A Novel Sooty Terns Algorithm for Deregulated MPC-LFC Installed in Multi-Interconnected System with Renewable Energy Plants

: This paper introduces a novel metaheuristic approach of sooty terns optimization algorithm (STOA) to determine the optimum parameters of model predictive control (MPC)-based deregulated load frequency control (LFC). The system structure consists of three interconnected plants with nonlinear multisources comprising wind turbine, photovoltaic model with maximum power point tracker, and superconducting magnetic energy storage under deregulated environment. The proposed objective function is the integral time absolute error (ITAE) of the deviations in frequencies and powers in tie-lines. The analysis aims at determining the optimum parameters of MPC via STOA such that ITAE is minimized. Moreover, the proposed STOA-MPC is examined under variation of the system parameters and random load disturbance. The time responses and performance speciﬁcations of the proposed STOA-MPC are compared to those obtained with MPC optimized via differential evolution, intelligent water drops algorithm, stain bower braid algorithm, and ﬁreﬂy algorithm. Furthermore, a practical case study of interconnected system comprising the Kuraymat solar thermal power station is analyzed based on actual recorded solar radiation. The obtained results via the proposed STOA-MPC-based deregulated LFC conﬁrmed the competence and robustness of the designed controller compared to the other algorithms.


Introduction
In the interconnected system, the frequency stabilization is very significant to keep the stability of the power system which is achieved by load frequency control (LFC). LFC aims at keeping the frequency at nominal value and vanishing the aberration in frequency and power flow in tie-lines to zero in case of sudden load disturbance. The objective of interconnecting multiplants is to share loads and maintain the system dependability in the event of curtailment of any generation plant. Recently, renewable energy sources (RESs) have been combined with conventional plants and installed in electric grids [1][2][3]. The deregulated power system is a conventional power system with modified structure. It consists of many autonomous entities such as transmission companies (TRANSCOs), distribution companies (DISCOs), and generation companies (GENCOs). The GENCOs, as autonomous power units, may contribute in the LFC task. Moreover, DISCOs may contract unilaterally

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A novel STOA approach is proposed to compute the MPC optimum parameters-based nonlinear deregulated LFC combined with conventional, RESs, and energy storage systems (ESSs).

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Wind turbine (WT), photovoltaic (PV) model with maximum power point tracker (MPPT), hydropower, diesel generator, and thermal plant are presented and modeled in deregulated LFC. • Practical case study of interconnected system comprising the Kuraymat solar thermal power station is analyzed based on actual recorded solar radiation.

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The proposed MPC-LFC optimized via STOA achieved robust performance under changing some parameters of the system and random load disturbance.
The paper is organized as follows: Section 2 introduces the mathematical model of the deregulated LFC, Section 3 presents the proposed methodology, Section 4 presents simulation results, and Section 5 introduces conclusions.

Mathematical Model of Deregulated LFC
The proposed system considered in this paper includes three interconnected plants; the first area comprises reheat thermal, wind power units, and DISCOs (DISCO 1 and DISCO 2 ). Area 2 includes hydro, diesel power units, and DISCOs (DISCO 3 and DISCO 4 ). Area 3 consists of reheat thermal, PV with MPPT, and DISCOs (DISCO 5 and DISCO 6 ). Each plant has SMES; Figure 1 shows the proposed multi-interconnected system topology in the deregulated LFC system. The system construction in the Simulink model is presented in Figure 2.
power station is analyzed based on actual recorded solar radiation.  The proposed MPC-LFC optimized via STOA achieved robust performance under changing some parameters of the system and random load disturbance.
The paper is organized as follows: Section 2 introduces the mathematical model of the deregulated LFC, Section 3 presents the proposed methodology, Section 4 presents simulation results, and Section 5 introduces conclusions.

