Background

An interconnected power system comprises several control areas connected by tie lines to exchange power among them. The load in power systems is never steady, it continually changes with rising and falling trends. Failing to match any small sudden load change in any control area in an interconnected power system will change the system frequency and power flows in the tie lines. Large deviations in frequency can lead to power system instability and large fluctuations in tie-line power flows. Therefore, it is important in an interconnected power system to maintain an active power balance in all control areas to keep the system frequency and the tie-line power flows as close as possible to their scheduled values. In a large power interconnected system, the load is matched at the control area level by regulating the active power produced by generators in the control area using load frequency control (LFC). The balance in a control area is reached when the scheduled power exchange with neighbouring control areas equals the actual power exchange. In the presence of continually varying load, balancing active power between generation and load is a very challenging task, requiring superior controllers for the load frequency control. Widely used controllers are the Proportional-Integral (PI), and Proportional-Integral-Derivative (PID) controllers because of their simple structure [1,2,3]. There are several traditional and modern methods for tuning these controllers to get the best performance out of them.

The integration of renewable energy sources (RES) into conventional power systems has become a defining imperative in the quest for sustainable and environmentally responsible energy generation [4, 5]. As the global community strives to reduce carbon emissions and transition towards cleaner energy sources, RES such as wind and solar have gained widespread attention for their potential to reshape the power generation landscape. However, integrating renewable energy sources into the power grid presents significant challenges [6,7,8]. Due to their intermittency and randomness, renewable energy sources complicate the balancing of active power between generation and load. This increased complexity necessitates the use of more advanced controllers for load frequency control [9, 10].

Literature review

Several different methods have been proposed in the literature to achieve more efficient LFC strategy capable of maintaining active power balance between generation and load in the presence of severe power system disturbances. These methods include state estimation techniques like Kalman filtering [1], Extended and Unscented Kalman Filter [11], data-driven modeling and system identification approaches [6], reinforcement learning-based control [12,13,14,15], fuzzy logic control for rule-based adaptability [1, 16,17,18], and signal processing methods such as the wavelet transform [19]. Among these diverse methodologies, H-infinity (H∞) control stands out as a control theory approach that seeks to design controllers to minimize the worst-case effects of uncertainty and disturbances in a system [20,21,22]. The H∞ control has proved to play a crucial role in achieving robust and optimal regulation of the power system’s frequency and tie-line power flow while accounting for uncertainties and disturbances [20,21,22,23,24]. Model predictive control (MPC) is another advanced control strategy that offers a predictive approach to control, allowing for real-time optimization of control actions based on predictions of system behavior [23,24,25]. However, these proposed methods in the literature to increase the control quality of RES-integrated power systems exhibit various limitations. Kalman filtering, while effective, can be computationally intensive [10, 26, 27]. Advanced state estimation techniques, such as the Extended Kalman Filter and Unscented Kalman Filter [11], can be computationally demanding, making real-time implementation challenging. Fuzzy logic control relies on expert knowledge and rule-based systems, potentially making it less adaptable to unforeseen changes [28, 29]. H-Infinity control is generally complex to implement and necessitates a clear understanding of system uncertainties and performance specifications. MPC comes with computational intensity, potential latency, complex implementation, and sensitivity to modelling errors [23, 30, 31].

In response to the challenges and computational demands posed by these advanced control strategies in load frequency control, researchers are actively focusing on the use of fixed gain controllers. The design method of this type of controller is a two-step procedure consisting of determining the controller structure and finding a suitable method to calculate its parameters. In this controller design approach, researchers are actively exploring the use of metaheuristic optimization techniques to obtain the optimal gain parameters for the fixed gain controllers. This pursuit extends to traditional controllers such as Proportional-Integral (PI) [32] and Proportional-Integral-Derivative (PID) [33, 34], along with innovative controller structures like the cascaded Tilt Integral Derivative–PID (TID-PID) tuned by Grey Wolf Optimization (GWO) algorithm [35] and Teaching Learning-Based Optimization (TLBO) algorithm [36]. The literature presents a diverse array of controller designs, including the introduction of a firefly-optimized fuzzy PID controller [37], the application of the Flower Pollination algorithm for gain parameter optimization in a cascaded PI-PD controller [38], and the optimal tuning of a PID controller with derivative noise filters (PIDN) using Particle Swarm Optimization (PSO) algorithm [39]. In 2018, Genetic Algorithm (GA) was employed to optimally tune a cascaded PD-PID controller with double derivative filters (PDPID plus DDF) [40]. In 2021, improved frequency deviation results were achieved with this same structure by employing the Symbiotic Organism Search (SOS) algorithm to tune it [41].

