In this study, a bald eagle optimizer (BEO) is used to get optimal parameters of the fractional-order proportional–integral–derivative (FOPID) controller for load frequency control (LFC). Since BEO takes only a very short time in finding the optimal solution, it is selected for designing the FOPID controller that improves the system stability and maintains the frequency within a satisfactory range at different loads. Simulations and demonstrations are carried out using MATLAB-R2020b. The performance of the BEO-FOPID controller is evaluated using a two-zone interlinked power system at different loads and under uncertainty of wind and solar energies. The robustness of the BEO-FOPID controller is examined by testing its performance under varying system time constants. The results obtained by the BEO-FOPID controller are compared with those obtained by BEO-PID and PID controllers based on recent metaheuristics optimization algorithms, namely the sine–cosine approach, Jaya approach, grey wolf optimizer, genetic algorithm, bacteria foraging optimizer, and equilibrium optimization algorithm. The results confirm that the BEO-FOPID controller obtains the finest result, with the lowest frequency deviation. The results also confirm that the BEO-FOPID controller is stable and robust at different loads, under varying system time constants, and under uncertainty of wind and solar energies.

Frequency stability is one of the most dynamic problems with adverse influences on the entire system [

Despite this succinct literature representation, the No Free Lunch Theorem steers us that the identification of the controller parameters is probably improved using modern optimization approaches. Hence, in this study, we proposed a bald eagle optimizer (BEO) algorithm—a meta-heuristic optimization technique—to design the FOPID controller for system frequency support. The BEO was created in 2020 [

The recent BEO is implemented to obtain the optimal gains of FOPID for enhancing the system frequency.

The gotten results using the BEO are compared with the results based on SCA, Jaya, GWO, GA, BFO, and EOA algorithms in order to confirm its robustness.

The supremacy of the FOPID over the PID controller is demonstrated.

The performance of BEO-FOPID is validated under step load perturbation and random load variation.

The impact of both system parameters and sources of renewable energy (RE) on the performance of the BEO-FOPID controller is investigated.

The remnant of the article is laid out as follows: system modeling is introduced in the second section. BEO is explained in the third section. The objective function (Fun_{Obj}) is formulated in the fourth section. The control strategy is shown in the fifth section. Simulation results and discussion are produced in the sixth section. The conclusion is drawn in the seventh section.

_{d1}), tie-line power error (ΔP_{t}), and the controller’s output (ΔP_{ref}) are the inputs of each zone. On the other hand, the generator frequency deviation (Δf) and zone control errors (ZCEs) are the outputs of each zone.

Parameter | Value | |
---|---|---|

K_{ge} |
Generator gain | 120 Hz/p.u. MW |

T_{ge} |
Generator time constant | 20 s |

T_{t} |
Turbine time constant | 0.3 s |

T_{go} |
Governor time constant | 0.08 s |

R | Governor speed regulation coefficient | 2.4 Hz/p.u. MW |

B | Frequency bias coefficient | 0.425 |

K_{12} |
Tie-line coefficient | 0.545 |

r_{12} |
Zone capacity ratio | −1 |

BEO is an algorithm that follows the same manner of bald eagles in their hunting [

During this phase, new positions will be created using the following equation:

_{k,new} is the updated position, L_{best} refers to the search space that bald eagles are exploring, dependent on the location they discovered throughout their prior search, r is the number whose value randomly ranges from 0 to 1, γ controls the changes of location via taking a number between 1.5 and 2, and L_{mean} is the mean location [

Once the best search space L_{best} is determined by the algorithm, then it updates the location of the eagles within that space. During this phase, the eagle position is updated using the following formula:

^{th} point, which are written as

In this phase, eagles move from the finest available position toward the target prey. This manner is mathematically illustrated as following [

_{1} and c_{2} range from 1 to 2 and x_{1} and y_{1} are the coordinates of direction given as:

The main targets of LFC are to take the frequency back to its rated value as speedily as possible and lessen the oscillations of P_{t} among the contiguous control zones throughout load perturbations. The peak undershoots (PeUn), settling time (t_{sett}), and steady-state error are the descriptions of ∆f and ∆P_{t} in the time-domain study to be enhanced. It was discovered that the finest criterion for all stated descriptions is the integral-time-absolute errors (ITAE) of f and P_{t}, so Fun_{Obj} is suggested to minimize the ITAE [

_{simu} is the time of the simulation. The constraints subjugate the gains of the PID controller inside the lower and upper limits and Fun_{Obj} is correspondingly subjugated.

_{p}, K_{i}, and K_{d} are the gains of proportional, integral, and derivative components, respectively, λ and μ are the orders of integral and derivative, respectively, ΔP_{t} is the tie-line power error, and e(t) is the error signal.

The effectiveness and robustness of the BEO-FOPID controller are validated by comparing it with some previous controllers at increasing load, disturbed random loading (DRL), and under uncertainty of wind and solar energies. The impact of system parameters T_{ge}, T_{go}, and T_{t} on the performance of the BEO-FOPID controller is also investigated.

Our results have been acquired using MATLAB-R2020b under Windows 10 running on a laptop with an Intel Core i7−1065G7 CPU at 1.3 GHz (8 CPUs) with 16 GB of RAM.

The values of BEO parameters are determined as suggested in [_{1}=2, c_{2}=2, γ=2,

In this section, a 10% step load perturbation is implemented in zone 1.