Mathematical Model of Deregulated LFC
The proposed system considered in this paper includes three interconnected plants; the first area comprises reheat thermal, wind power units, and DISCOs (DISCO 1 and DISCO 2). Area 2 includes hydro, diesel power units, and DISCOs (DISCO 3 and DISCO 4). Area 3 consists of reheat thermal, PV with MPPT, and DISCOs (DISCO 5 and DISCO 6). Each plant has SMES; Figure 1 shows the proposed multi-interconnected system topology in the deregulated LFC system. The system construction in the Simulink model is presented in Figure 2.  In deregulated LFC, contracts conducted via GENCOs with DISCOs are made based on the DISCOs Participation Matrix (DPM). The DISCOs number represents the column numbers of DPM, and the GENCOs number is the row numbers of DPM in interconnected systems, the sum of each column in the matrix should be equal to unity. The elements of the matrix depend on contract participation factor (cpf ), and the DPM is described by Equation (1).
The scheduled steady-state power flow on the tie-line from area i to j is defined as follows: where Dn is the DISCOs number, Gn is the GENCOs number, and dP Li is the load disturbance in area i. The actual power flow on tie-line (dP tie,ij_actual ) can be described as follows: where dF i and dF j are the frequency deviations in area i and area j. T ij is the coefficient of synchronizing between areas i and j. The error in tie-line power between area i and area j can be expressed as dP tie,ij_error = dP tie,ij_actual − dP tie,ij_scheduled  The input signal to MPC is the area control error (ACE) which can be written as follows: where B i is the bias factor of frequency in area i.

Sooty Terns Optimizer Characteristics
Gaurav Dhiman [35] presented the sooty terns optimization algorithm (STOA) in 2019. Sooty terns are wide range of types with variable sizes and weights, they are sea birds that eat amphibians, earthworms, insects, fish, reptiles, etc. Sooty terns (STs) establish the sound of rain, such as catching worms concealed underground by feet and using crumbs of baking to entice the fish. Generally, STs live in colonies and use their cleverness to locate their prey and attack it. Immigration and attacking the prey are prominent aspects of STs behaviors, and migration is identified as the movement of seasonal STs to search for food-rich areas that provide adequate energy. During migration, the STs move in groups following the strongest one and, therefore, they adjust their initial positions to avoid collision with each other. The behavior of STs during migration can be described as follows: where → C st is the position of a sooty tern that does not conflict with another one, → P st represents the ST's current position, z represents current iteration, S A is ST motion in a certain search area, while C f is a variable controlling to set S A . STs search for the best neighbor and converge with it after avoiding a clash based on the following equation: where → M st refers to STs' different positions, → P bst is the best ST, C B is a random variable, while R and refers to random number in scale of [0, 1]. The ST or search agent can refresh its location with regards to the best ST.
where → D st indicates the disparity between the ST and the fittest ST. When attacking the prey, STs change their speeds and create a spiral behavior which is defined as follows: where R adi refers to the radius of every spiral turn, i is variable in scale [0 ≤ k ≤ 2π], v and u identify the constant of spiral form, and e refers to normal logarithm. STs update their positions based on the following equation:

The Proposed Approach
This section presents the major structure of MPC. Additionally, it clarifies the proposed approach combining MPC and STOA.

Model-Predictive Control (MPC)
MPC is a modern control concept that relies on future predictions to resolve the trouble under study. MPC is commonly utilized in the manufacturing systems. The MPC has many advantages, such as combinations of direct variables, system delay compensation, the ability to handle limitations, and online optimization. Figure 3 presents the MPC structure, which has prediction and controller units [36,37]. The unit of prediction predicts the future results of the system according to its current output, while the control unit utilizes the forecast output to reduce the restrictive equation of the objective function. If restrictions exist, the objective function can be reduced by utilizing the performance prediction function via the control unit. The basic concept of MPC relies on the calculation of the difference between the reference signal and the plant's actual output. The future output is then estimated over time intervals, known as sampling, until the output matches the reference signal.  In the MPC algorithm, the system can be described as linear or nonlinear. The plant input and output are presented in the following formula: where A, B, C, and D represent the system state-space matrices, So and Si indicate the output and input diagonal array, respectively, while up refers to a nondimensional vector of input variables. The input of MPC can be calculated as (u(k) = u(k − 1) + Δu(k)); by solving the problem with respect to sequence of input, one can get the following expression: where M refers to the control horizon, P refers to the prediction horizon (1 ≤ M ≤ P), T is the sample time, Q and R represent weighting factors, while ( + | ) refers to the forecasted output.

