Stochastic Flexible Power System Expansion Planning, Based on the Demand Response Considering Consumption and Generation Uncertainties

Abstract

This paper presents the generation and transmission expansion planning (GTEP) considering the switched capacitive banks (SCBs) allocation in the power system, including the demand response program (DRP). This scheme is based on the system flexibility. The objective function of the scheme minimizes the expected planning cost that is equaled to the summation of the total construction costs of the SCBs, the generation units (GUs) and the transmission lines (TLs), and the operating cost of the GUs. It is concerned with the AC power flow constraints, the planning-operation model of the mentioned elements, the DRP operation formulation, and the operating and flexibility limits of the network. In the following, the scenario-based stochastic programming is used to model the uncertainty parameters, such as the load and renewable power of wind farms. Then, the hybrid evolutionary algorithm, based on the combination of the crow search algorithm and the grey wolf optimizer, is used to determine the optimal point with the approximate unique solution. Finally, the scheme is applied on the transmission networks, the numerical results confirm the capabilities of the proposed scheme in simultaneously improving the flexibility, operation, and economic situation of the transmission network, so that the hybrid algorithm achieves the optimal solution in a shorter computation time, compared with the non-hybrid algorithms. This algorithm has a low standard deviation of about 92% in the final response. The proposed scheme with the optimal planning of the lines, sources, and capacitor banks, together with the optimal operation of the DRP succeeded in improving the energy loss and the voltage deviation by about 30–36% and 25–30%, compared with those of the power flow studies.

1. Introduction

1.1. Motivation and Approach

Power system expansion planning (PSEP) will be necessary, with the growth of energy consumption, in the upcoming years [1]. One of these plans is the generation and transmission expansion planning (GTEP), which simultaneously provides the optimal location and size for the generation units (GUs) and the transmission lines (TLs) over a desirable planning horizon [2,3]. The optimal conditions are generally achieved by minimizing the installation and operation costs of the mentioned elements, subject to the technical constraints of the transmission network [4,5]. It is noteworthy that among the different GUs, wind farms (WFs) are considered in the PSEP, due to their very environmental pollution and low operating costs, and their use in the transmission network has been suggested in various studies [6]. However, it must be said that the generation of power of this GU is intermittent, due to its dependence on the meteorological conditions. Therefore, the results of the day-ahead and real-time operation of a system with WFs will not be the same [7,8]. Thus, it is possible that, in real-time operation, an imbalance between the supply and demand is established [7]. This is known as the low system flexibility [8]. To compensate for this, the flexibility sources, such as the demand response program (DRP), can be incorporated into the system, so that it can minimize the fluctuations in the power of WFs in real-time operation, compared to the day-ahead operation [9]. In this case, a power system with a high flexibility is expected. Note, however, that these capabilities can be evaluated if a suitable model for the PSEP is used on the network.

1.2. Literature Review

Various studies have been carried out in the field of PSEP. Transmission expansion planning (TEP) is solved using a mixed integer non-linear programming (MINLP) model in [10], while taking into account the demand-side uncertainties. The problem is optimized by adopting a bacterial foraging algorithm. The authors in [11] provide a mathematical formulation for the PSEP problem constrained by the supply and demand-side constraints. The paper adopts a two-step adaptive robust optimization, where the highest possible operating cost is considered while the uncertainties defined by the user are taken into account. There are two distinct differences between the proposed model and those of the literature. Firstly, it considers the correlation of uncertainties via an ellipsoidal uncertainty set that depends on the variance-covariance matrix of uncertainties. Secondly, it explains the similarity of the second-stage problem and a specific group of mathematical programs. Furthermore, the suggested robust optimization is solved using a new nested decomposition method. The overall capital and operating costs are minimized using a co-optimization planning model [12]. The paper adopts a linearized AC power flow (AC-PF) model that contains a probabilistic reliability index, i.e., loss of load expectation, so that the reliable and economic co-planning decisions are extracted. Mavalizadeh et al. [13] present a mixed-integer linear robust multi-objective model and an information-gap decision theory-based framework for the PSEP, considering the demand and equipment price uncertainties. The suggested model helps the power system planners to construct new paths for the fuel transportation, in case of the need. The GTEP is subject to security constraints, as presented in [14]. A two-stage optimization problem is used in the method. The upper level of the problem deals with the economy of the transmission and generation companies (TransCo and GenCo), so that the difference between the cost and revenue of these companies in the energy and reserve market is minimized. The lower level addresses the minimization of the sum of the operation and security costs, constrained by some constraints.

