Effective participation of wind turbines in frequency control of a two-area power system using coot optimization

Abstract

In this paper, load frequency control is performed for a two-area power system incorporating a high penetration of renewable energy sources. A droop controller for a type 3 wind turbine is used to extract the stored kinetic energy from the rotating masses during sudden load disturbances. An auxiliary storage controller is applied to achieve effective frequency response. The coot optimization algorithm (COA) is applied to allocate the optimum parameters of the fractional-order proportional integral derivative (FOPID), droop and auxiliary storage controllers. The fitness function is represented by the summation of integral square deviations in tie line power, and Areas 1 and 2 frequency errors. The robustness of the COA is proven by comparing the results with benchmarked optimizers including: atomic orbital search, honey badger algorithm, water cycle algorithm and particle swarm optimization. Performance assessment is confirmed in the following four scenarios: (i) optimization while including PID controllers; (ii) optimization while including FOPID controllers; (iii) validation of COA results under various load disturbances; and (iv) validation of the proposed controllers under varying weather conditions.

1 Introduction

Renewable energy sources (RES) are promising alternatives to fossil fuels due to the exhaustion of fossil fuels and their negative environmental effects [1]. The frequency of a power system deviates from its nominal value because of continuous load changes and other disturbances [2]. Stored kinetic energy (KE) in the rotors of conventional synchronous generators (SG) will tolerate normal load changes until the operation of primary and secondary frequency regulation loops [3]. Power systems with a high penetration of RES have less KE owing to the reduction of inherent inertia with the increase of RES [4]. Thus, problems in power system stability and frequency regulation may be initiated [3, 4]. In this paper, participation of RES and energy storage devices for frequency support is discussed for a two-area power system. Frequency support is performed by a new optimization algorithm, namely, the Coot Optimization Algorithm (COA). The COA is robust in allocating the optimal parameters of the fractional order proportional integral derivative (FOPID) controllers, washout filter controller and droop controller.

Energy storage systems play an essential role in the frequency regulation of a power system [5]. Battery energy storage (BES) is selected in [6, 7] to adjust the frequency of a microgrid integrated with wind turbines (WT) while superconducting magnetic energy storage (SMES) is proposed for frequency regulation in [8, 9]. In addition, supercapacitor energy storage (SCES) is proposed in [10, 11] to control the frequency of a microgrid integrated with PV and tidal energy.

One technique for RES participation in frequency support is de-loading, where some reserves of active power in RES is maintained to support sudden load increase. Overvoltage de-loading of PV is illustrated in [12, 13] to tolerate load disturbance while overspeed and pitch angle control of variable speed wind turbine (VSWT) are discussed in [14,15,16]. Another technique is inertial response, which momentarily supports a power system by additional active power extracted from a VSWT. Recently, extensive research has been done to enhance the inertia of a power system with high penetration of RES using virtual inertial response techniques. Inertial response is classified into fast power reserve, hidden inertia, and droop control. In [17, 18], fast power reserve is used to enhance power system inertia while hidden inertia is proposed in [19, 20]. A droop controller, which is proposed in this paper, is demonstrated for inertial response and effective frequency regulation in [21,22,23].

Power system stability during load disturbance is achieved by load frequency control (LFC) which is a secondary control loop used to eliminate errors in frequency and tie line power [24]. Proportional integral derivative (PID) controllers have been benchmarked in many LFC studies. In [25], the dynamic performance of an offshore grid-connected wind farm is enhanced by flywheel energy storage (FWES) based on a PID controller. In [8], LFC of a power system considering high penetration of wind power is illustrated by SMES based on a PID controller. In [26], LFC is performed by a PID droop controller for a two-area power system highly penetrated with wind power, whereas in [27], LFC is performed by an adaptive PID droop controller for an isolated power system also highly penetrated with wind power. Also, LFC is illustrated in [28] using a two-degrees-of-freedom PID for a three-area power system integrated with plug-in electric vehicles (EVs).