Optimal Deregulated LFC Solving Problem
This section introduces the deregulated LFC using MPC optimized via the proposed STOA. The MPC parameters (M, P, T, Q, and R) are identified via the proposed methodology of STOA to minimize the ITAE of aberrations in frequencies and powers in tie-lines as follows: where t and n are the time of simulation and area number, dFi is the frequency deviation in area i, and dPtie,i refers to the deviation in tie-line power of area i. In this work, the MPC In the MPC algorithm, the system can be described as linear or nonlinear. The plant input and output are presented in the following formula: where A, B, C, and D represent the system state-space matrices, S o and S i indicate the output and input diagonal array, respectively, while u p refers to a nondimensional vector of input variables. The input of MPC can be calculated as (u(k) = u(k − 1) + ∆u(k)); by solving the problem with respect to sequence of input, one can get the following expression: where M refers to the control horizon, P refers to the prediction horizon (1 ≤ M ≤ P), T is the sample time, Q and R represent weighting factors, while y(k + i|k) refers to the forecasted output.

Optimal Deregulated LFC Solving Problem
This section introduces the deregulated LFC using MPC optimized via the proposed STOA. The MPC parameters (M, P, T, Q, and R) are identified via the proposed methodol- ogy of STOA to minimize the ITAE of aberrations in frequencies and powers in tie-lines as follows: (19) where t and n are the time of simulation and area number, dF i is the frequency deviation in area i, and dP tie,i refers to the deviation in tie-line power of area i. In this work, the MPC design is based on linear time invariant (LTI) which can be determined through MPC toolbox for each area with the aid of Matlab/Simulink. Figure 4 shows the MPC adaptation mechanism implemented through the suggested STOA; the MPC parameters' constraints are selected as 1 ≤ M, 1 ≤ P, 1 ≤ R, Q ≤ 10, and 0.1 ≤ T≤ 10. The MPC is fed by three inputs which are reference signal, deviation in frequency of the LFC system, and load disturbance measurement. The ITAE is computed depending on current aberrations in frequencies and powers in tie-lines and then fed to the proposed STOA. The MPC optimum parameters can be identified by STOA through minimizing the ITAE. Figure 5 explains the steps for implementing the proposed STOA.

Simulation Results
In this work, the MPC optimal parameters are determined via the proposed STOAbased deregulated LFC installed in multi-interconnected plants with RESs and SMES. The controlling parameters of STOA are assigned as 50 for population size, and maximum

Simulation Results
In this work, the MPC optimal parameters are determined via the proposed STOAbased deregulated LFC installed in multi-interconnected plants with RESs and SMES. The controlling parameters of STOA are assigned as 50 for population size, and maximum iteration of 100. The proposed approach is applied on the system shown in Figure 2 which consists of nonlinear three areas with multi-sources and deregulated LFC environment through three cases. The proposed system parameters are tabulated in Table A1 in Appendix A, while governor dead band (GDB) and generation rate constraint (GRC) are

Simulation Results
In this work, the MPC optimal parameters are determined via the proposed STOAbased deregulated LFC installed in multi-interconnected plants with RESs and SMES. The controlling parameters of STOA are assigned as 50 for population size, and maximum iteration of 100. The proposed approach is applied on the system shown in Figure 2 which consists of nonlinear three areas with multi-sources and deregulated LFC environment through three cases. The proposed system parameters are tabulated in Table A1 in Appendix A, while governor dead band (GDB) and generation rate constraint (GRC) are specified to be 3%. The obtained results via the proposed approach are compared to those obtained by MPC optimized via differential evolution (DE), stain bower braid algorithm (SBO), firefly algorithm (FA), and intelligent water drops algorithm (IWD).