The adaptive robust optimization model proposed in [15] for the GTEP problem, is in the form of a two-step framework, so that the decisions on the capital costs and operation are decoupled during the planning period. The column and constraint generation approach is incorporated to effectively find a solution to the problem. The reliability constraints are incorporated into the optimal PSEP problem [16]. The paper employs new reliability indices, namely the value-at-risk and conditional value-at-risk, in addition to the loss of the load expectation and the expected energy not-supplied (EENS). Furthermore, Bender’s decomposition method is adopted to decouple the PSEP into three subproblems. Hamidpour et al. [17] introduce the GTEP subject to resiliency while taking into account the natural events. The model consists of two objective functions so that the construction and operating costs of the resiliency sources and the EENS is minimized. The model is subject to the AC optimal power flow (AC-OPF) equations, the limitation of the planning and operation of RSs. A mathematical expression for the load shedding is suggested [18]. It is then modified as an AC model so that the TEP problem is solved, permitting to find a solution to an integrated transmission line and shunt the compensation planning. Alhamrouni et al. [19] solve the TEP problem by adopting a non-convex MINLP model. The paper also takes into account some constraints relevant to the operation of the power system. Finally,

Table 1. Taxonomy of the recent research studies.

Ref.	Flexibility Modeling	Planning of the Switched Capacitor Bunk	Solver
[10]	No	No	NHEA
[11]	No	No	Solvers for linear model
[12]	No	No	Solvers for linear model
[13]	No	No	Solvers for linear model
[14]	No	No	Solvers for linear model
[15]	No	No	Solvers for linear model
[16]	No	No	Solvers for linear model
[17]	No	Yes	Solvers for linear model
[18]	No	No	NHEA
[19]	No	No	NHEA
Current paper	Yes	Yes	HEA

HEA: Hybrid evolutionary algorithm, NHEA: non-HEA.

presents the taxonomy of the recent research publications.

1.3. Research Gaps

Based on the literature and Table 1, these are the following major research gaps in the field of PSEP:

• In most studies focusing on PSEP, such as in [10,11,12,13,14,15,16,17,18,19], consider the economic planning and technical (operation) indices of the network. However, it should be noted that WFs, thanks to their low pollution and low operating costs, can be an important factor in improving the economic, environmental, and power network operation. However, their presence will reduce the flexibility of the network. To compensate for this, the flexibility sources, such as storage and DRP, are suitable approaches [9]. Although this has been stated in various studies, less research has focused on the mathematical model of flexibility, noting that to assess an index, its status should be measured and this can be derived by providing a mathematical model of the flexibility in the problem;
• Reactive power devices are generally able to improve the voltage profile and voltage security index in the network, by controlling the reactive power [17,18], among which the capacitor bank is economically viable and cost-effective. Note, however, that if the optimal location and size are achieved for these devices, it is expected that the optimal conditions will be obtained for the mentioned indices. This is achieved in the case of the capacitor bank placement problem. However, in other research, such as in [17], the location of the reactive power devices in the PSEP problem has been considered;
• The PSEP problem includes the AC-OPF equations, so it has a MINLP framework. To solve this problem, some research, such as in [10,18,19], use iterative numerical solutions or evolutionary algorithms to solve this problem. These algorithms suffer from a long computational time. They also have different optimal solutions, in other words, they are not unique. Therefore, the obtained solution has a low confidence factor. Moreover, some studies, such as in [11,13,15,16], have proposed a PSEP model, based on the DC optimal power flow (DC-OPF). Although this model has a linear formulation and by following it, the computational time will be low and a unique optimal solution will be obtained, variables, such as the voltage drop, reactive power, and power loss, are eliminated in this model. Consequently, the numerical results obtained from this model will not be accurate and will have a significant computational error. To compensate for the limitations of these two approaches, some studies, such as in [12,14,17], present the PSEP model, based on the linearized AC-OPF. Nonetheless, there is also a significant computational error in this technique, regarding some variables, such as a power loss.

1.4. Contributions

In this paper, to compensate for the first and second research gaps, the dynamic GTEP problem is considered by taking into account the optimal location of the switched capacitor banks (SCBs) in a transmission network with a DRP, as shown in Figure 1. The scheme aims to minimize the total construction costs of the power sources, lines, and SCBs, plus the expected operation costs of the sources, which are subject to the AC-OPF, the network flexibility constraints, and models of the sources, DRPs, and SCBs. In this case, the network load and generation of the power of WFs are considered uncertainties. Therefore, in this paper, the scenario-based stochastic programming (SBSP) is used to model them. This technique is also compatible with the combination of the Monte Carlo simulation (MCS) and the Kantorovich method, which are used to generate and reduce scenarios, respectively. The hybrid evolutionary algorithm (HEA), based on the combination of the crow search optimization (CSA) and the grey wolf optimizer (GWO), is employed to solve the problem and cover the third research gap. Note that in this algorithm, updating the decision variables is performed in two different processes. Therefore, the optimal solution obtained is expected to have a low standard deviation in the final response. Of course, to prove this, we can refer to solving an optimization problem using a genetic algorithm (GA) without/with mutation process [20]. According to [20], if a mutation is used in the GA, a more optimal solution can be obtained because the decision variables have been updated in two different processes [20]. In the end, the contributions of this paper are:

• Formulation of the dynamic power system planning with the optimal location of the switched capacitor bank in the power network with the demand response program;
• Obtaining the optimal placement and the time of installation of the GUs, TLs, and SCBs, based on the network flexibility model; and
• Using the combined CSA and GWO solver to obtain the optimal point with a low standard deviation in the final solution for the PSEP problem.