The traditional controllers optimized for a specified operating condition may sometimes not work efficiently where operating conditions change with continuously varying load demands [33] and high penetration of RES. Hence, fractional order controllers are gaining the attention of researchers to enhance further the efficacy of LFC. Fractional order controllers introduce greater flexibility through the incorporation of fractional order derivative and integral terms [42]. Studies have shown that they provide better performance in various power system structures [42]. In related works, Lion Algorithm optimized fractional order proportional integral (FOPI) controller was applied in a two-area power system in [43]. FOPID has also been used in a number of studies for LFC tuned with different metaheuristics like the Big Bang Big Crunch (BBBC) optimization algorithm [44], Bacteria Foraging Technique (BFT) [45], and a hybrid Genetic Algorithm-Firefly Algorithm (hGA-FA) [46]. Several different cost functions like the integral time absolute error (ITAE) [46], integral time square error (ITSE) [47], integral square error (ISE) [45], and integral absolute error (IAE) [44] have been used. More sophisticated cascaded fractional order controllers in the literature include FOPI-FOIDN optimally tuned by Crow Search Algorithm (CSA) [48], PIDN-FOID optimized by the Whale Optimization Technique (WOT) [49], WOT tuned IDN-FOPD controller [50], PIFOD-(1 + PI) tuned by Yellow Saddle Goatfish Algorithm (YSGA) [51] and FOPI-FOPD tuned by Sine Cosine Algorithm (SCA) [52].

Research gap and challenges

The presence of high-frequency noise introduces a vulnerability in fractional order controllers with derivatives, for example, when used for hydropower [42]. In response to this challenge, researchers have predominantly favoured the utilization of Fractional Order Proportional-Integral (FOPI) [43, 53] and Integral (FOI) controllers [6] particularly for hydro plants, while fractional order derivative controllers have predominantly found applications in the secondary control for thermal plants in the literature [49,50,51,52,53]. The FOPI and FOI controllers, which do not incorporate fractional order derivatives, may exhibit slightly more oscillations around the desired frequency as compared to fractional order controllers with derivative action included [42]. In an attempt to use fractional order controllers with derivative action on hydro plants, fuzzy logic controllers have been cascaded with ICA-tuned FOPI-FOPID in the study reported in [54] and PIDN-FOPIDN reported in [55]. The main drawback of fuzzy logic controllers in load frequency control is their inherent difficulty in precisely capturing and modeling complex nonlinear system dynamics [54, 56].

Contribution

To address the above challenges more effectively, this paper includes an optimized derivative filter in the FOPI-FOPID structure to obtain FOPI-FOPIDN for a control area, which contains hydro and thermal plants, as well as renewable energy sources (RES). The gain parameters of the proposed controller consisting of fractional order parameters, and the filter coefficient “N” are then optimized using a PSO variant, which like the standard PSO, has few parameters to tune and is easy to program. The PSO variant called ADIWACO, developed by the authors, combines adaptive and dynamic techniques to adjust the inertia weight and acceleration coefficients of the standard PSO (SPSO) [57]. ADIWACO has proved to be effective in tuning traditional PID controllers for load frequency control of interconnected power systems with RES [58].

Paper organization

The rest of the paper is structured as follows: “Test systems” section provides a detailed description of the test systems utilized in this study. In “FOPIFOPIDN controller” section, the proposed FOPI-FOPIDN controller is presented. “Improved PSO (ADIWACO)” section gives an overview of the PSO variant, ADIWACO, used to optimally tune the FOPI-FOPIDN controller. “Implementation” section presents the simulation results, accompanied by discussions highlighting the efficacy of the proposed methodology. Finally, “Optimal tuning of the controllers” section concludes the paper with a summary of key findings, implications, and avenues for future research.

Test systems

Two widely used test systems depicted in Figs. 1 and 2 are considered. Their comprehensive details are available in [6, 19, 34,35,36, 54, 55, 59, 60]. Test system 1 represents a two-area power system having thermal reheater and hydro plants as conventional sources in each area. In addition, wind and solar units are connected in area 1. The second test system, which is a three-area power system, is used to verify the scalability of the proposed LFC strategy. This power system has a thermal reheater plant in each control area, representing all conventional coherent generators within the control area. Additionally, wind and solar units are integrated in Area 1.