Unilateral-Based Transaction
In this case, there is unilateral contract between DISCOs and GENCOs in area 1; this can be represented as given in Equations (20) and (21). The demand power is 0.005 pu for DISCO 1 and DISCO 2 (DISCO 1 = DISCO 2 = 0.005), while the total load disturbance in area 1 (dP D1 ) is 0.01 pu, which presents the sum demand load in DISCO 1 and DISCO 2 . However, there is no demand for power by DISCO 3 , DISCO 4 , DISCO 5 , DISCO 6 , and load disturbance in areas 2 and 3.
The change in the response of the generation units for each GENCO can be written as follows: Table 3 illustrates the errors (integral absolute error (IAE), integral square error (ISE), integral time absolute error (ITAE), and integral time square error (ITSE)) that are obtained by the different algorithms compared to the proposed technique with/without SMES. The optimum parameters of MPC-based deregulated LFC obtained by the presented methodologies are illustrated in Table 4. The aberrations in frequencies and powers flow in tie-lines are shown in Figure 6, while Table 5 presents the system performance specifications including peak undershoot (PUs), peak overshoot (POs), and settling time (Ts) of the fluctuations in frequencies and powers in tie-lines. The settling time and overshoot are minimized by the proposed STOA with/without SMES.

Bilateral Transaction
In this case, the DISCOs contract with any GENCOs are bound by the terms of the contract concluded between them. Assume that the power demand for each DISCO is 0.005 (DISCO 1 = DISCO 2 = DISCO 3 = DISCO 4 = DISCO 5 = DISCO 6 = 0.005), while the total load disturbance in all areas is 0.01 pu (dP D1 = dP D2 = dP D3 = 0.01 pu), and the DPM is assigned as in Equation (24).
The power change of the generation units for each GENCO is illustrated as follows: The best obtained fitness function is via the proposed approach compared to IWD, FA, DE, and SBO, as tabulated in Table 6. MPC optimum parameters obtained by different approaches with deregulated LFC under bilateral transaction case are tabulated in Table 7. Aberrations in frequencies and powers in tie-lines are shown in Figure 7. Table 7 introduces the system performance specifications for curves presented in Figure 7. The effect of installed SMES in the system to minimize ITAE is clarified and given in Table 6, Table 8 and Figure 7.

Contract Violation Transaction
Usually, the demand for power increases and DISCOs strive to achieve the profits, therefore there is a violation of contracts with the GENCOs. The GENCOs must meet the increase of power demand from DISCOs. Given the contracting procedures mentioned in Section 5.2 and Equations (22) and (23), the power demand requested by the DISCO1 and DISCO2 are modified to 0.01, while the other DISCOs requests remain the same, at 0.005. Moreover, the power change of the GENCO1, GENCO2, and load disturbance in all areas are given as follows:

Contract Violation Transaction
Usually, the demand for power increases and DISCOs strive to achieve the profits, therefore there is a violation of contracts with the GENCOs. The GENCOs must meet the increase of power demand from DISCOs. Given the contracting procedures mentioned in Section 5.2 and Equations (22) and (23), the power demand requested by the DISCO 1 and DISCO 2 are modified to 0.01, while the other DISCOs requests remain the same, at 0.005. Moreover, the power change of the GENCO 1 , GENCO 2 , and load disturbance in all areas are given as follows: When the system given in Figure 2 is simulated under this case, the ITAE obtained via the proposed STOA is 1.0102. Table 9 presents a comparison between the values of errors obtained by the proposed approach and the other simulated algorithms. The MPC optimum parameters obtained by different approaches with deregulated LFC are presented in Table 10. The frequencies and tie-line powers' aberrations are displayed in Figure 8. The corresponding performance specifications for such cases are tabulated in Table 11. The settling time and overshoot are minimized by the proposed STOA.

Sensitivity Analysis
To confirm the robustness and reliability of the proposed approach-based deregulated LFC, the constructed MPC is investigated under changing of the system parameters and random load disturbance. Sensitivity analysis is conducted on deregulated three interconnected plants with LFC and SMES in bilateral and contract violation transactions cases through changing the system parameters, such as Tg, Kr, Tr, Tt, Kp, and Tp, to ±25% and ±50%. The obtained ITAEs in this case are given in Table 12. The proposed approach has the robust performance and competence under changing the system parameters.

