Dual degree branched type-2 fuzzy controller optimized with a hybrid algorithm for frequency regulation in a triple-area power system integrated with renewable sources

Abstract

The uncertainties associated with multi-area 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 type-2 fuzzy-based proportional–derivative branched with dual degree-of-freedom proportional–integral–derivative controller for the LFC of a three-area hybrid deregulated power system. Thermal, electric vehicle (EV), and DRG sources (including a solar panel and a wind turbine system) are connected in area-1. Area-2 involves thermal and gas-generating units (GUs), while thermal and geothermal units are linked in area-3. Practical restrictions such as thermo-boiler dynamics, thermal-governor dead-band, 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 closed-loop 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 AO-tuned type-2 fuzzy-based branched controller.

1 Introduction

1.1 Conspectus of load frequency control

Disparities between generation and load-demand 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 open-market, 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].

Table 1 A concise overview of published sources on LFC

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 degree-of-freedom (DOF) controllers, two/three DOF controllers are used in [16, 31, 32].

A fractional-order (FO) controller has many evident benefits over an integer-order 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 closed-loop 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 logic-based adaptive controllers [6, 19, 24]. Consequently, fuzzy-logic-aided controllers (FLCs) are frequently adapted in series with PID controllers [39, 40], or cascaded with multi-stage fractional controllers [10, 11, 18, 36]. A combination of FO and type-2 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 high-level 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 plug-in 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 thermo-boiler dynamics (TBD), thermal-governor dead-band (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 optimization-genetic algorithm (PSO-GA) [62], hybrid harmony search and cuckoo optimizer algorithm (HSCOA) [63], hybrid modified grey wolf optimized cuckoo search algorithm (MGWO-CSA) [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 tie-line 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 three-area, 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 type-1 fuzzy system (T1FS). Hence, type-2 fuzzy systems are used to solve this problem [19].

• 3. When subjected to an overwhelming computational burden, the performance of both type-1 and type-2 fuzzy systems declines. This drawback is eliminated with branched fuzzy structures by providing an extra path for control signals [23, 24]. However, type-2 fuzzy-based branched controllers are yet to be used for three-area 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 type-2, branched controller, AO and AVOA to develop an enhanced LFC strategy for a complex deregulated three-area PS involving nonlinearities of TDB, TBD, and GRC.

1.3 Contributions

In this study, thermal, EV and DRG units are considered in area-1. The DRG of area-1 includes a WTS and an SP system. Area-2 has thermal and gas units, while thermal and geothermal units are connected in area-3. Considering frequency and tie-line 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 type-2 fuzzy-based proportional–derivative branched with a dual degree-of-freedom proportional–integral–derivative (T2FPD + 2DOFPID) controller.

• 3. The performance and robustness of the proposed AOAVOA-tuned 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 three-area 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 three-area 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 Bode-based 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 three-area 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 area-2 [4, 16, 44]. Thermal sources contribute 60% of power generation in each area, represented by participation factors (pf₁₁ = pf₂₁ = pf₃₁ = 0.6), whereas gas contributes 40% of power generation in area-2 and geothermal contributes 40% of power generation in area-3 (pf₂₂ = pf₃₂ = 0.4). EV and DRG contribute 10% and 30% of power generation (pf₁₃ = 0.1, pf₁₂ = 0.3) in area-1. The DRG of area-1 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 tie-line 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 ‘S₁’, and ‘S₂’ for $cp{f}_{21}$ and $cp{f}_{22}$, etc. S₁, S₇, and S₁₃ are involved in GENCO-1 (thermal unit of the first area), whereas S₂, S₈, and S₁₄ are for GENCO-2. 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

Fig. 1

Schematic of a Deregulation b Investigated three-area system c Models of GENCOs

$$DPM=\left[\begin{array}{ccc}cp{f}_{11}& \cdots & cp{f}_{16}\\ \vdots & \ddots & \vdots \\ cp{f}_{61}& \cdots & cp{f}_{66}\end{array}\right]$$

(1)

The power flowing through the tie-line (scheduled power) is

$${\Delta P}_{\mathrm{tiescheduled}}={\Delta P}_{\mathrm{exp}}-{\Delta P}_{\mathrm{imp}}$$

(2)

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 tie-line power flow [23] can be expressed by:

$${\Delta P}_{\mathrm{tieerror}}={\Delta P}_{\mathrm{tiescheduled}}-{\Delta P}_{\mathrm{tierated}}$$