Fig. 1
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Test system 1

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Fig. 2
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Test system 2

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The transfer functions for the wind turbine plant, G(s)WTG, and solar PV plant, G(s)PV, given in Figs. 1 and 2, are defined as follows [6]:

$$G\left( s \right)_{WTG} = \frac{{K_{wTG} }}{{1 + sT_{wTG} }}$$
(1)
$$G\left( s \right)_{PV} = \frac{{K_{PV} }}{{1 + sT_{PV} }}$$
(2)

where KWTG and KPV are the respective gains of the wind turbine and solar PV plants and TWTG and TPV are their respective time constants.

FOPIFOPIDN controller

The proposed load frequency control strategy is based on a fractional-order Proportional Integral cascaded with a fractional-order Proportional Integral Derivative with a derivative filter (FOPI-FOPIDN). The structural representation of the cascaded controller is shown in Fig. 3.

Fig. 3
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Structure of the FOPI-FOPIDN cascaded controller

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The FOPI serves as the master control, addressing tie-line power and frequency deviations using fractional-order calculus [7]. Its parameter λ1 represents the non-integer order of the integrator and \(K_{P1}\) and \(K_{I1}\) are the gains of the proportional and integral terms. The output of the FOPI controller is sent to the FOPIDN controller for further fine-tuning and enhanced disturbance rejection. Its parameters λ2 and μ denote the non-integer order of the integrator and differentiator, respectively, while N represents the derivative filter coefficient. The parameters \(K_{P2}\), \(K_{I2}\) and \(K_{D}\) are the gains of the proportional, integral and the derivative terms. The complete transfer function of the cascaded FOPI-FOPIDN system is as follows:

$$C\left( s \right)_{FOPI - FOPIDN} = \left( {K_{P1} + \frac{{K_{I1} }}{{s^{\lambda 1} }}} \right)\left( {K_{P2} + \frac{{K_{I2} }}{{s^{\lambda 2} }} + \frac{{NK_{D} s^{\mu } }}{{s^{\mu } + N}}} \right)$$
(3)

The main contribution of this methodology is the inclusion of the derivative filter to alleviate the impact of high-frequency noise. The challenge of tuning the filter coefficient to strike a balance between noise attenuation and control responsiveness is effectively addressed through the application of a suitable metaheuristic algorithm. The primary objective of the load frequency control is to maintain a zero Area Control Error (ACE). It is given by (4) for the two-area power system and by (5) for the three-area power system [54].

$$\left\{ {\begin{array}{*{20}l} {ACE_{1} = B_{1} \Delta F_{1} + \Delta P_{tie } } \hfill \\ {ACE_{2} = B_{2} \Delta F_{2} + \alpha_{12} \Delta P_{tie } } \hfill \\ \end{array} } \right.$$
(4)
$$\left\{ {\begin{array}{*{20}l} {ACE_{1 } = B_{1} \Delta F_{1} + \Delta P_{tie,1 } } \hfill \\ {ACE_{2 } = B_{2} \Delta F_{2} + \Delta P_{tie,2 } } \hfill \\ {ACE_{3 } = B_{3} \Delta F_{3} + \Delta P_{tie,3 } } \hfill \\ \end{array} } \right.$$
(5)

and [54]

$$\left\{ {\begin{array}{*{20}l} {\Delta P_{tie,1} = \Delta P_{tie,12} + \Delta P_{tie,13} } \hfill \\ {\Delta P_{tie,2} = \alpha_{12} \Delta P_{tie,12} + \Delta P_{tie,23} } \hfill \\ {\Delta P_{tie,3} = \alpha_{13} \Delta P_{tie,13} + \alpha_{23} \Delta P_{tie23} } \hfill \\ \end{array} } \right.$$
(6)

where Bi is the bias coefficient of control area i, ΔFi is the change in frequency of control area i and ΔPtie,ij, is the change in tie-line power transported from control area i to control area j, \(\alpha_{ij}\) is the area rating ratio of control area i to control area j.