Sensitivity Analysis
To confirm the robustness and reliability of the proposed approach-based deregulated LFC, the constructed MPC is investigated under changing of the system parameters and random load disturbance. Sensitivity analysis is conducted on deregulated three interconnected plants with LFC and SMES in bilateral and contract violation transactions cases through changing the system parameters, such as T g , K r , T r , T t , K p , and T p , to ±25% and ±50%. The obtained ITAEs in this case are given in Table 12. The proposed approach has the robust performance and competence under changing the system parameters. The application of random load disturbance is vital as the load demand is not usually constant on the system all the time. To confirm the reliability of the proposed technique, the random load change shown in Figure 9 is applied through the DISCO 1 and DISCO 2 for the same control values and conditions in contract violation case described in Section 5.3, while the total load on area 1 is the sum of DISCO 1 and DISCO 2 . Figure 10 illustrates the aberrations in frequencies and powers in tie-lines under random load change. As the reader can see, the time responses of frequencies and tie-line powers' violations pass through four time intervals according to the load disturbance shown in Figure 10. The proposed MPC-LFC designed via STOA succeeded in vanishing the perturbations in frequencies and tie-line powers in all intervals, with less oscillations compared to the others. The overshoot and undershoot are minimized by the proposed STOA with/without SMES compared to DE and SBO.  The application of random load disturbance is vital as the load demand is not usually constant on the system all the time. To confirm the reliability of the proposed technique, the random load change shown in Figure 9 is applied through the DISCO1 and DISCO2 for the same control values and conditions in contract violation case described in Section 5.3, while the total load on area 1 is the sum of DISCO1 and DISCO2. Figure 10 illustrates the aberrations in frequencies and powers in tie-lines under random load change. As the reader can see, the time responses of frequencies and tie-line powers' violations pass through four time intervals according to the load disturbance shown in Figure 10. The proposed MPC-LFC designed via STOA succeeded in vanishing the perturbations in frequencies and tie-line powers in all intervals, with less oscillations compared to the others. The overshoot and undershoot are minimized by the proposed STOA with/without SMES compared to DE and SBO.

Practical Case Study
It is important to investigate the proposed MPC-LFC optimized via STOA on a practical plant, this is done by replacing the PV model with the Kuraymat solar thermal power station. Figure 11 shows the location of Kuraymat, which is 90 miles south of Cairo, Egypt. It is a combined cycle plant that has gas turbines with capacity of 80 MW and steam turbine of 40 MW, in addition to one parabolic trough solar system with rating of 20 MW. The solar field covers an area of about 130,800 m 2 and consists of 40 rows of collectors, with each row having four SKAL-ET 150 parabolic trough collectors, and each collector consists of 12 modules [38,39]. In this work, the solar thermal plant is represented in Matlab/Simulink, as shown in Figure 12, to clarify the effect of changing solar radiation on the system. This plant comprises a solar field which represents collectors of parabolic troughs, governor, and steam turbine; the combined heat by collectors is utilized to heat the fluid and water to produce steam and drive the turbine. The recorded solar radiation by the plant shown in Figure 13 is used, and these data are fed to the solar thermal energy unit. The solar radiation was transformed over the day to match the simulation time of the system, and  Table 13, which shows the errors obtained by different approaches at the Kuraymat solar thermal power station. The optimum parameters of MPC obtained by different methodology-based deregulated LFC with solar thermal plant are tabulated in Table 14. The aberrations in frequencies and powers in tie-lines are shown in Figure 14, while Table 15 presents the system performance specifications for curves presented in Figure 14. The settling time and peak overshoot are minimized by the proposed STOA with/without SMES. The obtained results confirm the robustness and competence of the proposed MPC-LFC optimized via STO in this such case.

