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Dual degree branched type2 fuzzy controller optimized with a hybrid algorithm for frequency regulation in a triplearea power system integrated with renewable sources
Protection and Control of Modern Power Systems volume 8, Article number: 48 (2023)
Abstract
The uncertainties associated with multiarea power systems comprising both thermal and distributed renewable generation (DRG) sources such as solar and wind necessitate the use of an efficient load frequency control (LFC) technique. Therefore, a hybrid version of two metaheuristic algorithms (arithmetic optimization and African vulture's optimization algorithm) is developed. It is called the ‘arithmetic optimized African vulture's optimization algorithm (AOAVOA)’. This algorithm is used to tune a novel type2 fuzzybased proportional–derivative branched with dual degreeoffreedom proportional–integral–derivative controller for the LFC of a threearea hybrid deregulated power system. Thermal, electric vehicle (EV), and DRG sources (including a solar panel and a wind turbine system) are connected in area1. Area2 involves thermal and gasgenerating units (GUs), while thermal and geothermal units are linked in area3. Practical restrictions such as thermoboiler dynamics, thermalgovernor deadband, and generation rate constraints are also considered. The proposed LFC method is compared to other controllers and optimizers to demonstrate its superiority in rejecting step and random load disturbances. By functioning as energy storage elements, EVs and DRG units can enhance dynamic responses during peak demand. As a result, the effect of the aforementioned units on dynamic reactions is also investigated. To validate its effectiveness, the closedloop system is subjected to robust stability analysis and is compared to various existing control schemes from the literature. It is determined that the suggested AOAVOA improves fitness by 40.20% over the arithmetic optimizer (AO), while frequency regulation is improved by 4.55% over an AOtuned type2 fuzzybased branched controller.
1 Introduction
1.1 Conspectus of load frequency control
Disparities between generation and loaddemand introduce power system frequency oscillations which can result in serious consequences. Hence, an efficient load frequency control (LFC) strategy is crucial for eliminating such frequency deviations and maintaining the system dynamics at specified values. Consequently, LFC has attracted a lot of interest over the past 20 years [1,2,3]. In modern interconnected power networks, even a small disruption in one location can affect the quality of the power in other areas. Since many different control regions are interconnected, these systems are exceedingly complicated, requiring careful planning and effective LFC approaches [4, 5]. Various electricity providers are competing with one another for customers by maximizing their efficiency and lowering the cost of power. The government has therefore designed a restructured regulation scheme known as "deregulation" for these power industries [6]. Under deregulated conditions, when generation firms (GENCOs) do/don’t participate, an independent system operator (ISO) is obliged to take care of the LFC. While operating in an openmarket, distribution firms (DISCOs) can enter into contracts with any GENCOs of any region based on a disco participation matrix (DPM) to satisfy the load demand [7]. Table 1 provides a brief comparative review of some LFC strategies [1, 4, 8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].
Conventionally, basic secondary controllers, such as I, PI, PID, and PIDN have been used for LFC [15, 29, 30]. However, these basic controllers have many drawbacks, e.g., long response time, high sensitivity to disturbances, inability to handle complicated systems, etc. Several attempts have been made recently to overcome the aforementioned disadvantages. For instance, to obtain better dynamic performance than single degreeoffreedom (DOF) controllers, two/three DOF controllers are used in [16, 31, 32].
A fractionalorder (FO) controller has many evident benefits over an integerorder controller in the frequency regulation domain. An additional DOF, more flexible parameters, and a larger range of slopes are a few of these benefits [33, 34]. As a result, many researchers have used FO controllers for LFC applications [11, 35,36,37]. Cascade controllers are also implemented to improve closedloop responses in LFC applications [5, 18, 38], whereas combinations of cascade and two DOF controllers are reported in [22]. Because of the uncertainties involved with renewable energy, modern power systems (PSs) are becoming increasingly complicated and hence require more advanced control strategies. Systems having nonlinearities can be handled more effectively by fuzzy logicbased adaptive controllers [6, 19, 24]. Consequently, fuzzylogicaided controllers (FLCs) are frequently adapted in series with PID controllers [39, 40], or cascaded with multistage fractional controllers [10, 11, 18, 36]. A combination of FO and type2 FLC is also reported in [41], while other LFC strategies include the use of a model predictive controller (MPC) [42, 43], linear active disturbance rejection control (LADRC) [44], sliding mode control (SMC) [45, 46], TID controller with filter (TIDN) [34, 47]. To improve frequency stability in an Egyptian power system with highlevel renewable penetration, a coordinated control strategy (CCS) is implemented in [12], whereas CCS has also been used in wind and battery PS [48], wind turbine generators and plugin type hybrid electric vehicles (PHEVs) [13], and virtual power plants [20].
Several investigators have considered thermal, hydro, and gas units for determining the efficacy of LFC on the system [9, 49,50,51]. However, very few studies have examined thermoboiler dynamics (TBD), thermalgovernor deadband (TDB), and generation rate constraint (GRC) nonlinearities [23, 52, 53]. Because of the popularity of renewables in recent times, hybrid PSs involving these sources are considered to study the efficacy of modern LFC strategies [54, 55]. Metaheuristic optimizers don't need the knowledge of the system dynamics, and thus, many such algorithms have been used to find optimal controller gains for LFC applications involving the aforementioned nonlinearities (TBD, TDB and GRC). In addition to those listed in Table 1, the previously used algorithms include particle swarm optimizer (PSO) [56], whale optimizer (WO) [57], SMA [58], Harris hawk optimizer (HHO) [59], teaching–learning based optimizer (TLBO) [60], fruit fly optimization (FFO) [61], hybrid particle swarm optimizationgenetic algorithm (PSOGA) [62], hybrid harmony search and cuckoo optimizer algorithm (HSCOA) [63], hybrid modified grey wolf optimized cuckoo search algorithm (MGWOCSA) [26], and African vultures optimizer algorithm (AVOA) [64]. Modern PSs need intelligent controllers that are optimized with such metaheuristic optimizers to eliminate frequency deviations and tieline power variances more effectively.
1.2 Key research gaps and aims
Based on Table 1 and the literature review carried out in Sect. 1.1, the following can be concluded:

1.
In the majority of the available literature on restructured threearea, a simple PS is considered [2, 4, 15].

2.
The high level of uncertainty connected to the rule base cannot be adequately handled by the type1 fuzzy system (T1FS). Hence, type2 fuzzy systems are used to solve this problem [19].

3.
When subjected to an overwhelming computational burden, the performance of both type1 and type2 fuzzy systems declines. This drawback is eliminated with branched fuzzy structures by providing an extra path for control signals [23, 24]. However, type2 fuzzybased branched controllers are yet to be used for threearea PSs.