To achieve the primary objective of the LFC, it will be necessary to find the optimal parameters of the FOPI-FOPIDN controller. This task becomes increasingly complex as the number of controllers needed increases with the number of control areas. This challenge is addressed in this study using the PSO variant called ADIWACO in [57]. The integral time-absolute error (ITAE) given by (7) for the two-area power system and by (8) for the three-area power system is used as its fitness functions [34, 54].

$$J_{ sys, 1} = \mathop \smallint \limits_{0}^{T} \left[ {\left| {\Delta f_{1} } \right| + \left| {\Delta f_{2} } \right| + \left| {\Delta P_{tie,1} } \right|} \right]t dt$$
(7)
$$J_{ sys, 2} = \mathop \smallint \limits_{0}^{T} \left[ {\left| {\Delta f_{1} } \right| + \left| {\Delta f_{2} } \right| + \left| {\Delta f_{3} } \right| + \left| {\Delta P_{tie,1} } \right| + \left| {\Delta P_{tie,2} } \right| + \left| {\Delta P_{tie,3} } \right|} \right]t dt$$
(8)

where \(\Delta f_{i}\) represents the change in frequency of control area i and \(\Delta P_{tie, i}\) the change in the total tie line power transported from control area i.

The parameters to be optimally determined for each controller are Kp1, Kp2, KI1, KI2, KD, N, λ1, λ2 and μ. These parameters are determined subject to the constraints given by (9) from [61]. One controller is required for each control area of a power system.

$$set constraints \left\{ {\begin{array}{*{20}c} {0 < K_{P} < 10} \\ {0 < K_{I} < 20} \\ {0 < K_{D} < 5} \\ {0 < \mu < 2} \\ {0 < \lambda < 2} \\ \end{array} } \right.$$
(9)

Improved PSO (ADIWACO) [56]

The standard PSO is a swarm-based optimization technique inspired by the collective behavior of bird flocks or fish schools. In the PSO, a population of potential solutions, represented as particles, explores the solution space by adjusting their positions based on their own best-known solutions and the globally best solution found by the entire population [62]. Mathematically, velocity, v and position, x of each particle are updated iteratively using the following equations [62]:

$$v_{i} \left( {t + 1} \right) = w v_{i} \left( t \right) + c_{1} \left( {p_{i} \left( t \right) - x_{i} \left( t \right)} \right)$$
(10)
$$x_{i} \left( {t + 1} \right) = x_{i} \left( t \right) + v_{i} \left( {t + 1} \right) + c_{2} \left( {g\left( t \right) - x_{i} \left( t \right)} \right)$$
(11)

where \(w =\) the inertia weight \(c_{1} , c_{2} =\) acceleration coefficients, \(x_{i} \left( t \right)\) = the current position of a particle, \(x_{i} \left( {t + 1} \right)\) = the updated position of a particle, \(p_{i} \left( t \right)\) = the personal best of a particle, \(g \left( t \right)\) = the global best of a particle,\(v_{i} \left( t \right)\) = the velocity of a particle and \(v_{i} \left( {t + 1} \right)\) = updated velocity of the updated particle with the position \(x_{i} \left( {t + 1} \right)\) [62].

The standard PSO uses constant inertia weight and acceleration coefficients [63]. The improved PSO in [57] enhances its performance by employing adaptive dynamic inertia weight and acceleration coefficients. The inertia weight, w is defined as follows:

$$w = \mu \tanh \delta$$
(12)

where

$$\mu = \frac{{Personal _{best} - Global_{best} }}{{Personal_{best} }}$$
(13)
$$\delta = W_{max} - \frac{{\left( {W_{max} - W_{min} } \right) \times the~ number~ of~ the ~current ~iteration }}{Maximum ~number~ of ~iterations }$$
(14)

where Wmax and Wmin represent the upper and lower limits of the inertia weight respectively. The parameter μ lies in the range [0, 1]. The acceleration coefficients are calculated at each iteration as follows:

$$c_{1} = c_{2} = \mu \cosh \psi$$
(15)

where

$$\psi = C_{max} - \frac{{\left( {C_{max} - C_{min} } \right) \times the\;number\;of\;the\;current\;iteration }}{Maximum\;number\;of\;iterations}$$
(16)

Tuning algorithm

The ADIWACO PSO is used as follows to obtain the optimal values of Kp1, Kp2, KI1, KI2, KD, N, λ1, λ2 and μ for each of the cascaded fractional order controllers (FOPI-FOPIDN).

figure a

Implementation

The performance of the proposed LFC strategy is evaluated on the test systems presented in Figs. 1 and 2 using MATLAB / Simulink Software (R2023a). The computer setup used for the testing has the following specifications: Windows 11 (64-bit) for the software environment and an Intel(R) Core (TM) i5-8250U CPU @ 1.60 GHz 1.80 GHz with 24.0 GB installed RAM for the hardware environment. The parameters of the two-area and the three-area power systems are presented in the “Appendix” section.