(3)

The area control error (ACE) [55] acts as the input for the controllers which is given as:

$$ACE=\sum {B}_{\mathrm{i}}\Delta {F}_{\mathrm{i}}+{\Delta P}_{\mathrm{tieerror}}$$

(4)

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 dead-band (TDB)

The thermal dead-band 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. Dead-band 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 Thermo-boiler dynamics (TBD)

A boiler is a machine used to generate steam under pressure. The block diagram of a drum-type boiler is shown in Fig. 1c. The transfer functions of the pressure control unit, fuel system and boiler storage are [70]:

$$\left.\begin{array}{c}{G}_{\mathrm{PC}}(s)=\frac{{K}_{\mathrm{IB}}(1+s{T}_{\mathrm{IB}})(1+s{T}_{\mathrm{RB}})}{s(1+0.1{T}_{\mathrm{RB}}s)}\\ {G}_{\mathrm{FS}}(s)=\frac{{e}^{-{T}_{\mathrm{D}}S}}{1+s{T}_{\mathrm{F}}}\\ {G}_{\mathrm{BS}}(s)=\frac{1}{s{C}_{\mathrm{B}}}\end{array}\right\}$$

(5)

${K}_{1}$, ${K}_{2}$ 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 low-pressure stage output to lag behind the high-pressure 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]:

$${G}_{\mathrm{RT}}\left(s\right)=\frac{1}{1+s{T}_{\mathrm{t}1}}$$

(6)

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]:

$${G}_{\mathrm{TT}}\left(s\right)=\frac{1+s{K}_{\mathrm{r}2}{T}_{\mathrm{r}2}}{1+s{T}_{\mathrm{r}2}}$$

(7)

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]:

$$\left.\begin{array}{c}{G}_{\mathrm{GTTPG}}(s)=\frac{1}{1+s{G}_{3}}\\ {G}_{\mathrm{GTTPT}}(s)=\frac{1}{1+s{t}_{3}}\end{array}\right\}$$

(8)

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 Bio-gas unit

A bio-gas unit is made up of a valve position, a fuel system, a gas governor, compressor discharge, and GRC. The block diagram of a bio-gas unit is shown in Fig. 1c. The transfer functions of the valve position, fuel system, gas governor, and compressor discharge are [72]:

$$\left.\begin{array}{c}{G}_{\mathrm{VP}}(s)=\frac{1}{{c}_{\mathrm{g}2}+s{b}_{\mathrm{g}2}}\\ {G}_{\mathrm{GG}}(s)=\frac{1+s{X}_{\mathrm{G}2}}{1+s{Y}_{\mathrm{G}2}}\\ {G}_{\mathrm{FS}}(s)=\frac{1-s{T}_{\mathrm{CR}2}}{1+s{T}_{\mathrm{F}2}}\\ {G}_{\mathrm{CD}}(s)=\frac{1}{1+s{T}_{\mathrm{CD}2}}\end{array}\right\}$$

(9)

where ${c}_{\mathrm{g}2}$ and ${b}_{\mathrm{g}2}$ are bio-gas valve operation parameters, while ${X}_{\mathrm{G}2}$ and ${Y}_{\mathrm{G}2}$ are bio-gas governor operation time-constants. ${T}_{\mathrm{CR}2}$ and ${T}_{\mathrm{F}2}$ are bio-gas fuel scheme time constants, and ${T}_{\mathrm{CD}2}$ is the bio-gas compressive discharge time-constant. 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]:

$${G}_{\mathrm{SPV}}\text{(s) = }\frac{1}{1+s{T}_{\mathrm{SPV}}}$$

(10)

${T}_{\mathrm{SPV}}$ = 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]:

$${G}_{WTP}\text{(s) = }\frac{1}{1+s{T}_{\mathrm{WTP}}}$$

(11)

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, dead-band 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]:

Fig. 2

Characteristic of a ${K}_{\mathrm{PEV}}$ vs SOC in idle mode of operation b ${K}_{\mathrm{av}}$ vs SOC_av

$$\left.\begin{array}{c}\Delta {P}_{\mathrm{PEV}}^{\mathrm{max}}=+\left[\Delta {P}_{\mathrm{PEV}i}\times \frac{1}{{N}_{\mathrm{PEV}}}\right]\\ \Delta {P}_{\mathrm{PEV}}^{\mathrm{min}}=-\left[\Delta {P}_{\mathrm{PEV}i}\times \frac{1}{{N}_{\mathrm{PEV}}}\right]\end{array}\right\}$$

