuzzy logic requires hard work to get the influential signal response from the power system (Abd-Elazim and Ali, 2018). Optimization techniques such as genetic algorithm (GA) (Milani and Mozafari, 2009) (Chang et al., 1998; Rerkpreedapong et al., 2003; Das et al., 2010; Das et al., 2012; Mallesham et al., 2012; Konar et al., 2014; Regad et al., 2019; Hemeida et al., 2020; Sidi Brahim et al., 2021), particle swarm optimization (PSO) (Pain and Acharjee, 2014; Abd-Elazim and Ali, 2018) (Gözde et al., 2008; Selvakumaran et al., 2012; Modi et al., 2013; Rao and Rama Krishna Reddy, 2015; Shankar et al., 2015; Hazlee Azil et al., 2016; Jeyalakshmi and Subburaj, 2016; Singh and Ramesh, 2019; Hemeida et al., 2020; Veerasamy et al., 2020; Safari et al., 2021) (Sidi Brahim et al., 2021) (Mallesham et al., 2012)- (Kumari and Jha, 2014), firefly algorithm (FA) (Naidu et al., 2013; Shakarami et al., 2013; Padhan et al., 2014; Chandra Sekhar et al., 2016; Abd-Elazim and Ali, 2018; Boddepalli and Navuri, 2018; Gupta et al., 2021), and artificial bee colony (Ghasemi and Shayeghi, 2011; Rathor et al., 2011; Gozde et al., 2012; Naidu et al., 2014; Elsisi et al., 2015; Kouba et al., 2015; Kumar et al., 2017; Abo-Elyousr, 2018), have been used and found to be effective for load frequency control problem. But proportional, integral and derivative gain range of PID controller during optimization problem has been narrowed. Renewable energy source, photovoltaic and wind turbine, transfer function model has been adopted in the previous studies. In the same way, load frequency control has also been done in previous studies using the energy storage systems but without efficient PID controller that ultimately require large energy storage capacity to damp the frequency (Kalyani et al., 2012)- (Aditya and Das, 2001). Taking into account the mentioned limitations, the current study discusses and solve the problem by taking real-time data to train the network for solar photovoltaic prediction, load deviation forecasting, widening proportional, integral, and derivative controller gain range, and integrating the BESS/PHEV with efficient tuned PID controller.

Power System Model

Power system control concentrate on steady state operation. This study presents how to deal with active power to keep the power system in steady state. The major objective of the control strategy is to provide qualitative and reliable power to customers within an interconnected system. Changes in the active power disturb the frequency of the system. Therefore, a control strategy is designed to regulate load frequency control using control loops. Two common approaches, transfer function and state variable, are adopted to transform the power system model into a mathematical model by making some proper assumptions (Prakash and Sinha, 2012; Pain and Acharjee, 2014; Prajapati and Parmar, 2016; Azeer et al., 2017; Lone et al., 2018; Yang-Wu et al., 2019). A single area power system model is shown in Figure 1.

FIGURE 1

Generator Model

The generator equation has been extracted from the swing equation as shown in Eq. 1.

Load Model

The power system contains resistive and inductive load that is frequency independent and dependent respectively (Sadat, 2012). Therefore, the net change in load power can be described as the combination of frequency sensitive and frequency non-sensitive load changes shown in Eq. 2.Where is the frequency independent load change and denotes the frequency-sensitive load change and represent the ratio of percent change in load to the percent change in frequency. So, the relationship for variation in load with respect to frequency fluctuation can be expressed as,

Frequency dependent load change has been observed from the Atlas Power Plant. The Atlas power plant is a coal fired thermal power plant located at the geographic location (Longitude:31.5204, Latitude:74.3587). The historical load demand data has been extracted from the power data reference book produced by the National Transmission and Dispatch Company Limited (NTDCL) (Power Data Reference Book, 2017). Historical daily load data from 2011 to 2016 has been taken as input along with weather and seasonal data. Seasonal and weather data consists of wind speed, surface pressure, temperature, precipitation factor, and humidity. Average load has been calculated for 365 days from load profile data between 2011 and 2016 (Alessandra et al., 2011). Frequency dependent change in load has been extracted from the average load for consecutive 365 days as shown in Figure 2. This change in load power has been taken as an output (Kotur and Žarković, 2016).