1.5. Paper Organization

The rest of the paper is organized as follows: Section 2 describes the dynamic GTEP formulation and modeling of the energy consumption and the generation uncertainties. The HEA-based problem-solving method is then presented in Section 3. Finally, in Section 4 and Section 5, the numerical results and conclusions are discussed, respectively.

2. Dynamic GTEP Model

2.1. Problem Formulation

The proposed scheme is an optimization problem. The optimization formulation includes the objective function [21,22,23,24,25,26,27], and constraints [28,29,30,31,32,33,34]. Furthermore, the optimization can be implemented on the energy network, if there are smart [35,36,37,38,39] and telecommunication [40,41,42,43,44] platforms in this system.

In this section, the dynamic GTEP formulation is expressed by considering the location model of the SCBs and the operation model of the DRP. This problem minimizes the total planning costs of the GUs, TLs, and SCBs by considering the AC-PF constraints, the planning-operating model of the GUs, SCBs, and TLs, the DRP operation model, and the operation and flexibility limits of the network. Therefore, the optimization formulation of this scheme can be written as follows:

Objective function: The objective function of this formulation is presented in Equation (1), which has four terms. In terms 1–3 of this equation, the investment costs in the construction of the TLs, SCBs, and GUs are presented, respectively. Note that the transmission line between buses b and l and buses l and b, is the same, so in the first term of Equation (1), the term 1/2 appears. In the last term of the equation, the expected operating cost of the GUs is formulated. This function includes the fuel cost of the GUs, which has a parabolic function [14].

$$\begin{array}{ll}\min PC& =\frac{1}{2}.{\displaystyle \sum _{\begin{array}{l}b,l\in {\mathsf{\Theta }}_B\\ h\in {\mathsf{\Theta }}_Y\end{array}}{C}_{blh}^L{x}_{blh}^L}+{\displaystyle \sum _{\begin{array}{l}c\in {\mathsf{\Theta }}_C\\ h\in {\mathsf{\Theta }}_Y\end{array}}{C}_{ch}^{SCB}{x}_{ch}^{SCB}}+{\displaystyle \sum _{\begin{array}{l}g\in {\mathsf{\Theta }}_G\\ h\in {\mathsf{\Theta }}_Y\end{array}}{C}_{gh}^G{x}_{gh}^G}+\\ & {\displaystyle \sum _{\begin{array}{l}\tau \in {\mathsf{\Theta }}_{OH}\\ h\in {\mathsf{\Theta }}_Y\\ w\in {\mathsf{\Theta }}_S\end{array}}du\times CF\times {\pi }_w{\displaystyle \sum _{g\in {\mathsf{\Theta }}_G}\left({\alpha }_g+{\beta }_g{P}_{g\tau hw}^G+{\chi }_g{\left(P_{g\tau hw}^G\right)}^2\right)}}\end{array}$$

(1)

Planning constraints: The planning model of the SCBs, GUs, and TLs, is described in constraints (2)–(11) [14,17]. Constraints (2)–(4) refer to the investment budget for the GUs, SCBs, and TLs, respectively, where the investment budget represents the maximum capital available for the construction of these elements. In the proposed design, the dynamic planning is considered for the mentioned elements. In other words, if the planning horizon is 6 years, the planning is performed for smaller intervals, such as 2 years. In this case, index h is selected from the set {1, 2, 3}. The elements of this set refer to the first to third planning periods, where the planning step is 2 years. Now, Equations (5)–(7) represent a logical constraint, stating that the referred power elements can be installed only in one planning period. In other words, each element can be constructed only in one of the first to third periods. Constraints (8)–(10) also determine the presence of the GUs, SCBs, and TLs in the planning periods, respectively. For example, if an element is constructed at h = 2, then that binary variable, corresponding to its presence at h = 1, is zero, and for other values, u has a value of unity. Finally, in Equation (11) a logical constraint is used which it indicates that the TL between buses b and l is the same as the TL between buses l and b.

$${\displaystyle \sum _{\begin{array}{l}g\in {\mathsf{\Theta }}_G\\ h\in {\mathsf{\Theta }}_Y\end{array}}{C}_{gh}^G{x}_{gh}^G}\le {\overline{C}}^G$$

(2)

$${\displaystyle \sum _{\begin{array}{l}c\in {\mathsf{\Theta }}_C\\ h\in {\mathsf{\Theta }}_Y\end{array}}{C}_{ch}^{SCB}{x}_{ch}^{SCB}}\le {\overline{C}}^{SCB}$$