(12)

3 Suggested type-2 fuzzy-based 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 rule-base and membership functions (MFs). A type-2 (T2) fuzzy system (FS) is preferred to avoid the rule-base uncertainties that are present in type-1 (T1) FS [24]. The three-dimensional surface view of the T1FS and T2FS rule-base is given in [28]. A T2FS is described by a 3-D MF [24] as opposed to the 2-D 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:

$$\widetilde{F}=\left.\{((x,\mu ),{\sigma }_{\widetilde{\mathrm{F}}}(x,\mu ))\right|\forall x\in X,\forall \mu \in {P}_{\mathrm{x}}\subseteq [\mathrm{0,1}]\}$$

(13)

where $0\le {\sigma }_{\widetilde{\mathrm{F}}}(x,\mu )\le 1$. The 3-D MF $\widetilde{F}$ can be expressed in the following integral form:

$$\widetilde{F}=\underset{x\in X}{\int }\underset{\mu \in {P}_{\mathrm{x}}}{\int }\frac{{\sigma }_{\widetilde{F}}(x,\mu )}{(x,\mu )}{P}_{\mathrm{x}}\in \left[\mathrm{0,1}\right]$$

(14)

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, rule-base, inferential system, T-reducer, and defuzzifier [74].

Fig. 3

Block diagram of the T2FPD + 2DOFPID controller

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 Rule-base

Fuzzy rules are constructed keeping in view the convergence nature of the problem. These rules are similar to the type-1 system with the only difference being that MFs are governed. A typical ${K}^{\mathrm{th}}$ rule of T2FLC [74] is given by

$$If\,\,ACE_{1} = \tilde{F}_{1}^{{\text{k}}} \,\,\,{\text{and}}\,\,\,{\raise0.7ex\hbox{$d$} \!\mathord{\left/ {\vphantom {d {dt}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${dt}$}}\left( {ACE} \right) = \tilde{F}_{1}^{{\text{k}}} ,\,\,\,{\text{then}}\,\,u = \tilde{G}^{{\text{k}}}$$

(15)

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:

$${F}^{k}\left(x\right)=\left[{f}^{\mathrm{k}}\left(x\right),{\overline{f} }^{\mathrm{k}}\left(x\right)\right]$$

(16)

where

$${f}^{\mathrm{k}}(x)={\underset{\_}{\mu }}_{{\widetilde{\mathrm{r}}}_{1}^{\mathrm{k}}}\left({x}_{1}\right)\times {\underset{\_}{\mu }}_{{\widetilde{\mathrm{r}}}_{2}^{\mathrm{k}}}\left({x}_{2}\right)$$

(17a)

and

$${\overline{f} }^{\mathrm{k}}\left(x\right)={\overline{\mu }}_{{\widetilde{\mathrm{r}}}_{1}^{\mathrm{k}}}\left({x}_{1}\right)\times {\overline{\mu }}_{{\widetilde{\mathrm{r}}}_{2}^{\mathrm{k}}}\left({x}_{2}\right)$$

(17b)

3.4 T-reducer

The T2 set obtained from the inferential system needs to be converted into a T1 set, which is done by the T-reducer. The centered set method of type reducing is [74]:

$${Y}_{\mathrm{centered}}={\int }_{{y}^{1}\in {Y}^{\mathrm{k}}}^{.}\dots {\int }_{{y}^{1}\in {Y}^{\mathrm{k}}}^{.}{\int }_{{f}^{1}\in {F}^{\mathrm{J}}(x)}^{.}\dots {\int }_{{f}^{\mathrm{N}}\in {F}^{\mathrm{J}}(x)}^{.}\left[\sum_{k=1}^{N}{f}^{\mathrm{k}}/\sum_{k=1}^{N}{f}^{\mathrm{k}}{y}^{\mathrm{k}}\right]=\left[{u}_{\mathrm{l}},{u}_{\mathrm{r}}\right]$$

(18)

The Karnik–Mendel method is used to compute the extremities ${u}_{\mathrm{l}}$ and ${u}_{\mathrm{r}}$ as [74]:

$${u}_{\mathrm{l}}=\sum_{\mathrm{j}=1}^{\mathrm{N}}{f}_{\mathrm{l}}^{\mathrm{k}}{y}_{\mathrm{l}}^{\mathrm{k}}/\sum_{\mathrm{j}=1}^{\mathrm{N}}{f}_{\mathrm{l}}^{\mathrm{k}}$$