Optimal tuning of the controllers

The parameters of the PSO variant (ADIWACO) are presented in the “Appendix” section. For successful implementation of the algorithm, a maximum number of iterations of 100, commonly used in the literature for metaheuristic algorithms for this type of application, is chosen. The tuning was done using a step load perturbation of 0.1 pu in area 1 of the power systems. The convergence rate curve given in Fig. 4 shows that the algorithm converged in fewer than 30 iterations for both test systems and that the maximum iterations of 100 was more than it was required.

Fig. 4
figure 4

Convergence profile of proposed algorithm for a test system 1 and b test system 2 with FOPI-FOPIDN controller

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Testing

The effectiveness of the LFC strategy using FOPI-FOPIDN controllers is assessed on both test systems, employing optimal parameters obtained through the ADIWACO algorithm. The assessment is done using step load perturbation, combined random load, PV and wind perturbations, and system parameters variation. The resulting ITAE values, frequency, and tie-line power responses are compared with those of various strategies from the literature to establish its superiority. Models of the test systems, with and without physical constraints, are considered. The parameters obtained for the FOPI-FOPIDN controllers deemed to be identical during tuning are presented in Table 1.

Table 1 Optimal gain parameters obtained
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On the test system 1, widely used in the literature to test PID controllers, the performance of the proposed FOPI-FOPIDN controllers is benchmarked against PID controllers with gain parameters determined by ADIWACO [58], Magnetotactic Bacteria Optimizer (MBO) [6], Grey Wolf Algorithm [34], and a hybrid Firefly Algorithm and Pattern Search Technique [64]. Additionally, it is compared with cascaded fuzzy PID-fractional-order PID with double derivative filters (FPIDN-FOPIDN), tuned by the Imperialist Competitive Algorithm (ICA) [59].

Similarly, on test system 2, the comparison controllers are PID tuned by ADIWACO [58], FPIDN-FOPIDN tuned by ICA [59], Cascaded Fuzzy FOPI–FOPID tuned by ICA [54], Fuzzy FOPI–FOIDN tuned by Crow Search Algorithm (CSA) [65], and FOPI–FOPID with no derivative filter tuned by ADIWACO.

Results and discussion

Models without physical constraints

Test Systems 1 and 2 are employed for various experimental scenarios.

Step load perturbation: test system 1

An incremental step load perturbation of 0.1 pu is applied in area 1 of Test System 1 as the sole disturbance. The responses of area one frequency, area two frequency, and tie line power flow for the six LFC strategies are compared in Figs. 5, 6 and 7, and ITAE values in Table 2. From the curves, the LFC strategy based on the proposed ADIWACO-tuned FOPI-FOPIDN controllers outperforms all the variously tuned PID controllers and the cascaded fuzzy PID-fractional-order PID with double derivative filters (FPIDN-FOPIDN) in terms of deviations in all the responses and thus yielding the best overshoot, undershoot and settling time. The steady-state errors are zero for all. In terms of the ITAE values, the ADIWACO-tuned FOPI-FOPIDN controller shows 10.7826% improvement on ICA-tuned FPIDN-FOPIDN, 30.96% on ADIWACO tuned PID, 96.29% on hFA-PS tuned PID, 96.3797% on GWO tuned PID and 99.99% on MBO tuned PID controllers. This confirms the superior performance of the proposed FOPI-FOPIDN-based LFC over the other LFC strategies in mitigating the impact of step load perturbations.

Fig. 5
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Area 1 change in frequency, test system 1, SLP = 10%

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Fig. 6
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Area 2 change in frequency, test system 1

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Fig. 7
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Change in tie line power, test system 1

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Table 2 ITAE values for step perturbation—test system 1 (sampling time = 10 s)
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Step load perturbation: test system 2