Lower limit | Upper limit | |
---|---|---|

0 | 3 | |

0.1 | 2 |

BEO-FOPID | 2.063 | 2.9999 | 1.0003 | 0.5721 | 1.0351 | 0.0132 | 0.0649 | 0.1006 | 2.6838 | 0.5044 |

BEO-PID | 1.7895 | 3 | — | 0.5434 | — | 2.9993 | 0.0013 | — | 0.8167 | — |

_{sett} have the least values. Thus, the time for frequency to arrive at its steady state is very short. The GA-PID and PFO-PI controllers cannot effectively improve the system dynamic response, where they consume a long time to drive the frequency to its steady state and result in high PeUn.

_{sett}, the particles population (N_{p}), and the number of iterations (N_{m}) are given. The results reveal that the smallest value of ITAE, PeUn, t_{sett}, and algorithm parameters are obtained by the BEO-FOPID controller, indicating its supremacy for LFC.

Optimized controller | Parameters | t_{sett} (s) |
PeUn | ITAE | |||||
---|---|---|---|---|---|---|---|---|---|

N_{p} |
N_{m} |
ΔP_{t} (p.u.) |
Δf_{1} (Hz) |
Δf_{2} (Hz) |
ΔP_{t} (p.u.) |
Δf_{1} (Hz) |
Δf_{2} (Hz) |
||

GA-PID [ |
30 | 100 | 6.08 | 6.87 | 3.48 | −0.0246 | −0.1039 | −0.065 | 0.6012 |

BFO-PI [ |
20 | 100 | 6.625 | 5.46 | 7.02 | −0.0806 | −0.2617 | −0.2261 | 1.827 |

GWO-PID [ |
40 | 100 | 3.34 | 1.06 | 3.17 | −0.021 | −0.1113 | −0.0551 | 0.134 |

SCA-PID [ |
10 | 100 | 3.4636 | 2.6162 | 3.5916 | −0.0229 | −0.1155 | −0.0676 | 0.1516 |

Jaya-PID [ |
10 | 100 | 5.1840 | 4.2747 | 4.2747 | −0.0247 | −0.12 | −0.0723 | 0.0935 |

EOA-PID [ |
20 | 50 | 2.08 | 1.736 | 2.96 | −0.0161 | −0.1114 | −0.0551 | 0.07682 |

BEO-PID | 20 | 50 | 2.0431 | 1.219 | 2.4114 | −0.01654 | −0.0968 | −0.0472 | 0.07675 |

BEO-FOPID | 20 | 50 | 2.1616 | 0.812 | 2.5378 | −0.01532 | −0.0909 | −0.0464 | 0.07445 |

To validate conclusively the implementation of the BEO-FOPID controller for system frequency support, a random load whose value ranges from 0% to 10% of the rated load is created, as revealed in _{1}, Δf_{2}, and ΔP_{t} recovered to zero regardless of the percentage of increasing or decreasing the load, as shown in

In this section, we investigate the performance of the BEO-FOPID controller when the system parameters T_{ge}, T_{go}, and T_{t} are changed by ±50% and a 10% step load perturbation is simultaneously implemented to zone 1, as indicated in

From

The decrease in T_{ge} and the increase in T_{go} harm the system stability where the system includes more oscillation, as shown in

The increase in T_{t} leads to larger values of PeUn and t_{sett} of Δf_{1}, Δf_{2}, and ΔP_{t}, as shown in

∆f_{1} has less t_{sett}, where the frequency reaches its nominal value after 1.8 s at most. On the other hand, ∆f_{2} and ∆P_{t} have greater t_{sett}, where frequency returns to its nominal value after 7 and 4 s at most, respectively.

The BEO-FOPID controller maintains the system frequency within its predefined range even under varying system time constants.

The wind farm chosen for this study, involves 100 wind turbines (WTs; G52/850-GAMESA model [_{WT}) because it relies on wind speed as depicted in the following equation [

_{c-i}, υ_{c-o}, and υ_{n} are the cut-in, cut-out, and rated speeds of the WT, respectively. The small-capacity wind generators are modeled by a huge-capacity generator whose transfer function (TF_{WTG}) is written as:

The wind farm is not contributing to basic frequency control as we suppose the wind generators are running at the maximum power point tracking (MPPT) [

The PV array (KC200GT-Kyocera model) we studied has 200 kW rated power at standard test conditions (STCs), that is, radiation (G) of 1000 W/m^{2} and ambient temperature (T_{amb}) of 25°C. One hundred PV arrays are parallelly linked to supply 20 MW (0.01 p.u.) at STCs. The PV energy is linked just after zone 1 thermal turbine shown in _{PV}) is mainly proportional to G and weakly inversely proportional to T_{amb} as described in

Here, T_{amb} is supposed to be constant at 25°C, so P_{PV} is proportional to G only. In this article, the efficiency of MPPT systems of PV (η_{MPPT}) is supposed to be 98%. The transfer functions of the inverter (TF_{inv}) and interconnection device (TF_{i_c}) are written in

In this study, we have suggested using BEO for tuning the FOPID controllers to support system frequency. The objective function has minimized the deviations of frequencies and tie-line power. A two-zone interlinked power system has been utilized to confirm the efficacy of the BEO-FOPID controller for LFC. Compared with other methods, the FOPID controller based on BEO has performed well for supporting system frequency when the load in zone 1 increased by 10% of the rated load. Furthermore, the BEO-FOPID controller is effective when loading in zone 1 disturbed randomly from 0% to 10% of the rated load. We also investigated the impact of variation of the system time constants by −50% and +50% with simultaneous load increase by 10% on the system performance. The results have proved that these time constants variations harm the system stability while the BEO-FOPID controller has retained its ability to maintain the system frequency at its set value. Moreover, the uncertainty of wind and solar energies has been also investigated where the simulation results have confirmed the capability of the BEO-FOPID controller for supporting the system frequency.

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number “IF_2020_NBU_434”.

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