The fractional order PID (FOPID) controller is validated and its robustness proved to be better than a traditional PID because of the higher number of its tunable parameters [24]. In [29], LFC is examined for a two-area power system by two-degrees-of-freedom FOPID. In addition, in [30], LFC is equipped for a microgrid integrated with VSWT using droop control-based FOPID, whereas in [31], LFC is demonstrated using FOPID for a single area containing hydro, reheat and non-reheat turbines. A lot of studies have been devoted to illustrating the optimization algorithms which are used to.

tune power system parameters. In [21], LFC is reinforced using BES for PV and extracting the KE from VSWT by tuning the parameters of the PID controller using a stochastic fractal optimizer. In [32], optimal charging of EVs for LFC is performed by allocating the parameters of a PI controller using a genetic algorithm. Optimum parameters of the PID controller are obtained in [33] using firefly and particle swarm optimization (PSO) to achieve optimal LFC using hybrid FWES and BES, while optimal parameters of the PI controller are obtained in [34] by a sine cosine algorithm to optimize the LFC using SCES. In [35], LFC is performed by optimizing the parameters of the PID controllers of SMES and BES using a social spider optimizer. Parameters of an FOPID controller for optimal LFC are obtained in [24] and [36] by a modified hunger games search optimizer and chaotic multi-objective optimizer, respectively.

Frequency regulation of a two-area power system is performed in this paper by COA which allocates the parameters of the FOPID controller. COA is a novel algorithm and has been validated as a robust optimizer in many studies [37, 38]. In [39], COA is validated for optimal parameter extraction of a lithium-ion battery when compared to other six benchmarked optimizers. Optimal sizing of the energy storage system required to support a wind power producer is obtained in [40] by COA, which proves its robustness compared to two other benchmarked optimizers. The paper seeks a novel algorithm while performing LFC for the proposed two-area power system, and COA, HBA and AOS are chosen as the novel robust optimizers, while their fitness is compared with PSO and WCA which are well known and are used as the benchmarked optimizers. From the results, COA proves its robustness over HBA, AOS, PSO and WCA.

The proposed model is a two-area power system. Area 1 contains steam SG and has 50% penetration of wind power, while Area 2 contains hydro SG and has 20% penetration of solar power in addition to an auxiliary storage system. COA is validated first by comparing the optimization results while including PID controller as the benchmark, and robust optimizers such as atomic orbital search optimizer (AOS) [41], honey badger algorithm (HBA) [42], water cycle algorithm (WCA) and PSO. Then PID controllers are replaced by FOPID controllers and the optimization is performed again by the same five optimizers to validate the robustness of COA. COA results are benchmarked and validated for its robustness under variable load disturbances and varying weather conditions.

The main outcomes of this paper are:

• (a) A novel application of COA is proposed for parameter extraction of FOPID frequency controllers.

• (b) LFC are supported via droop controller for inertial response of WT, pitch angle supplementary controller, and transient support of stored energy.

• (c) Benchmarking with challenging optimization methods is incorporated for the validation of the proposed method.

• (d) Different scenarios are studied to investigate the robustness of the designed controllers.

The remainder of the paper is structured as follows. Modelling of the two-area power system is discussed in Sect. 2, and Sect. 3 illustrates the algorithm of COA. Simulation results for different scenarios are demonstrated in Sect. 4, and the outcomes of this paper are discussed in Sect. 5.

2 System modelling

The intended system is a two-area power system shown in Fig. 1. It consists of conventional plants and different RES such as type 3 wind turbines and PV systems. Area 1 consists of 50% conventional steam turbine and 50% VSWT. The output power of the conventional steam turbine is represented by ${P}_{m1}$ while the output electrical power of the VSWT is represented by ${P}_{e}$ in Fig. 1. Area 2 consists of 80% conventional hydro turbine and 20% solar energy, with the output power of the conventional hydro turbine represented by ${P}_{m2}$ and the solar energy represented by ${P}_{PV}$ in Fig. 1.