The same incremental step load perturbation of 0.1 pu is applied in area 1 of the Test System 2. The responses of the area frequencies and the tie-line power flows for the six LFC strategies are compared in Figs. 8, 9, 10, 11, 12, 13 and 14. Their ITAE values are presented in Table 3. In this scenario also, the curves clearly show that the proposed LFC strategy gives the least settling time, overshoot and undershoot in the responses of the tie-line power flows and frequencies. The proposed strategy also demonstrates remarkable performance improvements over the comparison LFC strategies in terms of the ITAE values. Specifically, it shows 92.6864% improvement on the ICA tuned FFOPI-FOPID, 99.0593% over the ICA tuned FPIDN-FOPIDN and 99.7628% on ADIWACO tuned PID. The highest improvement, representing 99.9986%, is obtained over that of a recent CSA-tuned FFOPI-FOIDN. The least improvement of 74.4848% is obtained over that of ADIWACO-tuned FOPI-FOPID with no derivative filters. These results reaffirm the superiority of the proposed FOPI-FOPIDN-based LFC strategy over the comparison controller in mitigating the impact of step load perturbations.

Fig. 8
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Area 1 change in frequency for test system 2

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Fig. 9
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Area 2 change in frequency for test system 2

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Fig. 10
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Area 3 change in frequency for test system 2

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Fig. 11
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Area 1 change in tie line power for test system

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Fig. 12
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Area 2 change in tie line power for test system 2

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Fig. 13
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Area 3 change in tie line power for test system 2

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Fig. 14
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RLP and changes in solar and wind generations

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Table 3 ITAE values for step perturbation—test system 2 (sampling time = 20 s)
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Renewable energy source integration: test system 1

A random load perturbation combined with wind and solar generation perturbations as in [29] is applied to area 1 of Test System 1. The three perturbations are shown in Fig. 14. This scenario is used to verify the performance of the controllers in the presence of random load and variable renewable energy sources. The simulation results for the proposed LFC strategy and the comparison LFC strategies are presented in Figs. 15, 16 and 17. Additionally, Table 4 provides the corresponding ITAE values. The proposed LFC strategy gives the least deviations in all the responses. This is confirmed by its ITAE value which is the least closely followed by ICA tuned FPIDN-FOPIDN. The results clearly indicate its robustness in the presence of severe power disturbance.

Fig. 15
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Scenario 2 Δf1 for test system 1

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Fig. 16
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Scenario 2 Δf2 for test system 1

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Fig. 17
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Scenario 2 ΔPtie for test system 1

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Table 4 ITAE values for renewable source integration—test system 1 (sampling time = 20 s)
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Renewable energy source integration: test system 2

The disturbance in Fig. 14 is also applied in area 1 of Test System 2. The relevant responses are presented in Figs. 18, 19, 20, 21, 22 and 23 and the ITAE values in Table 5. Consistent with the previous results, the proposed LFC strategy demonstrates the least deviations in all responses and the lowest ITAE value. This reaffirms its superior performance when subjected to severe power disturbances and highlights its potential application to larger interconnected power systems.

Fig. 18
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Area 1 change in frequency for test system 2

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Fig. 19
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Area 2 change in frequency for test system 2

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Fig. 20
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Area 3 change in frequency for test system 2

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Fig. 21
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Area 1 change in tie line power for test system 2

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Fig. 22
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Area 2 change in tie line power for test system 2

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Fig. 23
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Area 3 change in tie line power for test system 2

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Table 5 ITAE values for renewable source integration—test system 2 (sampling time = 40 s)
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System parameters variation

To further assess the robustness of the strategy under uncertain system model parameters, the parameters of the Test System 1: Kps, Tps, Tt, Tp, Tg and R are varied by ± 50% as in [60]. Figures 24 and 25 present a comparison of the responses of area 1 and area 2 frequencies with the nominal and changed parameter values. As seen from the curves, a change in parameters within ± 50% range will not significantly affect the performance of the controller. This affirms that the controller is robust enough to perform just as predicted by the simulation results even if the parameters used for the system model are within ± 50% of the actual power system parameters.

Fig. 24
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Area 1 change in frequency

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Fig. 25
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Area 1 tie line power response

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System model of test system 2 with physical constraints included

Regardless of the controller type or the metaheuristic algorithm used in tuning the controller parameters, the LFC system responds more efficiently without physical constraints, showing quicker recovery to the nominal frequency and less deviation during transients [66]. When non-linear physical constraints are imposed, regular controllers may struggle to meet desired performance, taking longer to restore nominal frequency and damping oscillations after external disturbances [66]. To test the efficacy of the proposed LFC strategy on a more practical system, the following physical constraints are considered in the models of the test systems as shown in Fig. 26: communication time delay (TD), generation rate constraints (GRC) and governor dead band (GDB). These constraints are widely adopted in the literature for realistic assessment of the performance of LFC strategies [60, 64, 66,67,68,69].