(3)

$$\frac{1}{2}.{\displaystyle \sum _{\begin{array}{l}b,l\in {\mathsf{\Theta }}_B\\ h\in {\mathsf{\Theta }}_Y\end{array}}{C}_{blh}^L{x}_{blh}^L}\le {\overline{C}}^L$$

(4)

$${\displaystyle \sum _{h\in {\mathsf{\Theta }}_Y}{x}_{gh}^G}=1\forall g$$

(5)

$${\displaystyle \sum _{h\in {\mathsf{\Theta }}_Y}{x}_{ch}^{SCB}}=1\forall c$$

(6)

$${\displaystyle \sum _{h\in {\mathsf{\Theta }}_Y}{x}_{blh}^L}=1\forall b,l$$

(7)

$$u_{gh}^G-u_{g,h-1}^G=x_{gh}^G\forall g,h$$

(8)

$$u_{ch}^{SCB}-u_{c,h-1}^{SCB}=x_{ch}^{SCB}\forall c,h$$

(9)

$$u_{blh}^L-u_{bl,h-1}^L=x_{blh}^L\forall b,l,h$$

(10)

$$x_{blh}^L=x_{lbh}^L\forall b,l,h$$

(11)

Power flow constraints: The AC-PF constraints of the system are presented in Equations (12)–(15) [45,46,47], which express the power balance (active and reactive) for a bus, (12) and (13), and the power (active and reactive) passing from the TL, and Equations (14) and (15) [48,49,50]. Note, however, that Equations (14) and (15) represent a combined operation and planning model of the TL so that if the binary variable u^L is equal to unity, the TL is exploited, based on Equations (14) and (15). However, if u^L = 0, the TL is not present in the network, then there is no operating model for it.

$${\displaystyle \sum _{g\in {\mathsf{\Theta }}_G}{I}_{bg}^G{P}_{g\tau hw}^G}-{\displaystyle \sum _{l\in {\mathsf{\Theta }}_B}{I}_{bl}^L{P}_{bl\tau hw}^F}=P_{b\tau hw}^L-P_{b\tau hw}^{DRP}\forall b,\tau ,h,w$$

(12)

$${\displaystyle \sum _{g\in {\mathsf{\Theta }}_G}{I}_{bg}^G{Q}_{g\tau hw}^G}+{\displaystyle \sum _{c\in {\mathsf{\Theta }}_C}{I}_{bc}^{SCB}{Q}_{c\tau hw}^G}-{\displaystyle \sum _{l\in {\mathsf{\Theta }}_B}{I}_{bl}^L{Q}_{bl\tau hw}^F}=Q_{b\tau hw}^L\forall b,\tau ,h,w$$

(13)

$$\begin{array}{r}{P}_{bl\tau hw}^F=\left\{G_{bl}{\left(V_{b\tau hw}\right)}^2-V_{b\tau hw}\times V_{l\tau hw}\left\{G_{nbl}\cos \left({\gamma }_{b\tau hw}-{\gamma }_{l\tau hw}\right)+B_{bl}\sin \left({\gamma }_{b\tau hw}-{\gamma }_{l\tau hw}\right)\right\}\right\}{u}_{blh}^L\\ \forall b,l,\tau ,h,w\end{array}$$

(14)

$$\begin{array}{r}{Q}_{bl\tau hw}^F=\left\{-\left(B_{bl}+\frac{B_{bl}^0}{2}\right){\left(V_{b\tau hw}\right)}^2-V_{b\tau hw}\times V_{l\tau hw}\left\{G_{nbl}\sin \left({\gamma }_{b\tau hw}-{\gamma }_{l\tau hw}\right)-B_{bl}\cos \left({\gamma }_{b\tau hw}-{\gamma }_{l\tau hw}\right)\right\}\right\}{u}_{blh}^L\\ \forall b,l,\tau ,h,w\end{array}$$

(15)

Model of the GUs and SCBs: In the following, the combined planning-operation model of the GUs and SCBs is formulated in Constraints (16)–(20), respectively. Constraints (16) and (17) refer to the GU capability curve [51], which expresses the controllable range of power (active and reactive) in the GU. Constraint (18) is used only for the renewable Gus, such as WF [6]. This relationship points to the fact that the renewable GUs inject active power into the transmission system, where it is equal to their upper capacity, in proportion to the weather term (${\overline{P}}_{}^G$) [6]. This is because the operating cost of these types of GUs is very low [52]. In Constraints (19) and (20), the injected reactive power of the SCBs into the transmission network is calculated by Equation (19), and its step performance limit is formulated in Equation (20). In these constraints, the operation model of the GU (SCB) is applicable, only if u^G = 1 (u^SCB = 1).