Fig. 1

Block diagram model of the studied two-area power system

2.1 Model of type 3 wind turbine

A type 3 WT with a rating of 3.6 MW is proposed in this model. Its parameters are given in Table 1 [43]. The block diagram of the type 3 WT is illustrated in Figs. 2, 4 and 5 [43]. The PI controller in Fig. 2 gives an output torque signal depending on the WT speed deviation. This model is selected to estimate the dynamic operation of the intended two-area power system during any power disturbances. The output WT mechanical power is illustrated by:

$${P}_{m}=0.5\times \rho \times {A}_{wt}\times {{V}_{w}}^{3}\times {C}_{p}$$

(1)

where the air density is $\rho$ (${\text{kg}}/{\text{m}}^{3}$), the blade swept area is ${A}_{wt}$ (${\text{m}}^{2}$), the wind speed is ${V}_{w}$ (assumed 11 ${\text{m}}/{\text{s}}$), and the coefficient of performance is ${C}_{p}$ given as:

$${C}_{p}=\sum _{k=0}^{4}\sum _{n=0}^{4}{\alpha }_{k,n}{\lambda }^{n}{\beta }^{k}$$

(2)

where $\beta$ is the blade pitch angle and ${\alpha }_{k,n}$ are constants illustrated in [43]. $\lambda$ is the WT tip speed ratio given as:

$$\lambda =\frac{{K}_{b}\times {\omega }_{wt}}{{V}_{w}}$$

(3)

where the rotor speed is denoted by ${\omega }_{wt}$ and the WT radius is represented by ${K}_{b}$. The relations between the output mechanical power of the WT and its speed at different wind speeds are demonstrated in Fig. 3. Maximum power point tracking (MPPT) can be expressed by the curve fitting, as:

$${\omega }_{wt\_ref}=1.6{{P}_{e}}^{3}-2.7{{P}_{e}}^{2}+2.3{P}_{e}+0.45$$

(4)

Fig. 2

Block diagram model of the proposed type 3 wind turbine

Fig. 3

Output mechanical power of the proposed type 3 wind turbine at certain rotational speed for different wind speeds

Equation (4) relates the reference speed of the WT (${\omega }_{wt\_ref}$) to its output electrical power. The maximum and minimum limits of the WT speed are 1.2 pu and 0.7 pu, respectively.

Table 1 Parameters of the proposed two-area power system

The droop controller is emulated by Fig. 4, which demonstrates that the temporary power signal ${P}_{droop}$ is dependent on the frequency deviation ($\Delta F$) of Area 1 ${\Delta F}_{1}$. The droop controller performs the same function as the droop controller of the SG [3, 44], which accelerates and decelerates the rotor of type 3 WT in case of positive and negative ${\Delta F}_{1}$, respectively [45].

Fig. 4

Droop controller of the proposed type 3 wind turbine

The pitch angle controller, which avoids the output power corresponding to higher wind speed exceeding the generator rated power, is illustrated in Fig. 5. The dependent WT speed deviation signal ${\beta }_{ref}$ is the required pitch angle to avoid the speed of the rotor exceeding its limits. ${\beta }_{ref}$ is zero during normal wind speed, but for wind speed above the rated value, it has a value higher than zero. ${\beta }_{a}$ is an additional signal which is used to increase and decrease WT output power during negative and positive $\Delta F$, respectively [45]. ${P}_{droop}$ and $\beta$ are the outputs of Figs. 4 and 5, which are the VSWT model inputs in Fig. 2.