(19a)

and

$${u}_{\mathrm{r}}=\sum_{\mathrm{j}=1}^{\mathrm{N}}{f}_{\mathrm{r}}^{\mathrm{k}}{y}_{\mathrm{r}}^{\mathrm{k}}/\sum_{j=1}^{N}{f}_{r}^{k}$$

(19b)

3.5 Defuzzification

The final step of defuzzification reverts crisp output from the type-1 set. Of the several methods of defuzzification, the centroid of the area approach [36] is used which is given by:

$${u}_{{\mathrm{COA}}_{r}}=\frac{{\int }_{u}{\mu }_{A}\left(u\right)udu}{{\int }_{u}{\mu }_{A}\left(u\right)du}$$

(20)

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 high-negative (${rf}_{\mathrm{n}2}$), little-negative (${rf}_{\mathrm{n}1}$), zero (${rf}_{\mathrm{zo}}$), little-positive (${rf}_{\mathrm{p}1}$), and high-positive (${rf}_{\mathrm{p}2}$). 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 rule-base 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 dual-DOF 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).

Fig. 4

Various type-1/type-2 MFs

Table 2 Rule-base of the FLC (for T1FS and T2FS)

4 Optimization

4.1 Arithmetic optimizer

AO is a class of population-generated 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:

$$X=\left(\begin{array}{ccc}{x}_{\mathrm{1,1}}& \dots & {x}_{1,\mathrm{n}}\\ \vdots & \ddots & \vdots \\ {x}_{\mathrm{N},1}& \cdots & {x}_{\mathrm{N},\mathrm{n}}\end{array}\right)$$

(21)

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]:

$$MOA\left({C}_{\mathrm{Iter}}\right)=Min+{C}_{\mathrm{Iter}}\times \left(\frac{Max-Min}{{M}_{\mathrm{Iter}}}\right)$$

(22)

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]:

$$MOP\left({C}_{\mathrm{Iter}}\right)=1-\frac{{C}_{\mathrm{Iter}}^{\frac{1}{\mathrm{\alpha }}}}{{M}_{\mathrm{Iter}}^{\frac{1}{\mathrm{\alpha }}}}$$

(23)

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:

$${x}_{\mathrm{i},\mathrm{j}}\left({C}_{{\text{Iter}} \, }+1\right)=\left\{\begin{array}{ll} {\text{best}} \, \left({x}_{\mathrm{j}}\right)\div (MOP+\varepsilon )\times \left(\left(U{B}_{\mathrm{j}}-L{B}_{\mathrm{j}}\right)\times \mu +L{B}_{\mathrm{j}}\right);&\quad {r}_{2}<0.5\\ {\text{best}} \, \left({x}_{\mathrm{j}}\right)\times MOP\times \left(\left(U{B}_{\mathrm{j}}-L{B}_{\mathrm{j}}\right)\times \mu +L{B}_{\mathrm{j}}\right); &\quad {\text{Otherwise}} \, \end{array}\right.$$

(24)

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 beta-distributed 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]:

Fig. 5

Flow chart of hybrid AOAVOA

$${x}_{\mathrm{i},\mathrm{j}}\left({C}_{{\text{Iter}} \, }+1\right)=\left\{\begin{array}{ll} {\text{best}} \, \left({x}_{\mathrm{j}}\right)-(MOP+\varepsilon )\times \left(\left(U{B}_{\mathrm{j}}-L{B}_{\mathrm{j}}\right)\times \mu +L{B}_{\mathrm{j}}\right);&\quad {r}_{3}<0.5\\ {\text{best}} \, \left({x}_{\mathrm{j}}\right)+MOP\times \left(\left(U{B}_{\mathrm{j}}-L{B}_{\mathrm{j}}\right)\times \mu +L{B}_{\mathrm{j}}\right); &\quad {\text{Otherwise}} \, \end{array}\right.$$

(25)

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]:

$${Y}_{\mathrm{i}}^{\mathrm{v}}=\left[{y}_{\mathrm{i}1,}^{\mathrm{v}}\cdot \cdot \cdot \cdot \cdot ,{y}_{\mathrm{id}}^{\mathrm{v}},\cdot \cdot \cdot \cdot \cdot ,{y}_{\mathrm{iD}}^{\mathrm{v}}\right]$$

(26)

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 Step-1 (Population bunch)

The vulture representing the finest answer is positioned in the first group while the vulture representing the second-finest position is placed in the second group. The remaining vultures are classified as the third group.