Practical Case Study
It is important to investigate the proposed MPC-LFC optimized via STOA on a practical plant, this is done by replacing the PV model with the Kuraymat solar thermal power station. Figure 11 shows the location of Kuraymat, which is 90 miles south of Cairo, Egypt. It is a combined cycle plant that has gas turbines with capacity of 80 MW and steam turbine of 40 MW, in addition to one parabolic trough solar system with rating of 20 MW.
The solar field covers an area of about 130,800 m 2 and consists of 40 rows of collectors, with each row having four SKAL-ET 150 parabolic trough collectors, and each collector consists of 12 modules [38,39]. In this work, the solar thermal plant is represented in Matlab/Simulink, as shown in Figure 12, to clarify the effect of changing solar radiation on the system. This plant comprises a solar field which represents collectors of parabolic troughs, governor, and steam turbine; the combined heat by collectors is utilized to heat the fluid and water to produce steam and drive the turbine. The recorded solar radiation by the plant shown in Figure 13 is used, and these data are fed to the solar thermal energy unit. The solar radiation was transformed over the day to match the simulation time of the system, and all conditions and restrictions mentioned in Section 5.2 were applied to obtain the results in this case. The obtained results of the actual case are reported in Table  13, which shows the errors obtained by different approaches at the Kuraymat solar thermal power station. The optimum parameters of MPC obtained by different methodologybased deregulated LFC with solar thermal plant are tabulated in Table 14. The aberrations in frequencies and powers in tie-lines are shown in Figure 14, while Table 15 presents the system performance specifications for curves presented in Figure 14. The settling time and peak overshoot are minimized by the proposed STOA with/without SMES. The obtained results confirm the robustness and competence of the proposed MPC-LFC optimized via STO in this such case.

Conclusions
This paper proposed a novel structure of load frequency control (LFC) installed in a multi-interconnected system with renewable energy sources and storage systems. The proposed controller is represented by model predictive control (MPC) optimized via recent metaheuristic optimizer of sooty terns optimization algorithm (STOA). The proposed methodology that incorporated STOA was employed to determine the optimal parameters of MPC-LFC. The presented fitness function to be minimized is the integral time absolute error (ITAE), comprising the frequencies and in tie-lines powers' deviations. The constructed MPC-deregulated LFC was combined in an interconnected nonlinear system involving photovoltaic (PV) with maximum power point tracker (MPPT), wind turbine (WT), and superconducting magnetic energy source (SMES). The system was simulated under deregulated cases as unilateral, bilateral, and contract violation-based transactions with/without SMES. The performance specifications (undershoots, peak overshoot, and settling time) of the time responses for frequencies and tie-line powers' aberrations obtained by the proposed STOA were compared to those of different optimizers in all cases. Moreover, the constructed MPC was examined under changing of the system parameters and random load change. Furthermore, a practical case study interconnecting Kuraymat solar thermal power station with others was analyzed based on actual recorded solar radiation. The best fitness function in unilateral transactions case was 0.3736, obtained via STOA, and 0.1302, when SMES was used. In the bilateral transactions case, the best fitness function was 3.2369, obtained using STOA, and 0.6619, with STOA-SMES. On the other hand, the values of ITAE at the contract violation-based transactions case were 5.1892 and 1.0106 by STOA with/without SMES, respectively. The proposed control achieved minimum target of 1.9642 and 0.7647 by STOA with/without SMES for LFC with solar thermal plant. The obtained results confirm the robustness and reliability of the proposed approach incorporating STOA in minimizing the aberration in frequencies and powers in tie-lines and achieving the system stability during load disturbances in the least time. In future work, enhancement of the STOA algorithm to reduce the consumption time is mandatory.   The load disturbance in area i R adi The radius of every spiral turn T ij The coefficient of synchronizing between areas i and j R and The random number in scale of [0, 1] C B The random variable dP Di total load disturbance in area i →

C st
The position of ST that does not conflict with ST another x(k) The system state C f Controlling variable y(k) The system outputs The constant gain of diesel unit S o and S i the output and input diagonal array K g The gain of steam plant governor T Sample time of MPC K gh The gain of hydro plant governor M and P The control and prediction horizons K p The gain of generator and power system Q and R Weighting factors K PV1 and K PV2 The gains of PV system t Simulation time K pw1 , K pw2 and K pw3 Wind plant gains dF i The frequency deviation of i area K r The gain of reheater dP tie,i The power deviation of tie-line in area i K t The gain of steam turbine T g