Fig. 5

Pitch angle controller of the proposed type 3 wind turbine

2.2 Model of PV

Panels of PV consist of a few PV modules which are connected in parallel and in series to increase the current and voltage of the PV array [24]. The mathematical equations which describe the PV model are given in (5)–(10) [46], while the required PV parameters for electrical characteristic simulation are given in Table 1 [47].

$${V}_{t}=\frac{{K}_{Bz}T}{{Q}_{e}}$$

(5)

$${I}_{ph}=\left(\frac{G}{{G}_{ref}}\right)\left({I}_{phn}+{K}_{i}\left({T-T}_{ref}\right)\right)$$

(6)

$${I}_{o}={I}_{sc}\left(\frac{G}{{G}_{ref}}\right){\left(\frac{T}{{T}_{ref}}\right)}^{3}{e}^{\left(1-\frac{{V}_{oc}}{M{V}_{t}}\right)}$$

(7)

$${I}_{PV}={I}_{ph}-\left(\frac{{I}_{PV}{R}_{s}+{V}_{PV}}{{R}_{P}}\right)-{I}_{o}\left({e}^{\left(\frac{{I}_{PV}{R}_{s}+{V}_{PV}}{M{V}_{t}}\right)}-1\right)$$

(8)

$${R}_{P}={R}_{Ps}\left(\frac{G}{{G}_{ref}}\right)$$

(9)

$${P}_{PV}={{V}_{PV}I}_{PV}$$

(10)

where ${V}_{t}$ is the thermal PV voltage, ${K}_{Bz}$ is the Boltzmann constant, $T$ is the temperature of the PV, ${Q}_{e}$ is the electron charge, and $G$ is the actual irradiance. ${G}_{ref}$ and ${T}_{ref}$ are the irradiance and temperature at standard conditions, respectively. ${K}_{i}$ is the short circuit current temperature coefficient, ${I}_{ph}$ and ${I}_{phn}$ are the respective light produced current at the actual temperature and at standard conditions, ${I}_{o}$ is the reverse saturation diode current, and ${I}_{sc}$ is the short circuit current. ${V}_{oc}$ is the open circuit voltage, $M$ is the factor of diode ideality, ${I}_{PV}$ is the PV output current, ${V}_{PV}$ is the PV output voltage, ${R}_{P}$ is the shunt resistance, ${R}_{s}$ is the series resistance, ${R}_{Ps}$ is the standard condition shunt resistance, and ${P}_{PV}$ is the output power of the PV.

MPPT methods are required for bulk stations of PV [48]. Reference [49] illustrates a comprehensive review on various MPPT methods for PV. An artificial neural network (ANN) for MPPT as discussed in [50] is used in this paper. There are two input layers ($T$, $G$) and one output layer (${V}_{PV}$). The output current and power at MPP are detected once the MPPT voltage is estimated.

2.3 Model of conventional units

Hydro and steam power plants are incorporated into the proposed two-area power system. The transfer functions of the steam turbine are modelled mathematically as:

$${\text{Turbine model}}\quad \frac{{F}_{HP}{T}_{RH}s+1}{\left({T}_{RH}s+1\right)\left({T}_{CH}s+1\right)}$$

$${\text{Governor model}}\quad\frac{1}{{T}_{G}s+1}$$

$${\text{Generator model}}\quad\frac{1}{{2 H}_{1}s+{D}_{1}}$$

The steam turbine model is based on a single reheat stage as described in [51]. ${T}_{RH}$ and ${T}_{CH}$ represent the reheater and main inlet valve time constants, and ${F}_{HP}$ is the fractional power generated by the high-pressure turbine. The time lag of the governor is represented by the time constant ${T}_{G}$. The generator model is represented by its swing equation where ${H}_{1}$ represents its inertia and ${D}_{1}$ is the load damping constant.

The transfer functions of the hydro turbine are modelled as:

$${\text{Turbine model}}\quad \frac{-{T}_{W}s+1}{{0.5T}_{W}s+1}$$

$${\text{Governor model}}\quad\frac{{T}_{R}s+1}{\left({T}_{GH}s+1\right)\left({\left(\frac{{R}_{Td}}{{R}_{Pd}}\right)T}_{R}s+1\right)}$$

$${\text{Generator model}}\quad\frac{1}{{2 H}_{2}s+{D}_{2}}$$

The model of the hydro turbine is a non minimum phase system that represents the change in output power due to the change in gate opening, and ${T}_{W}$ represents the water starting time. Unlike a steam turbine governor, a hydro turbine governor requires an additional transient droop with long resetting time to limit the gate movement until water flow and output power have time to catch up [51]. ${R}_{Pd}$ represents the permanent droop and ${R}_{Td}$ is the transient droop, ${T}_{R}$ is the reset time, and ${T}_{GH}$ is a time constant representing the delay in the governor response. Also, the generator model is represented by its swing equation where ${H}_{2}$ represents its inertia and ${D}_{2}$ is the load damping constant.