4.2.2 Step-2 (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 long-distance 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 Step-3 (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 Step-4 (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 Step-5 (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 time-weighted 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 population-size and iteration-count 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:

$${J}_{\mathrm{ITAE}}={\int }_{0}^{{T}_{\mathrm{sim}}}\{|{\Delta F}_{1}|+|\Delta {F}_{2}|+|\Delta {F}_{3}|+|\Delta {P}_{\mathrm{tie}12}|+|\Delta {P}_{\mathrm{tie}23}|+|\Delta {P}_{\mathrm{tie}31}|\}t dt$$

(27)

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 areas-1, 2 and 3 and tie-line power fluctuations connecting areas-1&2, areas-2&3 and areas-3&1, respectively. ${T}_{\mathrm{sim}}$ is the simulation time.

5 Results and analysis

The performance of the AOAVOA-tuned 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 type-2 customized triangular-trapezoidal MF used in this work produces better performance than the triangular and Gaussian membership functions. In this section, the supremacy of AOAVOA-tuned T2FPD + 2DOFPID control strategy over other controllers and optimizers is established under poolco, bilateral transaction, and contract violation scenarios.

Table 3 GENCOs' involvement and GRC [23]

Fig. 6

Comparing MFs, the output of GENCOs, tie line actual derivations, FODs, and dynamic responses of optimizers for Poolco scenario

5.1 Poolco-based 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 area-1 (areas-2 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 steady-state 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 tie-line power deviations are shown in Fig. 6c. In steady state, all generations reach their specified values (Fig. 6b). Since no power is required by area-2 and 3, tie-line 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 population-size of 5 and an iteration-count 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 time-domain matrices including settling-time (T_s), maximum overshoot (MOS), and maximum undershoot (MUS)) for various algorithms is presented in Table 5 to further vindicate the effectiveness of AOAVOA.

Table 4 Settings of proposed T2FPD + 2DOFPID controller obtained from different optimizers

Table 5 Quantitative performance comparison of different optimizers

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, type-1 fuzzy PD branched with 2-DOFPID (T1FPD + 2DOFPID), type-2 fuzzy integral-derivative branched with 2DOF PID (T2FID + 2DOFPID), type-2 fuzzy ID + PD (T2FID + PD), type-1 fuzzy ID + PD (T1FID + PD) and PID. The AOAVOA-based 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 disturbance-rejection 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 AOAVOA-tuned T2FPD + 2DOFPID controller shows better dynamic performance than the other configurations. The AOAVOA-tuned 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 area-1, the dynamic responses of different controllers are compared in Fig. 7c. The AOAVOA-based 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.

Table 6 Settings of different controllers obtained from AOAVOA

Fig. 7

Dynamic response with SLD, RLD and required control efforts

Table 7 Quantitative performance comparison of different controllers

Table 8 Settings of T2FPD + 2DOFPID controller obtained from different optimizers with RLD

Table 9 Statistical results of independent runs

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 area-1 is operated in constant and variable input modes to investigate its impact on dynamic responses. For DRG analysis, the following step-change pattern is considered:

$$x\left(t\right)=\left[u\left(t\right)+u\left(t-10\right)+u\left(t-20\right)-2u\left(t-30\right)\right]\times 1{0}^{-3}$$

(28)

where u(t) is the unit step signal. The AOAVOA-tuned 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.

Fig. 8

Dynamic responses of different case studies

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 delay-free 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 AOAVOA-tuned 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 area-1 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.

Fig. 9

Dynamic response for a ± 30% perturbation in thermal parameters in area-1, b various optimizers during the bilateral scenario, c various optimizers during contract violation

Table 10 Quantitative performance measures with perturbed thermal parameters

5.2 Bilateral-based 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 steady-state 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 tie-line 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.