FIGURE 2

The weather data has been first normalized by taking the maximum value as base from each factor affecting the load and dividing the entirety of the weather data with base value within the range (0–1). Similarly, historical data has also been normalized by dividing with the base value 219 MW and converting the load data within the range of (0–1). After normalizing the input and target data, NNTOOL is used to train the network. A multi-layer feed forward neural network was trained using wind speed, surface pressure, temperature, precipitation factor, humidity, and historical load data as input and deviation in load power data as target as shown in Figure 1 of the load deviation model (Srinivasan et al., 1991). The number of neurons has been chosen as 10, Tan sigmoid activation function, and Levenberg Marquardt as learning algorithm as shown in Table 1. The trained model depicts its effectiveness showing the coefficient of regression as 1.0 and mean squared error (MSE) as at 525 epochs. After training model, the network model was deployed in Simulink, where the neural network can successfully forecast the deviation in load power.

Prime Mover Model

It is the origin of mechanical power whose energy is obtained from burning coal or gas, or nuclear fission. The transfer function of the turbine can be represented as the ratio of the change in mechanical output power to the change in steam valve position as represented by Eq. 4.Where is the turbine time constant.

Governor Model

The speed governor model can be expressed as Eq. 5.The speed governor operates as a comparator and its output can be written as difference between the reference set power and power , where R indicates the speed regulation. The relation between governor input and valve opening can be expressed by Eq. 6 (Kumari et al., 2016).Where is taken as the governor time constant in seconds.

Solar Photovoltaic Model

It is fact that photovoltaic output power is intermittent in nature. It depends on two main factors: solar irradiance and temperature (Rodríguez et al., 2018). Solar photovoltaic power is directly proportional to solar irradiance while inversely proportional to temperature (Abd-Elazim and Ali, 2018). Solar photovoltaic model of 5 MW is considered at location (Longitude:31.5204, Latitude:74.3587) near the thermal power plant. Solar PV system is integrated to the thermal power plant. The solar power plant is built by LONGI solar modules of power 540W. LONGI solar module characteristics are shown in Table 2. The solar PV module output (W) was considered at different solar irradiance (ranges 100–1100 ) and temperature (0°C–55°C). These changes were applied to the whole 5 MW power plant to see the effect of intermittency on the power plant. The solar PV generated power 5 MW has been normalized using the base power of 219 MW according to the thermal power plant. A multi-layer feed forward neural network was trained using solar irradiance and temperature as input data and solar power as target data (Sedaghati et al., 2012; Ncane and Saha, 2019). The number of neurons is chosen to be 10, Tan sigmoid activation function, and Levenberg Marquardt as learning algorithm as shown in Table 3. The trained model depicts its effectiveness showing the coefficient of regression as 1.0 and mean squared error as at epochs 22. After successfully training the model, the network model was deployed using Simulink, where neural network can successfully forecast the solar power using the input data taken from the National Aeronautics and Space Administration (NASA). The peak value of solar irradiance and its corresponding temperature at a specific time of the day is considered to forecast the solar photovoltaic output power for that day.

Battery Energy Storage System/Plug-in Hybrid Electric Vehicle Model

Energy storage systems are a great source of frequency damping in the power system. As discussed in the introduction section, power from battery energy storage systems and plug-in hybrid electric vehicle play an important role for maintaining the frequency. Energy from PHEV can be utilized for small residential systems. The circle of willingness of customers for utilizing PHEV can be enhanced. Figure 1 consists of the control system model for a battery energy storage system or plug-in hybrid electric vehicle integrated to the power system. The term depicts the mechanism of auto balancing charging loop. This close-loop control will maintain the energy of the BESS/PHEV near the initial state, which is considered 50% charged. is the feedback gain from the frequency fluctuation in the power grid. represent the first order conversion delay from DC to AC of the BESS/PHEV (Uehara et al., 2009; Liang et al., 2012).

PID Controller Model for Optimization Problem

The PID Controller model has been adopted in the power system model as shown in Figure 1. The transfer function of the controller is as follows.

The control signal for maintaining the system frequency is given by Eq. 8 (Abd-Elazim and Ali, 2018).where is the area control error of the power system and is the input signal to the governor for controlling the valve output according to load demand of the power system.