4.
AO and AVOA have shown better convergence than other contemporary algorithms such as PSO, WOA, TLBO, etc. Hence, AO and AVOA are potential candidates for developing a hybrid algorithm [26, 65, 66].
From the above considerations, it is worth combining the benefits of fuzzy type2, branched controller, AO and AVOA to develop an enhanced LFC strategy for a complex deregulated threearea PS involving nonlinearities of TDB, TBD, and GRC.
1.3 Contributions
In this study, thermal, EV and DRG units are considered in area1. The DRG of area1 includes a WTS and an SP system. Area2 has thermal and gas units, while thermal and geothermal units are connected in area3. Considering frequency and tieline power variations of all the three locations, the effects of step load disturbances (SLD) and random load disturbances (RLD) are investigated. The main contributions of this study are:

1.
A novel hybrid arithmetic optimized African vulture's optimization algorithm (AOAVOA) is developed.

2.
This AOAVOA is used to tune the novel type2 fuzzybased proportional–derivative branched with a dual degreeoffreedom proportional–integral–derivative (T2FPD + 2DOFPID) controller.

3.
The performance and robustness of the proposed AOAVOAtuned T2FPD + 2DOFPID controller are thoroughly studied under various practical constraints (plant model perturbation, communication delay, and deregulation) related to the reliable operation of the challenging threearea PS under study.

4.
Significant performance improvements in comparison with recently reported control strategies are quantified.
1.4 Objectives
The followings are the objectives of this work:

To simulate a threearea hybrid PS as stated in the previous subsection.

To design and simulate the suggested T2FPD + 2DOFPID controller.

To obtain the optimal settings of the T2FPD + 2DOFPID controller using AOAVOA and other recently reported algorithms (WOA, TLBO, SMA, AO, and AVOA) and compare the dynamic responses in the presence of SLD.

To obtain the optimal gains of various existing and T2FPD + 2DOFPID controllers using AOAVOA and to compare their dynamic responses in the presence of SLD and RLD.

To conduct a Bodebased robust stability study.

To analyze the impact of communication delay, including EV and variable DRG.