Table 11 Settings of proposed T2FPD + 2DOFPID controller obtained from different optimizers for bilateral/contract violation

5.3 Contract violation-based 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 DISCO-1 is requesting 0.005 pu of additional power, and thus the overall supply ($\Delta {P}_{\mathrm{L}1}$) in area-1 is the sum of the supplies for DISCO-1, DISCO-2, and is a violation. The total supply of area-2 ($\Delta {P}_{\mathrm{L}2}$) is the result of adding a supply of DISCO-3 and DISCO-4, whereas the total supply of area-3 ($\Delta {P}_{\mathrm{L}3}$) is the result of adding a supply of DISCO-5 and DISCO-6. 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 area-1 after requesting an extra 0.005 pu of power from DISCO1 is as follows: supply to DISCO-1 & DISCO-2 + violation = 0.005 + 0.005 + 0.005 = 0.015 pu. Because of higher power requirements in area-1, the steady-state generation for area-1 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 tie-line 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]:

$$\Delta {F}_{\mathrm{n}}\left(s\right)={G}_{\mathrm{pn}}\left(s\right){G}_{\mathrm{cn}}\left(s\right){U}_{\mathrm{n}}\left(s\right)+{G}_{\mathrm{D}}\left(s\right)\Delta {D}_{\mathrm{n}}\left(s\right)+{G}_{\mathrm{tie}}\left(s\right)\Delta {P}_{\mathrm{tie}}\left(s\right)$$

(29)

Fig. 10

Generalized block diagram of the ith area in a multi-area PS

Substituting ${\Delta D}_{\mathrm{n}}(s)$= 0 and ${\Delta P}_{\mathrm{tie}}(s)$ = 0 into (29) yields:

$$\frac{\Delta {F}_{\mathrm{n}}\left(s\right)}{{U}_{\mathrm{n}}\left(s\right)}={G}_{\mathrm{pn}}\left(s\right)\times {G}_{\mathrm{cn}}\left(s\right)$$

(30)

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 area-1 is calculated as:

$${G}_{\mathrm{C}1}{G}_{\mathrm{p}1}=\frac{143{s}^{6}+2455{s}^{5}+9053{s}^{4}+9109{s}^{3}+3128{s}^{2}+257.3s+2.675}{{s}^{8}+18.21{s}^{7}+81.16{s}^{6}+130{s}^{5}+90.34{s}^{4}+26.86{s}^{3}+2.676{s}^{2}+0.07716s}$$

(31)

Similar equations for area-2 and 3 are:

$${G}_{\mathrm{C}2}{G}_{\mathrm{p}2}=\frac{-9.609e-05{s}^{3}-0.0005166{s}^{2}+0.02091s+0.007024}{{s}^{5}+15.98{s}^{4}+44.05{s}^{3}+6.329{s}^{2}+0.2083s}$$

(32)

$${G}_{\mathrm{C}3}{G}_{\mathrm{p}3}=\frac{0.0004388 s + 0.0009032}{{s}^{4}+30.05{s}^{3}+201.5{s}^{2}+10s}$$

(33)

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 closed-loop 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 closed-loop system in all three areas.

Fig. 11

Bode graphs of loop transfer functions

Table 12 Stability margins and crossover frequencies

5.5 Comparison of AOAVOA-tuned T2FPD + 2DOFPID method with previously reported works.

The performance of the AOAVOA-tuned 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.

Table 13 Performance comparison with published literature

Fig. 12

Comparison of the suggested control scheme with reported strategies

5.5.1 Comparison-1

The PS version employed in [8] is used for the dynamic response comparison-1 shown in Fig. 12(a). It is a two-area deregulated PS. Reference [8] uses a state feedback controller while [51] used the integral secondary controller. In [23], the EO-optimized type-1 Fuzzy FOPI + PIDN strategy is deployed while [24] improves it further by using the AOA-tuned type-2 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 Comparison-2

Reference [6] investigates a two-area, 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 Comparison-3

In [83], a two-area interconnected PS with renewable integration such as SPV and WTP is investigated, where LQR-based I and PI controllers are designed. The intermittencies of renewables are seen in the dynamic responses to 1% SLD of both LQR-I and LQR-PI 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 type-2 fuzzy proportional–derivative bifurcated with a dual degree-of-freedom 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 three-area restructured power system (PS). Integration of time-weighted 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 area-1. Thermal and gas units are considered in area-2 whereas thermal-geothermal configuration is considered in area-3. The advantage of the suggested AOAVOA-tuned 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 robust-stability study is also performed. The maximum frequency deviation in area-1 is mitigated by 2.89% while that of area-2 is mitigated by over 4.55% and tie-line 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.