$${\underset{¯}{P}}_g^G{u}_{gh}^G\le P_{g\tau hw}^G\le {\overline{P}}_g^G{u}_{gh}^G\forall g\in {\mathsf{\Theta }}_G-{\mathsf{\Theta }}_{WF},\tau ,h,w$$

(16)

$${\underset{¯}{Q}}_g^G{u}_{gh}^G\le Q_{g\tau hw}^G\le {\overline{Q}}_g^G{u}_{gh}^G\forall g,\tau ,h,w$$

(17)

$$P_{g\tau hw}^G={\overline{P}}_{g\tau w}^G{u}_{gh}^G\forall g\in {\mathsf{\Theta }}_{WF},\tau ,h,w$$

(18)

$$Q_{c\tau hw}^{SCB}=Q_c^0{y}_{c\tau hw}^{SCB}\forall c,\tau ,h,w$$

(19)

$$y_{c\tau hw}^{SCB}\in u_{ch}^{SCB}.\left\{1,2,\dots ,{\overline{N}}_c\right\}\forall c,\tau ,h,w$$

(20)

Formulation of the DRP: Next, the operating formulation of the DRP is given in Equations (21) and (22) [53]. Equation (21) indicates the controllable range of power of consumers participating in the DRP. Constraint (22) also guarantees that the reduced energy of these consumers in a certain operating period, will be provided by the grid during other operation hours. In this DRP, consumers are expected to participate in the plan to shift their energy consumption at a peak load period to an off-peak time. This is because it is commensurate with minimizing the fuel cost of the GUs in the last part of Equation (1). So that if the load decreases during the peak hours, the GUs with low fuel costs will feed the consumers. Furthermore, because the energy demand is low during the off-peak hours, the increased consumption during this period may be met by inexpensive GUs. This corresponds to minimizing the operating costs of the GUs.

$$\left|P_{b\tau hw}^{DRP}\right|\le {\kappa }_b{P}_{b\tau hw}^L\forall b,\tau ,h,w$$

(21)

$${\displaystyle \sum _{\tau \in {\mathsf{\Theta }}_{OH}}{P}_{b\tau hw}^{DRP}}=0\forall b,h,w$$

(22)

Network limits: Transmission network operation and flexibility constraints are formulated in (23) and (24) [54] and (25) [7], respectively. Operation limitations of the network include the limit of the apparent power passing through the TL, (23), and the limit of the voltage magnitude of the buses, (24). The high voltage limits are used to prevent insulation damage to the network equipment, due to overvoltage. Its lower limit is also used to prevent the mains from shutting down due to severe voltage drop in the network [54]. In this section, in order to establish the desired flexibility in the transmission network, a flexibility constraint, such as Constraint (25) is used for the non-renewable GUs. Note that it is expected that the results of the real-time and day-ahead operation of a system will not be the same in the presence of the renewable GUs, which is due to a forecasting error in the output active power of these types of GUs. This is known as a low system flexibility, which can lead to damage to the network, due to an imbalance between the demand and supply at the real-time scheduling. To compensate for this, flexibility sources, such as the DRP [7] or storage systems [55,56,57,58,59], must be able to reduce the power fluctuations of the renewable GUs in the real-time scheduling. In this condition, the desired flexibility is established in the network [7]. However, considering this requires a mathematical formulation of the flexibility in the proposed framework, this section presents Constraint (25). In this case, the deviation of the generated active power for the non-renewable GUs in scenario s from the scenario related to the deterministic problem (with the forecasted values of the uncertainty parameters assumed to be equal to the first scenario) should be less than ΔF, where ΔF represents the flexibility tolerance. If ΔF = 0, then 100% flexibility for the network can be evaluated. Therefore, it is expected that with this flexibility model, the results of the real-time scheduling will be close to the results in the day-ahead operation.

$${\left(P_{bl\tau hw}^F\right)}^2+{\left(Q_{bl\tau hw}^F\right)}^2\le {\left({\overline{S}}_{bl}^F\right)}^2\forall b,l,\tau ,h,w$$

(23)

$${\underset{¯}{V}}_b\le V_{b\tau hw}\le {\overline{V}}_b\forall b,\tau ,h,w$$

(24)

$$-\Delta F\le P_{g\tau hw}^G-P_{g\tau hw|w=1}^G\le \Delta F\forall g\in {\mathsf{\Theta }}_G-{\mathsf{\Theta }}_{WF},\tau ,h,w$$

(25)

2.2. Uncertainty Model

In Problems (1)–(25), renewable power, ${\overline{P}}_{}^G$, load, P^L and Q^L, are uncertainties. In this section, the non-parametric methods, such as the SBSP, models the uncertainty situation of these parameters. This method is according to a combination of the Monte Carlo simulation (MCS) and the Kantorovich method. First, the MCS produces a large number of scenario samples [60]. In each scenario, the values of the mentioned uncertainty parameters are determined, according to their standard deviation and mean values. The load probability is found from the normal probability distribution function (PDF) [9], and it is accessible by the Weibull PDF for the third uncertainty parameter [7]. In next step, the probability for a generated scenario (π⁰) is based on the product of the probability parameters of uncertainty in that scenario. Then, the scenario reduction approach, based on the Kantorovich method, selects several generated scenarios that have the shortest distance to each other, and then applies it to the proposed GTEP scheme. The probability for a selected scenario (π) is equal to the ratio of π⁰, related to this scenario and the sum of π⁰, of the selected scenarios. More details on this approach are provided in [9].