Area Control Error (ACE) of studied single-area power system can be written by Eq. 9. is called bias factor and denoted with , and represents the frequency deviation.

Multi-objective function is used in (Naidu et al., 2014) for load frequency control using the artificial bee colony optimization approach considering ITAE and ITSE. It has been observed the performance of the PID controller can be compromised, i.e., Either frequency deviation or settling time is compromised on the cost of each other. Therefore, it is recommended to use objective functions separately. But, the most suitable objective function should be identified based on the performance indices and convergence profile.

Three objective functions are used to check the performance indices of the PID controller, which are the integral absolute error (IAE), integral square error (ISE), and integral time absolute Error (ITAE) and given by Eqs 10–12.Where is area control error (ACE).

Frequency deviation permissible limit is calculated using the following steady-state equation (Sadat, 2012).

The maximum steady-state frequency deviation in Hz has found to be −0.7932 Hz for maximum positive load change and 0.5468 Hz for maximum negative load change for studied model. So, the frequency deviation permissible limit range should be (−0.7932, 0.5468) in Hz.

Proposed Approaches for Tuning the PID Controller

Particle Swarm Optimization

Particle swarm optimization technique was developed by Kennedy and Eberhart in 1995. They used nature inspired optimization algorithms in their technique. In this technique, particles are flown through the search space and update the position of the particle at time step of . The expression for the velocity updates is given by the Eq. 14.

This technique is adopted to find the optimal values of the PID controller parameters. Initially, the PID controller proportional gain , integral gain , and derivative gain optimal values have been searched in the range (−10, 1000). The searched controller gains produce insufficient closed-loop stability of the power system in this range. Therefore, the gain parameters range has been narrowed iteratively until closed-loop stability condition of power system model is achieved within the range (−10, 150). PID controller produces best results and minimizes the error significantly within the selected range (−10, 150) as compared to very short range (0, 5) (Pain and Acharjee, 2014; Abd-Elazim and Ali, 2018) (Gözde et al., 2008; Selvakumaran et al., 2012; Modi et al., 2013; Rao and Rama Krishna Reddy, 2015; Shankar et al., 2015; Hazlee Azil et al., 2016; Jeyalakshmi and Subburaj, 2016; Singh and Ramesh, 2019; Hemeida et al., 2020; Veerasamy et al., 2020; Safari et al., 2021) (Sidi Brahim et al., 2021) (Mallesham et al., 2012)- (Kumari and Jha, 2014). Table 4 depicts the PSO operators used to find out the optimal PID controller parameters. The best performing PID controller parameters are mentioned in Table 5.

Genetic Algorithm

Genetic Algorithm is an optimization technique based on heredity and evolution. This technique was suggested by John Holland in 1960 to search for the best solution of complex problems. It is an iterative process which maintains the constant population size of desired solution. GA initially begins with a randomly selected population of function input that is represented as a bit of strings. The Population is evaluated in each iterative step called generations to give a new population of the desired solution. That means GA uses the current population to produce a new population such that the new population is better on average if compared to the previous population. Best elements are used from the current population to form the best population. The successful process will produce a population better than the old one. Three steps selection, mating, and mutation are used to produce new population from the old population. The basic genetic algorithm cycle is shown in Figure 3. New generations are produced iteratively with the repetition of these three steps. The process continues until the stopping criteria is reached like the maximum number of iterations is achieved or no improvements (Milani and Mozafari, 2009) (Chang et al., 1998; Rerkpreedapong et al., 2003; Das et al., 2010; Das et al., 2012; Mallesham et al., 2012; Konar et al., 2014; Regad et al., 2019; Hemeida et al., 2020; Sidi Brahim et al., 2021).

FIGURE 3

This technique is adopted to find the optimal values of the PID controller parameters. The PID controller proportional gain , integral gain , and derivative gain optimal values have been searched in the range (−10, 150). Different parameters and functions for applying GA technique are used in OPTIMTOOL toolbox of MATLAB and given in Table 4. The optimal parameters obtained are depicted in Table 5.