Fig. 26
figure 26

Block diagram representation of all three constraints [66]

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Including GRC, GDB, and CTD in test system 1

Various studies on Test System 2 use time delays of 5 ms or 10 ms [33] and typical GRC is quoted as 10% pu/min (equivalent to 0.0017 pu/s) for thermal reheater plants in [66]. It is also a common practice to set Governor Dead Band between 15 and 100 mHz across many countries [69, 70]. Therefore, in this study, values of 10% pu/min, 100 mHz, and 10 ms are chosen for GRC, GDB, and CTD respectively. An incremental step load perturbation of 10% is applied in area 1 of the test system. The simulation results obtained are compared with those of the ICA tuned cascaded fuzzy FOPI-FOPID, which has proved to be competitive in terms of ITAE values, in Figs. 27 and 28. Two cases are considered. In one case, the two controllers are tuned with the physical constraints included in the power system model, and in the other case without the physical constraints. From the two figures, the performance of the two controllers is poor when tuning is without constraints. When tuned with the constraints the proposed LFC strategy clearly shows far better performance than the LFC strategy based on the fuzzy FOPI-FOPID. With the fuzzy FOPI-FOPID, both the frequency and tie-line power flow become unstable.

Fig. 27
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Effects of incorporating constraints during tuning on area 1 frequency response

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Fig. 28
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Effects of incorporating constraints during tuning on Area 1 tie line power response

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Including GRC, GDB, CTD and RES in test system 2

In this study, power disturbances presented in Fig. 29 are simultaneously applied to in area 1 of Test System 2 with GRC = 10% pu/min, GDB = 100 mHz and CTD = 10 ms. The results presented in Fig. 30 again show that proposed control strategy performs better than the cascaded fuzzy FOPI-FOPID if the controllers are tuned with the constraints included in the test system model. These results confirm the robustness and efficacy of the proposed LFC strategy in the presence of variability and intermittency associated with RES and its potential application to real-world power systems with RES.

Fig. 29
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RLP and changes in solar and wind generations

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Fig. 30
figure 30

a Δ \(f_{1}\) for RES integration considering physical constraints on test system 2, b Δ \(P_{1}\) for RES integration considering physical constraints on test system 2

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Parameter variation with physical constraints, RES integration and random step load perturbation

The results in Figs. 31 and 32 show the responses of Area 1 frequency and tie-line power following simultaneous application of the perturbations in Fig. 14 in Area 1 of Test system 1 for three different parameter settings. GRC, GBD and CTD are included in the test system. The parameter variation has little effect on the tie line power response. In the case of the Area 1 frequency response, the effect of the parameter variation is significant between 8 and 15 s. Outside this period, the effect is negligible. The results show the robustness of the proposed controlling strategy against system parameter variations even when GRC, GBD and CTD are included in the test system.

Fig. 31
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Effects of System parameters variation on Δ \(P_{tie1}\)

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Fig. 32
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Effects of system parameters variation on \(f_{1}\)

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Conclusion

This paper presents an LFC strategy based on a cascaded fractional-order proportional integral-fractional-order proportional integral derivative (FOPI-FOPIDN) with a derivative filter. The controller is optimally tuned using a PSO variant known as ADIWACO. The robustness and scalability of the proposed LFC strategy are rigorously tested on a two-area test system and then a three-area test system using step load perturbation, and a combined random load, PV, and wind perturbations. The experimental results obtained are compared with those of recent LFC strategies in the literature in terms of ITAE values and deviations in frequencies and tie-line power flows. The results of the comparison analyses clearly show the superiority of the proposed LFC strategy over the comparison LFC strategies. Furthermore, the proposed control strategy is subjected to more stringent testing by incorporating key physical constraints, namely generator rate constraints, governor dead band, and communication time delays in the test system. The robustness of the controller under uncertain system model parameters is also verified through variations in the power system parameters. The simulation results obtained from the above stringent experimental setups and its results indicate the superior performance of the proposed LFC strategy. Overall, the study shows that the proposed LFC strategy is more effective and robust than the recent comparison strategies and has enormous potential for load frequency control in real-world RES-integrated power systems. The limitation of the proposed method is its reliance on fixed gains, which means it is not an adaptive strategy. Future work should explore ways to enhance the adaptiveness of the proposed method.