To compare the outcomes of the suggested LFC scheme against some recently reported strategies.
2 System investigated
A complex threearea PS as shown in Fig. 1 with different GENCOs is considered. The system under consideration contains 6 GENCOs with two in each area. In all the three areas, thermal units have reheated turbines, TDB, TBD, and GRC, while Gas plants also have GRC. This makes the power system under investigation more realistic. A GRC of 10% per min and TDB of 0.05% are considered in all thermal generating units. For the gas unit, a GRC of 20% per min is considered in area2 [4, 16, 44]. Thermal sources contribute 60% of power generation in each area, represented by participation factors (pf_{11} = pf_{21} = pf_{31} = 0.6), whereas gas contributes 40% of power generation in area2 and geothermal contributes 40% of power generation in area3 (pf_{22} = pf_{32} = 0.4). EV and DRG contribute 10% and 30% of power generation (pf_{13} = 0.1, pf_{12} = 0.3) in area1. The DRG of area1 includes a WTS and an SP, which are coupled with a steady power input of 0.0005 pu each. The presence of EVs in the system improves its performance by increasing its dynamic behavior. The total load demand of DISCO1 and DISCO2 is set as 0.01 pu. A DPM is used to make agreements between DISCOs and GENCOs. Since there is a controller for each area in the system under consideration, the tieline power and frequency variances are minimized, and eventually, the area control error (ACE) is also minimized. The combined contribution of \(cp{f}_{11}\) and \(cp{f}_{12}\) is referred to as ‘S_{1}’, and ‘S_{2}’ for \(cp{f}_{21}\) and \(cp{f}_{22}\), etc. S_{1}, S_{7}, and S_{13} are involved in GENCO1 (thermal unit of the first area), whereas S_{2}, S_{8}, and S_{14} are for GENCO2. A similar trend is followed for the rest of the GENCOs. Elements of the DPM indicate the fraction of power contracted by a DISCO from the associated GENCO. For the studied PS, the DPM [23] is
The power flowing through the tieline (scheduled power) is
where \({\Delta P}_{\mathrm{exp}}\) is the power exported by GENCO in an area from DISCOs of the other two areas [23]. Similarly, \({\Delta P}_{\mathrm{imp}}\) is the power imported by GENCO in an area from the rest of the DISCOs. Thus, the error in the tieline power flow [23] can be expressed by:
The area control error (ACE) [55] acts as the input for the controllers which is given as:
where \({B}_{\mathrm{i}}\) is the bias coefficient of the i^{th} area. The notations introduced in this section are appropriately subscripted in Fig. 1 to make them relevant to individual control areas. The GENCO models used in the studied PS are discussed below.
2.1 Thermal unit
The dynamics of a thermal GENCO consist of the following blocks: GRC, TDB, TBD, thermal and reheat turbine. The transfer function models of these blocks are depicted in Fig. 1c, and a brief discussion of the system is presented here.
2.1.1 Generation rate constraint (GRC)
It provides a practical limit to the rate of change in power generation. GRC is always considered as a limiter with the steam turbine as shown in Fig. 1c. It is 10% per min in this work. The detailed modeling of GRC can be found in [67, 68].
2.1.2 Thermal governor with a deadband (TDB)
The thermal deadband is another type of nonlinearity encountered in LFC systems. It is defined as the total magnitude of speed fluctuation in the steady state speed that does not cause the governor valve to change. The TDB is always expressed as a percentage of the rated speed and indicates the level of insensitivity to the speed governing mechanism. Deadband is depicted by \({N}_{1}+{N}_{2}s\), where \({N}_{1}\) and \({N}_{2}\) are the Fourier coefficients. The parameter values of the GDB system are given as \({N}_{1}\) = 0.8 and \({N}_{2}\) = − 0.064 [68, 69].
2.1.3 Thermoboiler dynamics (TBD)
A boiler is a machine used to generate steam under pressure. The block diagram of a drumtype boiler is shown in Fig. 1c. The transfer functions of the pressure control unit, fuel system and boiler storage are [70]:
\({K}_{1}\), \({K}_{2}\) and \({K}_{3}\) are boiler parameters in Fig. 1c. \({K}_{\mathrm{IB}}\) is the boiler integrator gain, \({T}_{\mathrm{IB}}\) is the proportional–integral ratio of the gains, \({T}_{\mathrm{RB}}\) is the recirculation boiler time constant, \({T}_{\mathrm{D}}\) is the fuel system delay, \({T}_{\mathrm{F}}\) is the fuel system time constant, and \({C}_{\mathrm{B}}\) is the boiler storage time constant. The parameter values of the TBD system are given as \({K}_{1}\) = 0.85, \({K}_{2}\) = 0.095, \({K}_{3}\) = 0.92, \({C}_{\mathrm{B}}\) = 200, \({T}_{\mathrm{D}}\) = 0, \({T}_{\mathrm{IB}}\) = 26, \({K}_{\mathrm{IB}}\) = 0.02, \({T}_{\mathrm{F}}\) =25 and \({T}_{\mathrm{RB}}\) = 69.
2.1.4 Thermal turbine
Two main elements affect the dynamic response: (1) entrained steam between the inlet steam pressure and the turbine's first stage; and (2) the storage action in the reheater that causes the lowpressure stage output to lag behind the highpressure stage output. Hence, two temporal constants define the turbine transfer function. For simplicity, the turbine is assumed to have a single equivalent time constant as shown in Fig. 1c. The transfer function of the thermal turbine is [68]:
where \({T}_{\mathrm{t}1}\) = 0.3 is the turbine time constant.
2.1.5 Reheat turbine
Reheaters are specifically designed to raise the saturated steam temperature and give support in regulating steam exit temperature. The transfer function for the reheat turbine is given by [68]:
where \({K}_{\mathrm{r}2}\) is the gain of the reheat turbine and \({T}_{\mathrm{r}2}\) is the reheat turbine time constant, and are given as \({K}_{\mathrm{r}2}\) = 0.3 and \({T}_{\mathrm{r}2}\) = 10 s.
2.2 Geothermal power plant (GTTP) unit
The block diagram of a GTPP is shown in Fig. 1c. The transfer functions of the GTTP governor and GTTP turbine are [71]:
where \({G}_{3}\) and \({t}_{3}\) are the geothermal governor and turbine time constants, respectively, and are given as \({G}_{3}\) = 0.05 and \({t}_{3}\) = 0.1 [23].
2.3 Biogas unit
A biogas unit is made up of a valve position, a fuel system, a gas governor, compressor discharge, and GRC. The block diagram of a biogas unit is shown in Fig. 1c. The transfer functions of the valve position, fuel system, gas governor, and compressor discharge are [72]:
where \({c}_{\mathrm{g}2}\) and \({b}_{\mathrm{g}2}\) are biogas valve operation parameters, while \({X}_{\mathrm{G}2}\) and \({Y}_{\mathrm{G}2}\) are biogas governor operation timeconstants. \({T}_{\mathrm{CR}2}\) and \({T}_{\mathrm{F}2}\) are biogas fuel scheme time constants, and \({T}_{\mathrm{CD}2}\) is the biogas compressive discharge timeconstant. They are given as \({b}_{\mathrm{g}2}\) = 0.049, \({c}_{\mathrm{g}2}\) _{=} 1, \({X}_{\mathrm{G}2}\) = 0.6, \({Y}_{\mathrm{G}2}\) = 1.1, \({T}_{\mathrm{CR}2}\) = 0.01, \({T}_{\mathrm{F}2}\) = 0.239 s, \({T}_{\mathrm{CD}2}\) = 0.2 s, and GRC = 20% per min [23].
2.4 DRG
The solar photovoltaic (SPV) and wind turbine plant (WTP) used in the DRG depicted in Fig. 1c are detailed below.
2.4.1 SPV
The PV system that is connected to the grid is made up of solar panels, converters, power conditioners, and other connecting equipment. The equivalent PV system transfer function comprising the aforementioned components is [23]:
\({T}_{\mathrm{SPV}}\) = 1.8 s is the solar PV time constant in this work.
2.4.2 WTP
The transfer function of the WTP system is [23]:
where \({T}_{\mathrm{WTP}}\) is the wind turbine time constant which is 1.5 s in this work.
2.5 Electric vehicle (EV)