Firefly Algorithm

The flashing light of fireflies is of great importance to study their pattern and rhythmic movement. Studies discussed the main aim of pattern of flashes is either to communicate with the mating partners, to attract the potential prey or to signal other fireflies for potential safety warning. This signal consists of rhythmic flash, rate of flash and amount of flashing. This behavior of flashing light from fireflies can be used to solve many optimization problems. This algorithm has been developed by Xin-She Yang in 2008.

The Euclidian distance between two fireflies and with respect to their positions and can be found with the flowing Eq. 15Where denotes the element of the geographical coordinates. The attractiveness between fireflies can be denoted with the following attractiveness parameter. is called the coefficient of absorption. It is used to control the flashing light concentration. Fireflies’ positions and movement can be described by the following equations. indicates the instantaneous position of the firefly whereas represents the random behavior of firefly when the firefly can’t see another brighter firefly (Naidu et al., 2013; Shakarami et al., 2013; Padhan et al., 2014; Chandra Sekhar et al., 2016; Abd-Elazim and Ali, 2018; Boddepalli and Navuri, 2018; Gupta et al., 2021).

The PID controller proportional gain , integral gain , and derivative gain optimal values have been searched in the range (−10, 150) using firefly algorithm. Firefly algorithm operators used to find out the PID gain values are given in Table 4 and obtained optimal PID parameters in Table 5.

Artificial Bee Colony

Artificial bee colony is meta-heuristic optimization algorithm developed by Karaboğa in 2005. Honey bees foraging behavior has been utilized in this technique searching for nectar at different positions. Usually, three kind of honey bees are considered to be responsible for finding nectar positions: employed, onlooker and scouts. This algorithm adopts some variables and steps to proceed for finding optimal solution. The variables of ABC algorithm include population of honey bees (SN), maximum cycle number (MCN), objective functions and some functions to test the probability, fitness and optimization of the objective functions. The steps in the ABC algorithm includes initialization, employed, onlooker and scout bees. The initialization step starts by generating the number of solutions equal to total number of bees (SN). Each solution is obtained within the upper and lower limit of the selected decision variable with the following equation.Where denotes the total number of decision variables used in the objective function. The next step starts by generating the new food source by employed and onlooker bees by updating their positions as given by,Where is random number within range (−1, 1) and used to change the position around . The last step includes the probability of finding the nectar by onlooker bees, which is given by following expression,Where is the fitness value corresponding to nectar position . The fitness value is checked for the next iteration, and if it is better than previous fitness value the latest one is stored and the algorithm continues until the stopping criteria is reached (Ghasemi and Shayeghi, 2011; Rathor et al., 2011; Gozde et al., 2012; Naidu et al., 2014; Elsisi et al., 2015; Kouba et al., 2015; Kumar et al., 2017; Abo-Elyousr, 2018). Using the artificial bee colony, optimized PID controller parameters are found as shown in Table 5.

Conventional PID Controller

The Simulink model of the single-area power system containing PID controller is tuned using the classical method to enhance the performance and robustness. PID Tuner toolbox is launched from the Simulink model that automatically computes the linear power system model to find gain parameters corresponding to initial controller conditions. The obtained PID controller gains can robustly stabilize the power system. The performance of controller could be checked using frequency deviation performance indicators, i.e., settling time, zero steady state error. If the desired response is not achieved by auto tunning, PID controller gains can be adjusted manually by varying the response time (seconds) and transient behavior. The PID controller gains are updated to the Simulink model after meeting the desired requirement (MathWorks, 2022a)- (MathWorks, 2022b). The compensator formula for PID controller is given as,

In the studied model, best performing PID controller gains have been found to be 0.8177, 0.1893, and 0.3908 within the range (0, 1) corresponding to initial controller conditions. PID controller has been tuned with response time value of 13.63 s and transient behavior value of 0.6 between aggressiveness and robustness. Filter coefficient value has been chosen as 100 for this case.