3 Coot optimization algorithm

COA is categorized as a novel swarm-based meta-heuristic algorithm, introduced by Iraj Naruei and Farshid Keynia [52]. Its behavior for food seeking can be described by the following four phases: random motion; chain motion; improving coot position by tracing the leaders; and leader motion toward optimal zone including the food. The algorithm is started by selecting an initial population randomly, and this population is frequently estimated by the objective function until the optimal value is obtained. The randomly generated population can be evaluated as:

$$coot\, position\left(i\right)=Rand\left(1,dim\right)\times \left(UB-LB\right)+LB$$

(11)

$$LB = {LB}_{1}, {LB}_{2}, {LB}_{3}, \dots , {LB}_{dim}$$

(12)

$${UB} = {UB}_{1}, {UB}_{2}, {UB}_{3}, \dots , {UB}_{dim}$$

(13)

where $i$ is the current coot index, $coot\, position\left(i\right)$ is the position of coot, $dim$ is the dimension of the problem, $LB$ is the lower bound matrix and $UB$ is the upper bound matrix.

The fitness of each coot is calculated by the objective function after determining the coot initial population and position. The objective function which is required to be optimized by COA is given in (14), which represents the integral square error ($ISE$) consisting of three small values of tie line power deviation $(\Delta {P}_{12}$), and the deviations in frequency of Area 1 ($\Delta {F}_{1}$) and Area 2 ($\Delta {F}_{2}$).

$$ISE={\int }_{0}^{t}\left({{\Delta P}_{12}}^{2}+{{\Delta F}_{1}}^{2}+{{\Delta F}_{2}}^{2}\right)dt$$

(14)

The random motion of a coot which helps the algorithm to explore various zones in the search space and converge to the global optimum is illustrated in (15), and the new coot position is estimated in (16).

$$Q=Rand\left(1,dim\right)\times \left(UB-LB\right)+LB$$

(15)

$$coot\, position\left(i\right)=coot\, position\left(i\right)+A\times {R}_{1}\times \left(Q-coot\, position\left(i\right)\right)$$

(16)

where ${R}_{1}$ is a random number between [0,1] and $A$ is estimated by:

$$A=1-\frac{Iter}{Iteration }$$

(17)

where the maximum iteration number is represented by $Iteration$ and the current iteration is represented by $Iter$.

The second phase is the chain movement which can be implemented by calculating the mean position of two coots as:

$$coot\, position\left(i\right)=0.5 \times \left(coot\, position\left(i-1\right)+coot\, position\left(i\right)\right)$$

(18)

where $coot\, position\left(i-1\right)$ is the position of the second coot.

Improving the position of coots by tracing leaders can be implemented by selecting a few leaders randomly and estimating their mean position. Then coots modify their positions according to the mean position of the leaders. The criteria for leader selection is provided in (19) and the modified coot position according to leader position is estimated according to (20).

$$K=1+\left(i MOD LD\right)$$

(19)

$$coot\, position\left(i\right)=LDP\left(K\right)+2\times {R}_{2}\times {cos}\, \left(2R\pi \right)\times \left(LDP\left(K\right)-coot\, position\left(i\right)\right)$$

(20)

where $K$ is the index number of the leader, $LD$ is the number of leaders, ${R}_{2}$ is a random number between [0,1], $R$ is a random number between [−1,1] and $LDP\left(K\right)$ is the chosen leader position.