In this work, an aggregate EV model [73] is considered. The power is directly transferred into the grid during peak hours even if the EV aggregator does not participate in the deregulation. Figure 1c displays the block diagram showing the dynamics of an aggregate EV. To prevent sharp spikes brought about by EVs being disconnected from the grid, deadband nonlinearity is considered with primary droop settings thereby limiting the bandwidth of operation between ± 20 mHz. \({K}_{\mathrm{PEV}}\) is the EV gain whose value is determined by the state of charge (SOC) of the EV battery and the time constant is denoted by \({T}_{\mathrm{PEV}}\). \({K}_{\mathrm{PEV}}\) is set to 1 for the idle mode of operation (Fig. 2) while N_{PEV} is the number of EVs in the aggregator. SOC plots are shown in Fig. 2. The parameter values of EV are: \({R}_{\mathrm{PEV}}\) = 2.399, \({K}_{\mathrm{PEV}}\) = 1, \({T}_{\mathrm{PEV}}\) = 1 s, \({N}_{\mathrm{PEV}}\) = 10,000 and \({\Delta P}_{\mathrm{PEV}}\)= ± 5 × 10^{–7} [23]. \({\Delta P}_{\mathrm{PEVi}}\) denotes the incremental power injected in the i^{th} area bounded within \({\Delta P}_{\mathrm{PEV}}^{\mathrm{max}}\) and \({\Delta P}_{\mathrm{PEV}}^{\mathrm{min}}\) which are [73]:
3 Suggested type2 fuzzybased control strategy
FLCs are efficient in dealing with nonlinear systems and a variety of operating circumstances [10]. The block diagram of the proposed T2FPD + 2DOFPID controller is given in Fig. 3. As shown, there are two bifurcations (FPD and 2DOFPID) in parallel. The FLC part aids in adjusting to nonlinear dynamics and variations in operative environments subject to the choice of rulebase and membership functions (MFs). A type2 (T2) fuzzy system (FS) is preferred to avoid the rulebase uncertainties that are present in type1 (T1) FS [24]. The threedimensional surface view of the T1FS and T2FS rulebase is given in [28]. A T2FS is described by a 3D MF [24] as opposed to the 2D MF of T1FS. It is described in terms of MF \({\sigma }_{\widetilde{\mathrm{A}}}(x,\mu )\) subjecting to \(x\in X\) and \(\mu \in {P}_{\mathrm{x}}\subseteq [\mathrm{0,1}]\) such that:
where \(0\le {\sigma }_{\widetilde{\mathrm{F}}}(x,\mu )\le 1\). The 3D MF \(\widetilde{F}\) can be expressed in the following integral form:
where the dual integral is used for the T2 set which represents the inclusion of all allowable x and µ. The uncertainties in MF of the T2 fuzzy set \(\widetilde{F}\) is depicted by a region called the uncertainty footprint (UF), which is the union of all elementary MF lying under \(\widetilde{F}=1\forall \mu \in {P}_{\mathrm{x}}\subseteq [\mathrm{0,1}]\). The complete T2FS mechanism is governed by the operators comprising a fuzzifier, rulebase, inferential system, Treducer, and defuzzifier [74].
3.1 Fuzzifier
A fuzzifier transforms crisp input sets \({[AC{E}_{1},AC{E}_{2}\cdots AC{E}_{\mathrm{n}}]}^{\mathrm{T}}\) into T2 set intervals \(\widetilde{F}\).
3.2 Rulebase
Fuzzy rules are constructed keeping in view the convergence nature of the problem. These rules are similar to the type1 system with the only difference being that MFs are governed. A typical \({K}^{\mathrm{th}}\) rule of T2FLC [74] is given by
where \(\mathrm{k}=\mathrm{1,2}\dots ..\mathrm{N}\) and \({\widetilde{G}}^{k}\) is the corresponding output. \(N\) is the maximum number of rules corresponding to the five MFs (5 × 5).
3.3 Inferential system
The inferential system combines the fuzzy rules to map input T2FS with output T2FS. It evaluates the k^{th} rule to establish the relationship between \({f}^{\mathrm{k}}(x)\) and \({\overline{f} }^{\mathrm{k}}(x)\) as:
where
and
3.4 Treducer
The T2 set obtained from the inferential system needs to be converted into a T1 set, which is done by the Treducer. The centered set method of type reducing is [74]:
The Karnik–Mendel method is used to compute the extremities \({u}_{\mathrm{l}}\) and \({u}_{\mathrm{r}}\) as [74]:
and
3.5 Defuzzification
The final step of defuzzification reverts crisp output from the type1 set. Of the several methods of defuzzification, the centroid of the area approach [36] is used which is given by:
The orientation of various T1 and T2 MFs are given in Fig. 4. The triangular and trapezoidal MFs (rightmost in the upper row of Fig. 4) in [75] for T1FS are modified as presented in the lower rightmost portion of Fig. 4. In this MF, the five linguistic variables are highnegative (\({rf}_{\mathrm{n}2}\)), littlenegative (\({rf}_{\mathrm{n}1}\)), zero (\({rf}_{\mathrm{zo}}\)), littlepositive (\({rf}_{\mathrm{p}1}\)), and highpositive (\({rf}_{\mathrm{p}2}\)). This MF is considered because of its ease of use and rapid computational speed. For this study, the range of MF is chosen from − 1 to 1 [76], and the scaling factors for the input and output are adopted from [25]. The fuzzy rulebase for the proposed controller (Table 2) is adopted from [6]. Reference [55] has demonstrated the merits of a branched controller, while such controller can be further enhanced by adding extra DOF in the branched path. The dualDOF results in the reduction of overshoot, settling time, and rejection of noise, and it comprises two input signals (reference signal and measured output with frequency fluctuations).
4 Optimization
4.1 Arithmetic optimizer
AO is a class of populationgenerated metaheuristic (PGM) approach [77]. The majority of PGM approaches initialize with a set of random solutions, which are incrementally improved through repetitive iterations and evaluated regularly by a given cost function. As PGM approaches are uncertain, solving in a solitary run is rare. As a result, having a large number of repeats increases the likelihood of getting a global optimum. AO starts with some candidate solutions (\(X\)) generated randomly in each iteration [78], shown as:
From (21), the best/finest solution is determined. Once X is initialized, the optimizer decides the optimization phase to enter using the math optimizer acceleration (MOA) function computed as [77]:
The above equation gives the function value \(MOA\left({C}_{\mathrm{Iter}}\right)\) at the ith iteration. \({M}_{\mathrm{Iter}}\) denotes the maximum iteration count, while \(Min/Max\) represent the minimum/maximum MOA values, respectively. The exploration and exploitation phases are elaborated below.
4.1.1 Exploration phase
The math optimization probability (MOP) is expressed by [77]:
where \(MOP\left({C}_{\mathrm{Iter}}\right)\) is the value of MOP in the current iteration (\({C}_{\mathrm{Iter}}\)). \(\alpha\)=5 (in this work) is responsible for the exploitation accuracy. Depending on whether MOA is greater than \({r}_{1}\) or not, AO enters the exploration or exploitation phase. This determines the phase that AO must enter. Exploration is carried out by MOP and other functions and thereby the candidate solutions are updated using these operators in this phase. The following rule [77] is used to repeat the behavior of arithmetic operators by updating the positions:
where the ith solution in the next iteration is specified by \({x}_{\mathrm{i}}\left({C}_{{\text{Iter}} \, }+1\right)\). The jth position of the ith solution in the existing iteration is \({x}_{\mathrm{i}}\left({C}_{\text{Iter}}\right)\), and the \({\text{best}} \, \left({x}_{\mathrm{j}}\right)\) is the jth position in the best‐obtained solution until then. \(\varepsilon\) takes an integer value, and \(U{B}_{\mathrm{j}}/L{B}_{\mathrm{j}}\) represent the upper/lower jth positions, respectively. \(\mu\)=0.5 (in this work) is a parameter that controls the search process. If \({r}_{1}\)<MOA, the exploration phase begins as per (24). A second betadistributed random number (\({r}_{2}\)) is generated. If \({r}_{2}\)<0.5, the division operator is executed thereby ignoring the multiplication operator, whereas if \({r}_{2}\)≥0.5, multiplication is executed thereby ignoring the division. μ enhances the diversity of both operator values and the search space, and hence it is ensured that AO is not stuck at any local optima.
4.1.2 Exploitation phase
When \({r}_{1}\)>MOA, the exploitation phase is initialized. In this phase, either the ‘addition’ or the ‘subtraction’ operator is executed as per Fig. 5. The candidate solutions are updated as follows [77]:
Unlike both operators of (24), Eq. (25) will be able to converge at the optimal solution. Another random number (\({r}_{3}\)) is generated. If \({r}_{3}\)<0.5, the ‘–’ operator is executed, while on the other hand, the ‘+’ operator will be at work when \({r}_{3}\)≥0.5. When the ‘–’ operator is at work, the ‘+’ operator is ignored and vice versa. The scaling coefficient (μ) plays the same role as in the exploration phase to enhance the diversity of the search space. This helps AO to avoid getting stuck in local optima. The candidate solutions are further updated using (25).
4.2 African vulture’s optimizer
The AVOA algorithm was first reported in [64], and mimics the foraging behavior and dwelling behavior of Africa’s vultures. The African vulture population consists of ‘\(N\)’ vultures and the algorithm user determines the size of ‘\(N\)’ based on the requirement. Each vulture has a \(D\)dimensional position space, and the size of \(D\) is determined by the problem's dimension. Similarly, depending on the difficulty of the task, \(T\) is the maximum iterations/vulture actions. Therefore, each vulture's position i (\(1\le i\le N\)) at a particular iteration v (\(1\le v\le T\)) can be expressed with the following position vector [79]:
The African vulture population is divided into three groups based on their living patterns. If the vulture’s superior position is measured using the fitness value (of the likely solution), the first flock is tasked with locating the finest possible position among all vultures, while the subsequent flock believes the aforementioned likely solution to be the best among all the vulture positions. In the foraging stage, AVOA can be separated into the following five steps to simulate different vulture behaviors.
4.2.1 Step1 (Population bunch)
The vulture representing the finest answer is positioned in the first group while the vulture representing the secondfinest position is placed in the second group. The remaining vultures are classified as the third group.
4.2.2 Step2 (Hunger of vultures)
If the vulture is not starving, it has sufficient strength to travel great distances in quest of food. Alternatively, if the vulture is extremely famished, it lacks the physical strength to sustain a longdistance journey. As a result, hungry vultures will become more rampant, and instead of seeking food on their own, they will stick close to the vultures that have food. The exploration and exploitation stages of vultures can thus be formed based on the above behavior. The degree of hunger indicates when vultures are transitioning from the stage of exploration to exploitation.
4.2.3 Step3 (Exploration stage)
With their exceptional vision, vultures possess the ability to find their prey quickly. Therefore, when searching for food, vultures spend more time probing their surroundings and fly long distances in search of prey. In AVOA, vultures can examine a variety of random regions using two separate strategies, while the parameter \({P}_{1}\) decides the selection of any one of the strategies.
4.2.4 Step4 (Exploitation stage)
If \(\left{f}_{\mathrm{i}}^{\mathrm{v}}\right\) has a value between 0.5 and 1, the vulture will reach the intermediate stage. In this stage, \({P}_{2}\) is the variable which is in the range [0,1], and determines whether the vulture competes for food or flies in a circular pattern. Hence, when reaching this stage, \(ran{d}_{\mathrm{P}2}^{\mathrm{v}}\) is a random number in the range [0,1], which is generated before the vulture acts. When \(ran{d}_{\mathrm{P}2}^{\mathrm{v}}\le {P}_{2}\) the vultures perform a food contest, whereas when \(ran{d}_{\mathrm{P}2}^{\mathrm{v}}>{P}_{2}\), the revolving flight action is carried out. When \(\left{f}_{\mathrm{i}}^{\mathrm{v}}\right\ge 0.5\) the vulture is filled with food and energized. Hence, while the vultures are all together at the same moment, the strong vultures will not share their food. To gather food, the feeble ones try to form a swarm and attack the stronger ones. When the vulture is energized, it will not only demonstrate prey contest behavior but also fly at a high altitude. This behavior is modeled by AVOA using a helical model.
4.2.5 Step5 (Exploitation stage)
When \(\left{f}_{\mathrm{i}}^{\mathrm{v}}\right<0.5\), almost the whole population of vultures is fed. However, the best two kinds of vultures have gone starving and feeble after a lengthy period of hunting. The prey will be targeted such that different vultures will concentrate on an identical source of prey. Hence, in this stage, there is a variable \({P}_{3}\) in the range [0,1] that determines aggressive behavior. Before entering this stage, a random number \(ran{d}_{\mathrm{P}3}^{\mathrm{v}}\) in the range [0,1] is generated. When \(ran{d}_{\mathrm{P}3}^{\mathrm{v}}\le {P}_{3}\), the vultures exhibit aggressive behaviour, and when \(ran{d}_{\mathrm{P}3}^{\mathrm{v}}>{P}_{3}\), they exhibit attack behavior. More mathematical details about the aforementioned AVOA steps are presented in [28].
4.3 Proposed AOAVOA
The motivation for developing a hybrid AOAVOA by combining the merits of AO and AVOA is drawn from [66, 80]. Reference [66] hybridizes GWO and CSA whereas HHO and ICA are hybridized in [80]. Both AO and AVOA have their respective strengths and weaknesses. AO shows better convergence than other contemporary algorithms such as PSO, EO, HHO, SMA, TLBO and WOA, as demonstrated in [24]. On the other hand, AVOA also shows better convergence than some contemporary algorithms such as PSO, SMA, AO, and TLBO, as given in [28]. However, AO has some drawbacks, e.g., its location update based on the ideal value, premature convergence, and low solution precision put AO at risk of entering a local optimum [81]. Similarly, AVOA also has some disadvantages such as the existence of poor exploration–exploitation tradeoff and quickly reaching a local optimum [79]. In this study, it aims to achieve an effective optimal solution (parameters of the suggested controller using the hybrid AOAVOA by combining the merits of AO and AVOA). As seen in the flow chart described in Fig. 5, AOAVOA consists of two main phases (AO and AVOA) in cascade. The steps of the AO algorithm [24] are executed initially, and the best solution of AO is then given for initializing vultures in AVOA. Finally, the steps of the AVOA algorithm [28] are executed. Integration of timeweighted absolute error (ITAE) given in (25) is chosen as the cost function and the justification for the same can be found in the subsequent section. For 0.001 ≤ \({K}_{\mathrm{i}}\) ≤ 2, the populationsize and iterationcount are set to 5 and 20, respectively. The goal of employing AOAVOA for this study is to find the optimal controller parameters such that the following objective function (OF) [28] is minimized:
where \({\Delta F}_{1}\); \({\Delta F}_{2}\); \({\Delta F}_{3}\) and \(\Delta {P}_{\mathrm{tie}12}\); \(\Delta {P}_{\mathrm{tie}23}\); \(\Delta {P}_{\mathrm{tie}31}\) are the frequency fluctuations in areas1, 2 and 3 and tieline power fluctuations connecting areas1&2, areas2&3 and areas3&1, respectively. \({T}_{\mathrm{sim}}\) is the simulation time.
5 Results and analysis
The performance of the AOAVOAtuned T2FPD + 2DOFPID control strategy is demonstrated by simulating the interconnected system elucidated in Fig. 1. Nonlinearities (in GRCs of GENCOs) shown in Table 3 are considered to investigate the system's realistic operation. The DISCO’s load demand is chosen as 0.01 pu whereas the DPM used in this study is adopted from [28]. The dynamic responses resulting from controllers designed using various MFs discussed in Fig. 4 (subjected to SLD) are compared in Fig. 6a. The type2 customized triangulartrapezoidal MF used in this work produces better performance than the triangular and Gaussian membership functions. In this section, the supremacy of AOAVOAtuned T2FPD + 2DOFPID control strategy over other controllers and optimizers is established under poolco, bilateral transaction, and contract violation scenarios.
5.1 Poolcobased scheme