Results and Discussion

Three objective functions IAE, ISE, and ITAE have been applied to the genetic algorithm, particle swarm optimization, artificial bee colony, and firefly algorithm to find out the optimal proportional gain , integral gain , and derivative gain . Table 5 illustrates the PID gain parameters for GA, PSO, ABC, and FA using three objective functions. GA, PSO, and ABC techniques have found to be competitive and efficient than FA and conventional methods. The convergence characteristics of GA, PSO, and ABC are shown in Figure 4. The graph represents the objective function vs. number of iterations. The graph depicts almost two iterations are required to reach the best fitness 0.0219, 0.00034, and 0.0166 using IAE, ISE, and ITAE objective function respectively for genetic algorithm. It can also be seen that almost 40 iterations are required to reach the best fitness 2.20, 0.0344, and 44.0306 using IAE, ISE, and ITAE cost function respectively for particle swarm optimization. No significant change has been observed after 40 iterations. The fitness value has been found 0.73, 0.0115, and 14.55 using IAE, ISE, and ITAE objective function respectively in less than 20 iterations for artificial bee colony. Convergence properties describe the effective genetic algorithm that requires very few iterations to reach its best fitness value.

FIGURE 4

Several parameters are used to check the performance indices of hybrid power system for frequency deviation including integral absolute error, integral square error, and integral time absolute error. Rise time , settling time , peak overshoot , and maximum frequency deviation are also used to check the transient response of the power system. These performance indices are also checked by integrating power from BESS/PHEV for different capacities to see the frequency deviation suppression.

Frequency of the power system deviates either positive or negative depending upon the increase or decrease in load and photovoltaic power. The mismatch between load demand and photovoltaic power give rise to changes in frequency. Five key days, 1st January ( = 0.3394 pu), 30th March ( = 0.2374 pu), 13th December ( = 0.1575 pu), 24th May ( = −0.2275 pu), and 28th September ( = −0.1499 pu), showing significant load changes have been selected to check the conventional PID, GA optimized PID, PSO optimized PID, ABC optimized PID, and FA optimized PID responses against frequency deviation. Figures 5–7 depicts the frequency deviation responses of the power system in the same order as mentioned above from top to bottom.

FIGURE 5

Frequency deviation responses of the GA optimized PID, PSO optimized PID, ABC optimized PID, and FA optimized PID using the IAE objective function is shown in Table 6 and Figure 5. Their performance indices show that GA optimized PID, PSO optimized PID, and ABC optimized PID produce approximately same results as their proportional, integral, and derivative values have been found to be same. IAE, ISE, ITAE and are higher for higher mismatches between load and photovoltaic generated power and lower for lower mismatches, i.e., The values of IAE, ISE, ISE and are 0.02199, 0.0003987, 0.01991, and −0.1148 for GA, PSO, and ABC considering the 0.3394 pu load deviation and 0.01351 pu solar PV power for 1st January loading situation in single-area power system. These values of IAE, ISE, ITAE and are lowered to 0.01469, 0.000178, 0.01331, and −0.07668 for GA, PSO, and ABC for 0.2374 pu load deviation and 0.0147 pu solar PV power on 13th December loading conditions. FA optimized PID produces relatively poor result as the IAE, ISE, and ITAE values 0.1106, 0.001322, 0.61 for 1st January loading conditions indicate much higher than GA, PSO and ABC. The values of rise time , settling , and peak overshoot are same for all loading conditions. For example, the values of , settling , and peak overshoot are 0.0673, 2.87, and 78.7 for GA and PSO, 0.069, 2.77, and 78 for ABC on 1st January.

Considering the effect of BESS/PHEV with its different capacities, BESS/PHEV of 10 and 50 are used to check frequency deviation response. Results indicate that settling , peak overshoot are reduced to 2.3, 70 for GA, PSO, and ABC for of 10. These values are further reduced to 1.75 and 40 for of 50. The values of IAE, ISE, ISE and are reduced to 0.01873, 0.0003203, 0.01696, and −0.1103 for GA, PSO, ABC for of 10, and 0.01353, 0.0001902, 0.01399, and −0.0946 for 50 for 1st January loading conditions. Rise time remains unchanged in any case.