Finally, leaders’ position is modified to find a new optimal point near the best position that has been found, as:

$$LDP\left(i\right)=\left\{\begin{array}{c}B\times {R}_{3}\times {cos}\left(2R\pi \right)\times \left(GB-LDP\left(i\right)\right)+GB, {R}_{4}<0.5\\ B\times {R}_{3}\times {cos}\left(2R\pi \right)\times \left(GB-LDP\left(i\right)\right)-GB, {R}_{4}\ge 0.5\end{array}\right.$$

(21)

where ${R}_{3}$ and ${R}_{4}$ are random numbers between [0,1], $GB$ is the best position and $B$ is estimated by:

$$B=2-\frac{Iter}{Iteration }$$

(22)

$B\times {R}_{3}$ prevents the COA from blocking in a local optimum by performing larger movements randomly, which means that the COA performs exploitation and exploration at the same time. On the other hand, ${cos}\left(2R\pi \right)$ helps to seek a better position near the best-found position with various radii. Figure 6 illustrates the flow chart which demonstrates the procedures of COA.

Fig. 6

Flowchart of COA optimizer

4 Simulation results

This section illustrates various scenarios that are performed on the proposed system discussed earlier in Sect. 2. These scenarios are illustrated by MATLAB/SIMULINK 2018b. Optimization processes are applied on the 1st and 2nd scenarios for COA benchmarking and comparisons. COA results are validated in the 3rd and 4th scenarios.

For fair comparison, lower bound, upper bound and the number of populations are maintained constant for all optimizers and the number of iterations is selected as 100.

4.1 Scenario 1: optimization of conventional PID controller parameters

In this scenario, optimization is performed to allocate the optimal parameters of PID controllers, time of washout filter (${\tau }_{W}$), pitch angle controller gain (${K}_{\beta }$) and WT droop (${R}_{wt}$). The objective function of the $ISE$ is optimized for 100 s by the 5 optimizers, i.e.: PSO, WCA, HBA, AOS and COA. It is found that the corresponding $ISE$ are $2.6307\times {10}^{-5}$, $3.1371\times {10}^{-5}$, $3.3435\times {10}^{-5}$, $5.1316\times {10}^{-5}$ and $5.3451\times {10}^{-5}$ for COA, AOS, HBA, WCA and PSO, respectively. The convergence curves of the 5 optimizers for 100 iterations are shown in Fig. 7. Optimal settings of the optimized parameters for the 5 optimizers are demonstrated in Table 2, while the system responses ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ due to 0.05 pu load increase in Area 1 are illustrated in Fig. 8. The maximum deviations in ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ in the case of COA are $-1.97\times {10}^{-3}$ pu, $-7.4\times {10}^{-4}$ pu and $-4.13\times {10}^{-3}$ pu, respectively. It is also found that the maximum deviation in ${\Delta F}_{1}$ in case the of COA is less than WCA by 72.6% while the maximum deviation in ${\Delta F}_{2}$ in COA is less than PSO and WCA by 20.27% and 32.43%, respectively. In addition, the maximum deviation in ${\Delta P}_{12}$ in the case of COA is less than PSO, WCA, HBA and AOS by 30.5%, 69.98%, 5.09% and 3.9%, respectively. It proves that $ISE$ for COA is the smallest of the optimizers. The reductions in the maximum deviations of ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ in the case of COA are greater than other optimizers. The overall conclusion is that COA is the most robust of the optimizers for PID controllers.

Fig. 7

Comparison of $\text{I}\text{S}\text{E}$ convergence for the five used optimizers of the two-area power system including PID controllers

Fig. 8

System responses due to + 0.05 pu step load change in Area 1 while including PID controller a ${\Delta \text{F}}_{1}$, b ${\Delta \text{F}}_{2}$, c ${\Delta \text{P}}_{12}$