3. Problem Solution

The problem presented in the previous section has a MINLP model. In this section, the HEA, based on the combination of CSA [61] and GWO [62], is used to solve this problem. In this algorithm, the solution steps are as follows:

• Step 1: In this step, the solver provides N, which refers to the population size, values as random for the decision variables including x^G, x^SCB, x^L, P^G, for the non-renewable GUs, and Q^G, P^DRP, and y^SCB, for Constraints (26)–(32).

$$x_{gh}^G\in \left\{0,1\right\}\forall g,h$$

(26)

$$x_{ch}^{SCB}\in \left\{0,1\right\}\forall c,h$$

(27)

$$x_{blh}^L\in \left\{0,1\right\}\forall b,l,h$$

(28)

$$P_{g\tau hw}^G\in \mathrm{Equation}(16)\forall g\in {\mathsf{\Theta }}_G-{\mathsf{\Theta }}_{WF},\tau ,h,w$$

(29)

$$Q_{g\tau hw}^G\in \mathrm{Equation}(17)\forall g,\tau ,h,w$$

(30)

$$P_{b\tau hw}^{DRP}\in \mathrm{Equation}(21)\forall b,\tau ,h,w$$

(31)

$$y_{c\tau hw}^{SCB}\in \mathrm{Equation}(20)\forall c,\tau ,h,w$$

(32)

• Step 2: Dependent variables, including u^G, u^SCB, u^L, P^F, Q^F, Q^SCB, V, γ, and P^G, for the renewable GUs, based on Constraints (8)–(10), (12)–(15), (18), and (19) are calculated for the N values of the decision variables. Note, the Newton–Raphson technique [63] is adopted to solve the AC-PF constraints, (12)–(15).
• Step 3: In this step, the fitness function (FF) is first calculated for the N values of the decision variables. The FF, as given by (33), is the summation of the PC in Equation (1) and the sum of the penalty functions (PeF) resulting from the planning constraints, (2)–(7), the DRP constraint, (22), and the operation and flexibility limitations of the network (23)–(25). In other words, in this section, the penalty function technique was used to estimate the mentioned constraints [30]. The function of the penalty for the equation a = b and the limit a ≤ b are formulated as λ.(b − a) and μ.max (0, a − b), respectively [64]. λ ∈ (−∞, +∞) and μ ≥ 0 are the Lagrange multipliers, which are known as the decision variable, and their values are determined in this step, similar to the first step. In the next step, the optimal value of the FF is determined.

$$\min FF=Cost+PeF$$

(33)

• Step 4: In this step, the decision variables are updated by the CSA algorithm. First, the new N position for the decision variables is determined, based on Constraints (26)–(32) and the optimal value of the FF in the previous step is specified by the CSA algorithm. Then, the second and third steps are performed.
• Step 5: The fourth step is performed, with the difference that, in this step, the GWO is responsible for updating the decision variables. Furthermore, at this stage, the best value from the fourth step is considered.
• Step 6: If the convergence point is reached, the problem is solved. Otherwise, the fourth step is performed. In this section, it is assumed that the convergence is achieved if the problem solution iteration reaches I_max (the maximum number of iterations).

Finally, the flowchart of the proposed solver is shown in Figure 2.

4. Numerical Results and Discussion

In this section, the numerical results obtained from the application of the proposed design on the modified IEEE 6-bus and 118-bus transmission networks are presented. The proposed problem simulators are implemented in the MATLAB software environment, in accordance with the solution process presented in Section 3.

4.1. Modified IEEE 6-Bus Network

(A)