Frequency deviation responses of the GA optimized PID, PSO optimized PID, ABC optimized PID, and FA optimized PID using the ISE objective function is shown in Table 7 and Figure 6. Their performance indices show that GA optimized PID, PSO optimized PID, and ABC optimized PID produce efficient and competitive results. IAE, ISE, ITAE and are higher for higher mismatches between load and photovoltaic generated power and lower for lower mismatches, i.e., The values of IAE, ISE, ISE and are 0.02719, 0.0003375, 0.04641, and −0.09178 for GA, 0.02339, 0.0003489, 0.02866, and −0.1014 for PSO, 0.02629, 0.000327, 0.04707, and −0.0966 for ABC considering the 0.3394 pu load deviation and 0.01351 pu solar PV power on 1st January in a single-area power system. These values of IAE, ISE, ITAE and are lowered to 0.01817, 0.0001507, 0.03101, and −0.06102 for GA, 0.01563, 0.0001557, 0.01915, and −0.06781 for PSO, 0.01758, 0.000146, 0.03145 and −0.0644 for ABC for 0.2374 pu load deviation and 0.0147 pu solar PV power on 13th December loading conditions. The values of rise time , settling , and peak overshoot remain same for different loading conditions. For example, the values of rise time , settling , and peak overshoot are 0.0518, 4.51, and 87 for GA, 0.0581, 3.35, and 81.6 for PSO, 0.0551, 3.37, and 84 for ABC on 1st January loading conditions. FA optimized PID produces relatively poor result as the IAE, ISE, and ITAE values are much higher than GA, PSO and ABC as can be seen their values are 0.275, 0.005127, and 1.896.

FIGURE 6

Considering the effect of BESS/PHEV with its different capacities, gain for 0f 10 and 50 are taken to check the stability of power system. Results indicate that settling time , peak overshoot are reduced to 3.4, 80 for GA, 2.64, 74 for PSO, and 2.9, 76.7 for ABC for of 10. These values are further reduced to 2.77, 55.9 for GA, 2.2, 47.8 for PSO, and 2.7, 51.6 for ABC for of 50. Similarly, the values of IAE, ISE, ITAE and are reduced to 0.02258, 0.0002532, 0.03997, and −0.08901 for GA, 0.01997, 0.0002766, 0.02497, and −0.09794 for PSO, 0.02323, 0.00026, 0.0437, −0.09344 for ABC, for of 10, and further reduced to 0.01686, 0.0001398, 0.03458, and −0.07942 for GA, 0.01475, 0.0001646, 0.02049, and −0.085859 for PSO, 0.01835, 0.000155, 0.03927, and −0.08275 for ABC, for of 50 for 1st January loading conditions. Rise time remains almost unchanged in any case. FA optimized PID produces high error values although it is connected to BESS/PHEV support. Instead, the settling time reached to 488 s when BESS/PHEV gain is 50. It is concluded that GA optimized PID produces comparatively better result that PSO, ABC, and FA when ISE cost function is used for the load frequency control.

Frequency deviation responses of the GA optimized PID, PSO optimized PID, ABC optimized PID, and FA optimized PID using the ITAE objective function is shown in Table 8 and Figure 7. Their performance indices show that GA optimized PID, PSO optimized PID, and ABC optimized PID produce approximately efficient competitive results. IAE, ISE, ITAE and are higher for higher mismatches between load and photovoltaic generated power and lower for lower mismatches, i.e., The values of IAE, ISE, ITAE and are 0.02375, 0.0005794, 0.01643, and −0.1374 for GA, 0.02201, 0.0003988, 0.01991, and −0.1136 for PSO, 0.022, 0.000409, 0.01936, and −0.1165 for ABC, considering the 0.3394 pu load deviation and 0.01351 pu solar PV power on 1st January in a single-area power system. These values of IAE, ISE, ITAE and are lowered to 0.01587, 0.0002587, 0.01098, and −0.09176 for GA, 0.0147, 0.000178, 0.0133, and −0.07594 for PSO, 0.0147, 0.0001827, 0.01293 and −0.0778 for ABC, for 0.2374 pu load deviation and 0.0147 pu solar PV power on 13th December loading conditions. The values of rise time , settling , and peak overshoot remain same for different loading conditions. For example, the values of rise , settling , and peak overshoot are 0.0827, 2.55, and 73.4 for GA, 0.0665, 3.03, and 78.4 for PSO, 0.0686, 2.76, and 78.2 for ABC on 1st January loading conditions. FA optimized PID produces relatively poor result as the IAE, ISE, ITAE, and values are much higher than GA, PSO as can be seen the values are 0.07121, 0.005486, 0.04984, and −0.3024.