In this case, only one customer is considered. This is a government entity that grants power to everyone. Poolco is responsible for maintaining the system and connecting the buyer and vendor by selecting the bidder with the lowest price. A region's GENCOs and DISCOs (with equitable GENCOs) are eligible for LFC participation. Each GENCO is now able to take part in LFC using the following participation factors:\({pf}_{11}\)= 0.1, \({pf}_{12}\) = 0.3, \({pf}_{13}\) = 0.6, \({pf}_{21}\) = 0.6, \({pf}_{22}\) = 0.4, \({pf}_{31}\) = 0.6, and \({pf}_{32}\) = 0.4. The load change is considered only in area1 (areas2 and 3 are unchanged). It is assumed that the load is demanded only by DISCOs from its own area GENCOs. The load demand of all DISCOs is assumed to be 0.005 (\({P}_{\mathrm{L}1}\) = \({P}_{\mathrm{L}2}\) = \({P}_{\mathrm{L}3}\) = \({P}_{\mathrm{L}4}\) = \({P}_{\mathrm{L}5}\) = \({P}_{\mathrm{L}6}\) = 0.005 pu). The DISCO’s load demand is chosen as 0.01 pu whereas the DPM used in this study is adopted from [28]. For this case, the equations for \({P}_{\mathrm{tiescheduled}}\) and power generation in steady state are adopted from [82]. GENCOs generate steadystate power (\({\Delta P}_{\mathrm{G}1}\) = 0.0055, \({\Delta P}_{\mathrm{G}2}\) = 0.0065, \({\Delta P}_{\mathrm{G}3}\) = 0.0030, \({\Delta P}_{\mathrm{G}4}\) = \({\Delta P}_{\mathrm{G}5}\) = \({\Delta P}_{\mathrm{G}6}\) = 0, \({\Delta P}_{\mathrm{tie}12\mathrm{scheduled}}\) = 0.0020, \({\Delta P}_{\mathrm{tie}23\mathrm{scheduled}}\) = \({\Delta P}_{\mathrm{tie}31\mathrm{scheduled}}\) = 0). The GENCO outputs are presented in Fig. 6b whereas the actual tieline power deviations are shown in Fig. 6c. In steady state, all generations reach their specified values (Fig. 6b). Since no power is required by area2 and 3, tieline power flow in the steady state is set to zero (Fig. 6c).
5.1.1 Performance of suggested AOAVOA
To compare the effectiveness of AOAVOA with other algorithms (WOA, EO, SMA, AO, AVOA) in tuning the T2FPD + 2DOFPID controller, a populationsize of 5 and an iterationcount of 20 are chosen for 0.01 ≤ K_{i} ≤ 2 with minimizing ITAE as the cost function. As shown in Fig. 6d, choosing ITAE as the cost function results in better dynamic responses than IAE, ISE, and ITAE.
Hence, ITAE is used as OF in this work. The convergence characteristics of different algorithms are represented by their respective figure of demerits (FODs) in Fig. 6e. It is proved that AOAVOA converges more quickly and has a higher fitness value than other algorithms at the end of 20 iterations. Table 4 displays the T2FPD + 2DOFPID controller settings produced using various optimization strategies. It is evident from Table 4 and the responses to 1% SLD shown in Fig. 6f that the ITAE value of the dynamic response is better for AOAVOA. The comparison of timedomain matrices including settlingtime (T_{s}), maximum overshoot (MOS), and maximum undershoot (MUS)) for various algorithms is presented in Table 5 to further vindicate the effectiveness of AOAVOA.
5.1.2 Performance of T2FPD + 2DOFPID controller optimized by AOAVOA
The suggested AOAVOA is used to find the optimal settings of controllers such as T2FPD + 2DOFPID, type1 fuzzy PD branched with 2DOFPID (T1FPD + 2DOFPID), type2 fuzzy integralderivative branched with 2DOF PID (T2FID + 2DOFPID), type2 fuzzy ID + PD (T2FID + PD), type1 fuzzy ID + PD (T1FID + PD) and PID. The AOAVOAbased settings of various configurations are listed in Table 6. As is evident from Table 6, the suggested T2FPD + 2DOFPID controller results in dynamic responses with the smallest ITAE value. To explore the disturbancerejection capabilities of the system, an SLD of 1% (0.01 pu) is applied to all areas. The responses of different controllers are shown in Fig. 7a. It is clear that the AOAVOAtuned T2FPD + 2DOFPID controller shows better dynamic performance than the other configurations. The AOAVOAtuned T2FPD + 2DOFPID controller considerably reduces the undershoots, overshoots, and settling time as seen in Table 7. Figure 7b illustrates the comparison of the control efforts. In a practical PS, RLD is inevitable. Hence, with the help of RLD applied in area1, the dynamic responses of different controllers are compared in Fig. 7c. The AOAVOAbased controller gains in the presence of this RLD pattern are shown in Table 8. From Fig. 7c, it is evident that the suggested controller restores normalcy more quickly and smoothly than other controller configurations. It is worth mentioning that system stability is maintained even under random loads. Table 9 describes the statistics of 5 independent runs with different algorithms, comprising the maximum (\({ITAE}_{\mathrm{MAX}}\)), minimum (\({ITAE}_{\mathrm{MIN}}\)), average (\({ITAE}_{\mathrm{AVG}}\)) and standard deviations (\({ITAE}_{\mathrm{STD}}\)) of ITAE values obtained from the five independent runs. From Table 9, the superiority of the proposed AOAVOA over the EO, AO, and AVOA algorithms is further affirmed.
5.1.3 Case studies
5.1.3.1 Impact of DRG on system performance
DRG units contribute to controlling power output and frequency deviation. To assess the effects of DRG units on system performance, the DRG in area1 is operated in constant and variable input modes to investigate its impact on dynamic responses. For DRG analysis, the following stepchange pattern is considered:
where u(t) is the unit step signal. The AOAVOAtuned T2FPD + 2DOFPID settings with constant DRG are given in Table 4. The plots for variations in the dynamics of renewable energy sources (using the settings in Table 4) are shown in Fig. 8a. From Fig. 8a, it is clear that frequency fluctuations are mitigated and system stability is maintained even in the presence of variable DRG.
5.1.3.2 Impact of communication delay
Communication time delays (CTDs) of \(\theta\) = 50 ms and \(\theta\) =100 ms are applied to the controller output. The resulting responses are compared with that of a delayfree system as shown in Fig. 8b. As seen, the ability of the suggested strategy to yield satisfactory control action in the presence of CTD is evident.
5.1.3.3 Effect of plugging in EV
EVs contribute toward enhancing grid management infrastructure in the power sector. It is a unique form of distributive energy that can help in balancing unaccounted supply and demand in any area. Figure 8c shows the responses before and after the addition of EVs to the system using the AOAVOAtuned T2FPD + 2DOFPID settings given in Table 4. From the decreased undershoots, overshoots, and settling time shown in Fig. 8c, it is clear that EVs improve system performance appreciably.
5.1.4 Sensitivity analysis
Since it is difficult to have perfect models of PS components, analyzing the impact of parametric variations on the dynamic response is important. Thus, the goal of this analysis is to determine the robustness of the tuned controller against changes in PS parameters. Thermal parameters in area1 are changed by ± 30% in the studies. The time domain specifications are not significantly affected by parametric modifications in dynamic responses as shown in Fig. 9a and Table 10.
5.2 Bilateralbased scheme