Case study: The structure of the network is drawn by taking into account the elements in Figure 3. It contains a base power of 100 MVA and a base voltage of 230 kV. The allowable maximum/minimum voltage magnitude is 1.05/0.95 per unit (p.u) [65,66,67]. In this system, seven transmission lines can be constructed between the buses (1, 2), (1, 4), (2, 3), (2, 4), (3, 6), (4, 5), and (5, 6). These lines are marked with the symbols TL1 to TL7, respectively. Their specifications are the same as the existing parallel lines, and the various data of all TLs, including capacity (maximum apparent power), reactance and resistance, and the TL installing costs are presented in [14,17,68]. In addition, the network consists of four GUs, which are marked as AE1 to AE3 and BE1. Seventeen other GUs can be installed in this network, i.e., four GUs as WF and 13 GUs as non-renewable. WFs with titles W1–W4 can also be constructed in buses 6, 1, 1, and 2, respectively. Other GUs are designated as B1–B8 and A1–A5, which are constructed in buses 1, 2, 3, 6, 1, 1, 2, 3, 1, 1, 2, 3, and 6, respectively. The specifications of these GUs include the coefficients of the fuel cost function, the construction cost, and the allowable active and reactive power reported in [14,17,68]. It is noteworthy that the hourly power output data of the WFs are based on the product of the its capacity and the daily curve of the WF power rate. This curve is shown in Figure 4 [69]. In this network, three MVAr SCBs, with 20 steps, are able to be constructed in various network buses. Their construction cost is also assumed to be USD 3 /kVAr/year. The reported construction cost for the GUs, TLs, and SCBs, is for the first planning period, and it is multiplied by 1.1 and 1.2 for the second and third planning periods, respectively. The budget of investment for the SCBs, TLs and GUs is also estimated to be, USD 10 million, USD 20 million and USD 100 million.

There are consumers in buses 3 to 5, which occupy 0.4, 0.3, and 0.3 of the total load of the network, respectively. These consumers have a power factor of 0.9. In the first planning period, the total active load peak is equal to 30 MW, which, in the second (third) period, increases to 32 (34) MW. The daily load curve is based on the product of the daily curve of the load factor and the peak load, where this curve is shown in Figure 4 [69]. Since the planning period is assumed to be 2 years, du is equal to 730 days. Furthermore, in this section, the CF is assumed to be 0.7 [68]. It is further considered that the participation rate of consumers is 0.3 in the DRP. In order to achieve a high flexibility in this system, ΔF of 0.05 p.u. has been selected. In the proposed plan, the standard deviation of uncertainties is considered to be 10%. The MCS in the proposed scheme generates 2000 scenario samples for renewable power and load uncertainties, and then the Kantorovich method applies 80 scenario samples from the generated scenarios to the proposed GTEP problem.

(B)

Assessing the benefits of the mentioned solver: In this section, the numerical results obtained from the problem-solving by CSA+GWO, CSA, GWO, the differential evolution (DE) algorithm [70], and the genetic algorithm (GA) [20] are presented. The population size (N) and maximum iteration (I_max) are 50 and 5000, respectively. Other parameters of the different algorithms have been selected, based on [20,61,62,70]. In order to perform the statistical calculations, such as calculating the standard deviation (SD) of the optimal response at the convergence point, each algorithm solves the problem 20 times. The results of the problem-solving convergence are presented in

Table 2. Convergence results in the different solution processes.

Solver	PC (M$)	CI	CT (Min)	SD (%)
CSA + GWO	90.1	347	3.8	0.92
CSA	91.3	422	4.3	1.12
GWO	91.5	451	4.9	1.39
DE	93.1	632	7.3	2.23
GA	94.2	811	8.7	3.45

. Accordingly, the CSA+GWO algorithm obtains USD 90.1 million for the cost objective function, Equation (1), while other algorithms calculate a value of more than USD 91 million for this function. The hybrid CSA+GWO also obtains the optimum point in the 347th convergence iteration (CI), which corresponds to a computational time (CT) of 3.8 min. However, the CSA, GWO, DE, and GA algorithms find the optimal point in the CI (CT) above 420 (4.3 min). The GA algorithm achieves this point in a CT of 8.7 min. Therefore, the hybrid CSA+GWO solver has a higher convergence rate than the non-hybrid algorithms. Furthermore, note that =Problems (1)–(25), due to the presence of the AC-PF, are non-convex. Hence, its solvers generally obtain the locally optimal solution. With this in mind, it can be said that CSA+GWO, compared to other mentioned algorithms, has been able to find the local optimal point that is closer to the absolute optimal point. As another point, according to Table 2, it can be seen that the SD in the last response of the CSA+GWO algorithm at the convergence point, is about 0.92%, so the dispersion of the final response in this algorithm is low, so it has approximate unique response conditions. Nonetheless, this is not the case with non-hybrid algorithms, and they have almost a significant dispersion in the convergence point response.

(C)

Transmission network expansion status: The planning results of the GUs, TLs, and SCBs, based on the proposed GTEP for the different load levels, are expressed in

Table 3. Planning results of the GUs, TLs, and SCBs for the different load levels, based on the proposed GTEP scheme.