Table 2 Parameters of PID controllers

4.2 Scenario 2: optimization of FOPID controller parameters

In this scenario, PID controllers are replaced by FOPID controllers and optimization is performed to allocate the optimal parameters of these controllers in addition to ${\tau }_{W}$,${K}_{\beta }$ and ${R}_{wt}$. It is found that $ISE$ are $1.8708\times {10}^{-5}$, $2.3697\times {10}^{-5}$, $3.2711\times {10}^{-5}$, $3.7009\times {10}^{-5}$ and $4.7540\times {10}^{-5}$ for COA, AOS, HBA, PSO and WCA, respectively. The convergence curves of the 5 optimizers for 100 iterations are shown in Fig. 9. Optimal settings of the optimized parameters for the 5 optimizers are demonstrated in Table 3, and the system responses ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ due to 0.05 pu load increase in Area 1 are illustrated in Fig. 10. As seen, the maximum deviations in ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ in the case of COA are $-2.79\times {10}^{-3}$ pu, $-6.7\times {10}^{-4}$ pu and $-3.7\times {10}^{-3}$ pu, respectively. It is found that the maximum deviation in ${\Delta F}_{1}$ in the case of COA is less than PSO and WCA by 4.3% and 21.15%, respectively, while the maximum deviation in ${\Delta F}_{2}$ for the COA is less than PSO, HBA and AOS by 5.97%, 44.78% and 25.37%, respectively. Also, the maximum deviation in ${\Delta P}_{12}$ for the COA is less than PSO, WCA, HBA and AOS by 28.1%, 80.27%, 51.08% and 0.54%, respectively. It proves that $ISE$ for the COA is the smallest of the optimizers. Moreover, the reductions in the maximum deviations of ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ in the case of the COA are greater than the others. In addition, $ISE$ in this scenario are smaller than $ISE$ in the 1st scenario. Thus, it can be concluded that the COA is the most robust optimizer for FOPID controller when compared to the other four optimizers and PID. The advantage of FOPID over the conventional PID can be considered as FOPID having more optimized parameters than conventional PID. Thus, FOPID is included in the next two scenarios for further investigation.

Fig. 9

Comparison of $\text{I}\text{S}\text{E}$ convergence for the five optimizers of the two-area power system including FOPID controllers

Fig. 10

System responses due to + 0.05 pu step load change in Area 1 while including FOPID controller a ${\Delta \text{F}}_{1}$, b ${\Delta \text{F}}_{2}$, c ${\Delta \text{P}}_{12}$

Table 3 Optimal parameters of FOPID controllers

4.3 Scenario 3: robustness of the COA under variable load disturbances

In this scenario, the COA is validated (including FOPID) under various load disturbances in Area 1 which are shown in Fig. 11. It is observed that $ISE$ are $2.7045\times {10}^{-4}$, $3.7125\times {10}^{-4}$, $4.4567\times {10}^{-4}$, $3.3989\times {10}^{-4}$ and $4.1807\times {10}^{-4}$ for COA, AOS, HBA, PSO and WCA, respectively. The system responses ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ due to various load disturbances (shown in Fig. 11) are illustrated in Fig. 12. Looking at the deviations in the case of COA, the maximum deviations in ${\Delta F}_{1}$ are $-2.79\times {10}^{-3}$ pu, $5.28\times {10}^{-3}$ pu and $-5.36\times {10}^{-3}$ pu for the 1st, 2nd and 3rd disturbances, respectively. The maximum deviations in ${\Delta F}_{2}$ are $-6.7\times {10}^{-4}$ pu, $1.183\times {10}^{-3}$ pu and $-1.24\times {10}^{-3}$ pu for the 1st, 2nd and 3rd disturbances respectively. Also, the maximum deviations in ${\Delta P}_{12}$ are $-3.7\times {10}^{-3}$ pu, $8.47\times {10}^{-3}$ pu and $-7.97\times {10}^{-3}$ pu for the 1st, 2nd and 3rd disturbances, respectively. The percentage reductions in ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ compared to others are shown in Table 4. It can be seen that $ISE$ for COA are the smallest. For COA, the reductions in the maximum deviations of ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ are greater than other optimizers (Table 4). Thus, the overall conclusion is that COA is the most robust and benchmarked optimizer. COA is tested in the next scenario in real weather conditions.