In this case, GENCOs can exchange power with any DISCO. DPM is a representation of the power exchange pact among GENCOs and DISCOs. The DPM used for this scenario is adopted from [82]. For bilateral transactions, participation factors are: \({pf}_{11}\) = 0.1, \({pf}_{12}\) = 0.3, \({pf}_{13}\) = 0.6, \({pf}_{21}\) = 0.6, \({pf}_{22}\) = 0.4, \({pf}_{31}\) = 0.6 and \({pf}_{32}\) = 0.4. Table 11 displays the T2FPD + 2DOFPID controller settings produced using various optimization strategies. The load is assumed to be demanded only by DISCOs from its own area GENCOs. The load demand of all DISCOs is assumed to be 0.005 (i.e., \({P}_{\mathrm{L}1}\) = \({P}_{\mathrm{L}2}\) = \({P}_{\mathrm{L}3}\) = \({P}_{\mathrm{L}4}\) = \({P}_{\mathrm{L}5}\) = \({P}_{\mathrm{L}6}\) = 0.005 pu). GENCOs essentially generate steadystate power (\({\Delta P}_{\mathrm{G}1}\) = 0.005, \({\Delta P}_{\mathrm{G}2}\)= 0.007, \({\Delta P}_{\mathrm{G}3}\) = 0.004, \({\Delta P}_{\mathrm{G}4}\) = 0.005, \({\Delta P}_{\mathrm{G}5}\) = 0.005 and \({\Delta P}_{\mathrm{G}6}\) = 0.004, \({\Delta P}_{\mathrm{tie}12\mathrm{scheduled}}\) = 0.001, \({\Delta P}_{\mathrm{tie}23\mathrm{scheduled}}\) = 0.001, and \({\Delta P}_{\mathrm{tie}31\mathrm{scheduled}}\) = 0). The GENCO outputs are presented in Fig. 6b whereas the actual tieline power deviations are shown in Fig. 6c. Moreover, \({\Delta P}_{\mathrm{tie}12\mathrm{scheduled}}\) = \({\Delta P}_{\mathrm{tie}12\mathrm{actual}}\), \({\Delta P}_{\mathrm{tie}31\mathrm{scheduled}}\) = \({\Delta P}_{\mathrm{tie}31\mathrm{actual}}\), and \({\Delta P}_{\mathrm{tie}23\mathrm{scheduled}}\) = \({\Delta P}_{\mathrm{tie}23\mathrm{actual}}\). Hence, \({\Delta P}_{\mathrm{tie}12\mathrm{error}}\)= \({\Delta P}_{\mathrm{tie}23\mathrm{error}}\) = \({\Delta P}_{\mathrm{tie}31\mathrm{error}}\) = 0 [82]. The dynamic responses resulting from different optimizers used to tune the suggested controller are compared in Fig. 9b to vindicate the effectiveness of AOAVOA.
5.3 Contract violationbased scheme.
Contract violations are defined as demanding more electricity from DISCO than was bargained in the agreement. The GENCOs in the same region as the DISCO provide this uncontracted power. Let us assume that DISCO1 is requesting 0.005 pu of additional power, and thus the overall supply (\(\Delta {P}_{\mathrm{L}1}\)) in area1 is the sum of the supplies for DISCO1, DISCO2, and is a violation. The total supply of area2 (\(\Delta {P}_{\mathrm{L}2}\)) is the result of adding a supply of DISCO3 and DISCO4, whereas the total supply of area3 (\(\Delta {P}_{\mathrm{L}3}\)) is the result of adding a supply of DISCO5 and DISCO6. For the case of contract violation, the values of the DPM matrix and the apfs are similar to those of a bilateral transaction. The overall load demand for area1 after requesting an extra 0.005 pu of power from DISCO1 is as follows: supply to DISCO1 & DISCO2 + violation = 0.005 + 0.005 + 0.005 = 0.015 pu. Because of higher power requirements in area1, the steadystate generation for area1 will differ from the bilateral case. \(\Delta {P}_{\mathrm{G}1}\) = 0.005 + (0.005 × 0.005) = 0.005025, \(\Delta {P}_{\mathrm{G}2}\)= 0.007 + (0.005 × 0.005) = 0.007025. The GENCO outputs are presented in Fig. 6b whereas the actual tieline power deviations are shown in Fig. 6c. Table 11 displays the T2FPD + 2DOFPID controller settings produced using various optimization strategies, while Fig. 9c demonstrates the effectiveness of the proposed scheme through the enhanced dynamic responses.
5.4 Stability and robustness
The Laplace transform of the nth area PS shown in Fig. 10 is [23]:
Substituting \({\Delta D}_{\mathrm{n}}(s)\)= 0 and \({\Delta P}_{\mathrm{tie}}(s)\) = 0 into (29) yields:
The loop transfer function for the nth area is given by the above equation. As \({G}_{\mathrm{pn}}\) has nonlinearities such as TDB, TBD, and GRC, it is linearized using a linear analysis toolbox. Using MATLAB/SIMULINK's linear analysis tool, Eq. (30) for area1 is calculated as:
Similar equations for area2 and 3 are:
The Bode diagrams of (31), (32), and (33) are shown in Fig. 11. For the robust stability study, the following three scenarios are considered: (1) with \({G}_{\mathrm{C}1}\), \({G}_{\mathrm{C}2}\), and \({G}_{\mathrm{C}3}\) (controlled) and without perturbation in PS parameters; (2) with \({G}_{\mathrm{C}1}\), \({G}_{\mathrm{C}2}\), and \({G}_{\mathrm{C}3}\) (controlled) and 30% perturbation in PS parameters; and (3) without \({G}_{\mathrm{C}1}\), \({G}_{\mathrm{C}2}\), and \({G}_{\mathrm{C}3}\) (uncontrolled) and perturbation in PS parameters. The stability margins (gain and phase margins: \({GM}_{\mathrm{R}}\) and \({PM}_{\mathrm{R}}\)) and crossover frequencies (\({W}_{\mathrm{gc}}\) and \({W}_{\mathrm{pc}}\)) of Fig. 11 are given in Table 12. Figure 11 shows that the considered perturbation in PS parameters does not disturb the system’s closedloop stability. Because the plots for the uncontrolled system do not exceed the 0 dB line, there is no \({W}_{\mathrm{gc}}\) and associated \({PM}_{\mathrm{R}}\) in Table 12. It is also evident that the proposed controller stabilizes the closedloop system in all three areas.
5.5 Comparison of AOAVOAtuned T2FPD + 2DOFPID method with previously reported works.
The performance of the AOAVOAtuned T2FPD + 2DOF strategy is compared with some contemporary LFC strategies of Parmar et al. [8], Shankar et al. [51], Aryan and Raja [23], Anand et al. [24], Nayak et al. [6], Prakash and Parida [83] in Table 13 and Fig. 12.
5.5.1 Comparison1
The PS version employed in [8] is used for the dynamic response comparison1 shown in Fig. 12(a). It is a twoarea deregulated PS. Reference [8] uses a state feedback controller while [51] used the integral secondary controller. In [23], the EOoptimized type1 Fuzzy FOPI + PIDN strategy is deployed while [24] improves it further by using the AOAtuned type2 fuzzy ID + P controller. The proposed controller is responding with a significantly better performance than the other control schemes as seen in Fig. 12a and Table 13.
5.5.2 Comparison2
Reference [6] investigates a twoarea, two GENCO (thermal + gas) deregulated PS with modified sine–cosine algorithm (MSCA)tuned PID and fuzzy PID controllers. The suggested controller delivers much ameliorated dynamic responses to 10% SLD as is evident in Fig. 12b and Table 13.
5.5.3 Comparison3
In [83], a twoarea interconnected PS with renewable integration such as SPV and WTP is investigated, where LQRbased I and PI controllers are designed. The intermittencies of renewables are seen in the dynamic responses to 1% SLD of both LQRI and LQRPI schemes in Fig. 12c. The suggested controller on the other hand provides an enhanced and stable response. Also, the maximum frequency deviations are significantly improved with the suggested controller as is evident from Table 13.
6 Conclusion
A novel type2 fuzzy proportional–derivative bifurcated with a dual degreeoffreedom proportional–integral–derivative (T2FPD + 2DOFPID) controller is proposed. This controller is tuned using the proposed hybrid arithmetic optimized African vulture’s optimization algorithm (AOAVOA) for a threearea restructured power system (PS). Integration of timeweighted absolute error (ITAE) is chosen as the cost function. The PS chosen for this study includes thermal, electric vehicle (EV), and distributed renewable generation (DRG) sources in area1. Thermal and gas units are considered in area2 whereas thermalgeothermal configuration is considered in area3. The advantage of the suggested AOAVOAtuned T2FPD + 2DOFPID controller is recognized by conducting comparative studies with other controller configurations and optimizers (amid step and random load disturbances). The effect of EV discharge on system dynamics is also studied. A statistical comparison of AOAVOA, arithmetic optimizer (AO), and African vulture’s optimization algorithm (AVOA) is also performed with five independent runs. It is discovered that the maximum ITAE is 1.129 for AOAVOA, 1.219 for AO, and 4.161 for AVOA, the minimum ITAE is 0.023 for AOAVOA, 0.025 for AO and 0.206 for AVOA, the average ITAE is 0.819 for AOAVOA, 1.987 for AO and 3.035 for AVOA, the standard deviation is 0.661 for AOAVOA, 0.700 for AO and 1.623 for AVOA. Finally, the proposed scheme's outcomes are evaluated against a few recently published works, and a robuststability study is also performed. The maximum frequency deviation in area1 is mitigated by 2.89% while that of area2 is mitigated by over 4.55% and tieline power deviation is mitigated by 7.89% as compared to the recent works. The present work can be further enhanced with proper load forecasting with the integration of renewable sources in all control areas.
Availability of data and materials
The authors confirm that data and materials that support the results or analyses presented in this paper are freely available upon request.
Abbreviations
 FOFPID:

Fractional Order FuzzyProportional Integral Derivative
 FPIDNFOI:

Fuzzy Proportional Integral Derivative with filterFractional Order Integral
 MPC:

Model Predictive Control
 FFOPIFOPD:

Fuzzy Fractional Order Proportional IntegralFractional Order Proportional Derivative
 FOD:

Fractional Order Derivative
 2DOFPIDN:

2Degree freedomPID with filter
 IDNFOPD:

Integral Derivative with FilterFractional Order Proportional Derivative
 IT2FPID:

Interval Type2 Fuzzy PID
 CC2DOF (PI)PDF:

Cascaded 2DOF (PI)proportional derivative with filter
 FFOPI + PIDN:

Fuzzy Fractional Order PI + PIDN
 CFPD^{μ} FPI:

Cascade Fuzzy Fractional Order Proportional Derivative with FilterPI
 T2FID + I:

Type2 Fuzzy Integral Derivative + Integral
 STFPI:

Selftuning Fuzzy PI
 TID:

Tilt Integral Derivative
 T2FID + PD:

T2FID + Proportional Derivative
 OHS:

Oppositionbased Harmony Search
 BFOA:

Bacterial Foraging Optimization Algorithm
 ICA:

Imperialist Competitive Algorithm
 MOOF:

Multiobjective Optimization Framework
 MSCA:

Modified Sine Cosine Algorithm
 VPLA:

Volleyball Premier League Algorithm
 WO:

Whale Optimizer
 ISA:

Interactive Search Algorithm
 SA:

Sunflower Algorithm
 MEO:

Modified Equilibrium Optimization
 OVPLA:

Oppositionbased Volleyball Premier League Algorithm
 QOEO:

Quasioppositionbased Equilibrium Optimizer
 SMA:

Slime Mold Algorithm
 AO:

Arithmetic Optimizer
 MGWOCSA:

Modified Grey Wolf OptimizationCuckoo Search Algorithm
 SMES:

Superconducting Magnetic Energy Storage
 SSA:

Salp Swarm Algorithm
 WTS:

Wind Turbine System
 BESSs:

Battery Energy Storage Systems
 HPWHs:

Heat Pump Water Heaters
 HESS:

Hybrid Energy Storage System
 RFB:

Redox Flow Battery
 HAE FC:

Hydrogen Aqua Equalizer Fuel Cells
 SSGT:

Single Shaft Gas Turbine
 SP:

Solar Panel
 TD:

Time Delay
 DRG:

Distributed Renewable Generation
 TBD:

Thermo Boiler Dynamics
 TDB:

Thermal Governor Deadband
 GRC:

Generating Rate Constraint
 ITAE:

Integrated Time Absolute Error
 ITSE:

Integrated Time Squared Error
 ISE:

Integrated Squared Error
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The authors would like to sincerely acknowledge the valuable time devoted by the Reviewers and the EditorinChief to give suggestions to improve the quality of this work.
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Nisha Kumari received her B.Tech. degree in electrical and electronics engineering from Nalanda college of engineering Chandi, Nalanda, India and Master’s degree from National Institute of Technology, Patna in 2018 and 2022. She is currently working towards her Ph.D. in control system with the Indian Institute of Technology (Indian School of Mines), Dhanbad, India. Her research interests include load frequency control, type2 fuzzy control, branched controller, renewable energy, electric vehicles, and optimization. She has authored some peerreviewed papers in international conference and reputed journals.
Pulakraj Aryan (Graduate Student Member, IEEE) received the bachelor’s degree in electrical and electronics engineering from Birla Institute of Technology, Mesra, Ranchi, India in 2016 and Master’s degree from National Institute of Technology, Patna, India in 2020. He is currently working towards the Ph.D. degree in control systems with the National Institute of Technology Patna, India. His current research focuses on optimal control strategies for industrial processes. He has authored several book chapters and has more than 15 publications in peerreviewed international conference proceedings and reputed journals.
G. Lloyds Raja (Member, IEEE and IFACACDOS) received his Bachelors (in Electronics and Communication Engineering) and Master’s (in Embedded System Technologies) degrees from Anna University in 2009 and 2011, respectively. He was awarded Ph.D. from the Electrical Engineering Department of Indian Institute of Technology Patna in 2018. While serving as an Assistant Professor at Kalinga Institute of Industrial Technology (between 2017 and 2020), he visited the department of Automation at Shanghai Jiao Tong University (China) for a short duration as a postdoctoral researcher. Since 2020, he is with the Electrical Engineering Department of National Institute of Technology Patna where he serves as an Assistant Professor. His current research interests include chemical process control, advanced load frequency control strategies, adaptive control and applications of metaheuristic optimization techniques for controller tuning. He has edited two books on control engineering, authored several book chapters and has more than 40 publications in peerreviewed international conference proceedings and reputed journals.
Yogendra Arya (Senior Member, IEEE) received his A.M.I.E. in Electrical Engineering from The Institution of Engineers (India), in 2008, M.Tech. (Instrumentation & Control) from Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Haryana, India, in 2010 and Ph.D. from Delhi Technological University, Delhi, India in 2018. Currently, he is working as Associate Professor with J.C. Bose University of Science & Technology, YMCA, Faridabad, India. He has published several research papers and served as the member of editorial board in various reputed journals. He was awarded best associate editor award for 2021 and 2022 by Journal of Electrical Engineering & Technology (SCIE Journal published by Springer). He was placed in “Top 2% of Researchers in the World” list published by Stanford University, USA for 2020, 2021 and 2022. His research interests include AGC/LFC of conventional/restructured power systems.
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NK: Formal analysis; writing—original draft. PA: Investigation; resources; software; presentation. GLR: Conceptualization; data curation; supervision; writing—review and editing. YA: Validation; writing—review and editing.
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Kumari, N., Aryan, P., Raja, G.L. et al. Dual degree branched type2 fuzzy controller optimized with a hybrid algorithm for frequency regulation in a triplearea power system integrated with renewable sources. Prot Control Mod Power Syst 8, 48 (2023). https://doi.org/10.1186/s41601023003177
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DOI: https://doi.org/10.1186/s41601023003177