FIGURE 7

BESS/PHEV with its different capacities, gain 10 and 50 are taken into consideration. Results depict that settling time , peak overshoot are reduced to 2.04, 63.1 for GA, 2.25, 69.8 for PSO, and 2.3, 69.4 for ABC for of 10. These values are further reduced to 0.94, 29.9 for GA, 1.97, 41.4 for PSO, and 1.75, 40.3 for ABC for of 50. Similarly, the values of IAE, ISE, ITAE and are reduced to 0.01968, 0.0004687, 0.01284, and −0.1302 for GA, 0.01347, 0.0001919, 0.01673, and −0.1092 for PSO, 0.01877, 0.000329, 0.01653, and −0.116 for ABC, for of 10, and further reduced to 0.01412, 0.0002801, 0.009743, and −0.1080 for GA, 0.01475, 0.0001646, 0.01334, and −0.09385 for PSO, 0.01352, 0.0001955, 0.01357, and −0.095 for ABC, for of 50 for 1st January loading conditions. Rise time remains almost unchanged in any case. FA optimized PID produces high error values and frequency deviation although it is connected to BESS/PHEV support. It is concluded that GA optimized produces better results with respect to settling time and peak overshoot but PSO optimized PID produces minimum IAE, ISE, ITAE and using ITAE objective function.

The conventional PID controller is tuned using the PID tuner toolbox within MATLAB. Tables 6–8 depict that performance indices IAE, ISE, ITAE and increase with the higher load changes and decrease with lower load changes. But rise time , settling time , and peak overshoot remains same regardless of the load and photovoltaic power changes. It is depicted that performance indices IAE and ISE decreases when BESS/PHEV is integrated to the system at the gain levels 10 and 50, while ITAE increases when the BESS/PHEV is integrated. Frequency deviation decreases with the attachment of BESS/PHEV support. Rise time , settling time , and peak overshoot increase significantly for gain 10, but are infinite at gain 50 because of the unstable response of conventional PID controller. Therefore, the values are replaced by – (dash) where the conventional PID controller response is unstable. It is considered to be major drawback of the conventional PID controller. The conventional PID controller response is shown in Figures 5–7.

The maximum steady state frequency deviation permissible limit in Hz is found to be in the range (−0.7932, 0.5468) for maximum positive and negative load deviation respectively. It can be seen from Table 6, Table 7 and Table 8 that the frequency deviation in Hz has found to be within permissible range using the GA, PSO, ABC, and FA optimization methods. The frequency deviation has not found to be within steady state limit for Conventional PID controller. The frequency deviation range in Hz is (−0.9184, 0.4634) for Conventional-PID. The frequency deviation is suppressed by adding the BESS/PHEV source. But, the PID controller response becomes unstable if the large capacity of BESS/PHEV source is attached to the power system model. The power delivered to power system by BESS and/or PHEV is 0.025 pu for gain of 10, and 0.105 pu for gain of 50 taking into account the maximum positive loading condition on 1st January.

After comparison between the applied techniques and objective functions, it is concluded that PSO optimized PID controller using the ITAE objective function performs well just to mitigate frequency deviation of the power system but GA optimized PID controller using the ISE objective function gives better overall results considering all the performance indices. The convergence characteristics and parameter indices indicate the effectiveness, sustainability and consistency of proposed GA optimized PID tested within the loading range (−0.2275, 0.3394).

Conclusion

The load deviation of single area power system and solar photovoltaic power is forecasted using the state-of-the art artificial neural network. PSO, GA, ABC, and FA metaheuristic techniques have been employed to find out the optimal PID controller gain parameters using the IAE, ISE and ITAE objective functions. Some key days, showing significant load changes, have been chosen to see the performance of the single area hybrid power system. Results indicate the efficiency and competency of GA, PSO, and ABC among each other that sufficiently reduce frequency fluctuations. It has been found that PSO optimized PID along with ITAE cost function can sufficiently minimize the frequency. The performance indices and convergence characteristics show overall better results produced from GA optimized PID than others using ISE objective function. These frequency fluctuations are further improved by integrating power from BESS/PHEV source. Small BESS/PHEV capacity is required to damp the frequency when integrated with GA optimized PID.

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