Fig. 11

Variable step load change in Area 1

Fig. 12

System responses due to various step load changes in Area 1 while including FOPID controller a ${\Delta \text{F}}_{1}$, b ${\Delta \text{F}}_{2}$, c ${\Delta \text{P}}_{12}$

Table 4 Reductions in ${\Delta \text{F}}_{1}$, ${\Delta \text{F}}_{2}$ and ${\Delta \text{P}}_{12}$ deviations due to 3 step load changes using COA

4.4 Scenario 4: robustness of COA in varying weather conditions

Here, COA is validated (including FOPID) using real measurements at Zafarana in Egypt [35] for temperature, irradiance and wind speed. Four-day samples of temperature, irradiance and wind speed are recorded in Fig. 13, and the system responses ${\Delta F}_{1}$, ${\Delta F}_{2}$ and ${\Delta P}_{12}$ due to real weather data are illustrated in Fig. 14. The maximum deviation in frequency for the two-area power system is −0.001281 pu (−0.06405 Hz) which is a secure value according to the operation of the under frequency load shedding relays [53]. It can be seen that the system behaves satisfactorily in real weather conditions. Thus, COA proves its robustness and can be a promising and benchmarked optimizer.

Fig. 13

Real weather conditions at Zafarana a $\text{T}$, b $\text{G}$, c ${\text{V}}_{\text{w}}$

Fig. 14

System responses due to real weather conditions a ${\Delta \text{F}}_{1}$, b ${\Delta \text{F}}_{2}$, c ${\Delta \text{P}}_{12}$

In summary of the performance of COA and from the optimization results of PID controllers, we see that COA gives the best objective function. In addition, COA gives the best fitness function for the optimization of FOPID controllers. The fitness function of FOPID is 28.88% less than that of PID because the FOPID controller has more decision parameters and hence a higher degree of freedom. Therefore, FOPID is included in the validation and benchmarking scenarios. COA also shows its robustness under variable stiff load disturbances in the 3rd scenario, while it also behaves satisfactorily in real weather conditions in the 4th scenario.

In the studied cases, COA optimizer behaves better than other optimizers. This is because COA has four different coots moving strategies on the water surface: random to this side and that side; chain movement; movement adjustment according to the leader; and leader movement. The random movement helps to explore the search space, while the algorithm has immunity to being stuck in local minima by updating the new position, as described by (16). Extensive movement toward the optimum area are assured by the leader’s movement. The coots may move in a chain or toward group leaders randomly, and this helps to preserve the random nature of the algorithm.

5 Conclusion

In this paper, an efficient LFC has been performed on a two-area power system using robust FOPID controllers. This two-area power system contains steam and hydro generators integrated with a high penetration of RES such as PV panels and WTs. The five optimal parameters of the FOPID controllers which satisfy the best value of $ISE$ are obtained by the COA. The robustness of the COA is validated through four scenarios with comparisons to other benchmarked optimizers including PSO, WCA, HBA and AOS. Optimization is first performed while including traditional PID controllers and the results confirm that the COA results in the smallest $ISE$. The results of the 1st scenario show that $ISE$ with COA is less than AOS, HBA, WCA and PSO by 20.01%, 27.86%, 95.8% and 103.94%, respectively. Then, optimization is performed while traditional PID controllers are replaced by FOPID controllers. The results of the optimization shed light on the robustness of FOPID controllers based on the COA approach. The results of the 2nd scenario show that $ISE$ with COA is less than AOS, HBA, PSO and WCA by 26.67%, 74.85%, 97.82% and 154.12%, respectively. In the 3rd scenario, the system responses are observed while the two-area power system is subjected to variable load disturbances. The results illustrate that $ISE$ with COA is less than PSO, AOS, WCA and HBA by 25.68%, 37.3%, 54.58% and 64.8%, respectively. Finally, the performance assessment of FOPID optimized by the COA is examined in real weather conditions. This results in a maximum frequency deviation of −0.06405 Hz. The results illustrate the efficiency and applicability of the proposed FOPID controllers based on a COA approach.