Load Level	1	1.2	1.4
Installing time	h = 1
Installing TLs	TL3, TL7	TL3, TL7	TL3, TL5, TL7
Installing GUs	A1, all WFs	A1, B7, all WFs	A1, B5, B7, all WFs
Installing SCBs in bus	3 to 5

. Accordingly, for load level 1 (100%), two TLs between buses (2, 3) and (5, 6), all wind farms, W1–W4, the non-renewable GU A1, and the three SCBs in bus-bars, 3–5, is added to the modified IEEE 6-bus network, to ensure the load growth on the planning horizon. By increasing the load level by 20%, only one non-renewable GU, i.e., B7, is added to the network, while by increasing the load level by 40%, two non-renewable GUs, namely, B5 and B7, and one TL5 (between buses 3 and 6) are added to the network, with respect to load level 1. It is noteworthy that the selection of these elements is commensurate with the minimization of the objective Function (1) and the estimation of the operation and flexibility constraints (23)–(25). According to Table 3, the mentioned elements are generally installed in the first planning period (h = 1), because in this period the construction cost is lower than in previous years. In the following, the total planning cost and the planning cost of the different elements for the different load levels are plotted in Figure 5. Based on this figure, with the increasing load level, the construction cost of the SCBs is constant, because according to Table 3, their planning results are the same at different load levels. The construction cost of the TLs is the same for 100% and 120% load levels, but increases for a load level of 140%. This case is due to the installation of TL 3 (TL 2) at the load level of 140% (100% to 120%). According to Table 3, since newer GUs are added to the network as the load level increases, according to Figure 5, their construction and operation costs also increase in proportion to the increased load. Finally, these conditions increase the expected total planning cost, according to Figure 5.

(D)

Assessing the economic and technical capabilities of the proposed plan: In this section, in order to evaluate the economic and technical capabilities of the GTEP scheme, the numerical results of the following five studies are examined:

• Case I: Considering only the AC-PF studies;
• Case II: Implementation of the proposed GTEP framework without the DRP and SCB;
• Case III: coupling the GTEP and SCB placement problems;
• Case IV: Implementation of the GTEP in a network with the DRP without considering the SCB;
• Case V: GTEP implementation and the SCB location in a network with the DRP.

The simulation results are expressed in

Table 4. Technical and economic results of the GTEP in different cases.

Case	I	II	III	IV	V
PC (M USD)	43.8	105.9	104.2	91.3	90.1
TEL (MWh)	31,394	26,734	26,089	25,211	24,721
MVD (p.u)	0.059	0.054	0.048	0.046	0.044
PLCC (MW)	25	97	101	118	121

, which presents the values of cost, the peak load carrying capability (PLCC) [68], the total energy loss (TEL), and the maximum voltage deviation (MVD). Accordingly, Case I has the lowest cost, compared to other study cases, because it has only the expected fuel cost of BE1, AE1, AE2, and AE3. In other words, no element is constructed in this case. However, in this case, the TEL and MVD are greater than in other cases. Furthermore, in this case, the PLCC is up to 25 MW, the peak load is up to 25 MW, and the network can supply its consumers. Nevertheless, since the peak load in the planning intervals is more than 30 MW, according to Section 4.1.A, the network will suffer from outages in Case I, due to the operation limitations, (23) and (24). Thus, there is a need to expand the network. With the network expansion in Cases II-V, it can be seen that the lowest cost is obtained in Case V. It also has the minimum TEL and MVD. These parameters, in Case V, compared to Case I, are reduced by about 21.2% ((3,139,431 − 24,721)/31,394) and 25.4% ((0.059–0.044)/0.059)), respectively. Case V has also been able to obtain the highest PLCC, compared to the other case studies, by installing the GUs, TLs, and SCBs along with the DRP implementation. Therefore, in Case V, the network can supply 121 MW of the peak load to consumers. Since the peak load at the planning horizon after 6 years is 34 MW, and the PLCC for Case V is 121 MW, it can be said that the implementation of Case V on the modified IEEE 6-bus network prevents the re-planning of the network for many years.

Comparing Cases II to IV, it can be seen that the simultaneous planning of the GU and TL has a significant effect on reducing energy losses, voltage deviation, and the PLCC. The TEL (MVD) is reduced by about 15% (8%) in Case II, in comparison with Case I. In this case, the PLCC is increased to 97 MW, while in Case I, it is 25 MW. The SCB planning has little effect on reducing the TEL and increasing the PLCC, but they do have a significant effect on reducing the MVD. The SCB planning in Case III, with respect to Case II, was able to decrease the MVD by about 11.1%. The DRP operation has a significant effect on reducing the TEL and MVD and increasing the PLCC. It has been able to reduce the TEL (MVD) by about 5.7% (14.8%) and increase the PLCC by about 21.6% in Case IV, compared to Case II. Note, however, that the simultaneous planning of the GUs, TLs, and SCBs and the use of the DRP in Case V have been able to achieve a more favorable technical condition than in Cases II-IV, along with incurring the minimum cost.

4.2. IEEE 118-Bus Network

This system includes 186 TLs, 54 non-renewable GUs, and 91 load buses. Its details are reported in [17]. It is assumed that with each existing TL, another new TL can be placed in parallel with the network expansion. This also applies to the installation of new non-renewable GUs. In addition, five renewable Gus, as WF type, including a maximum apparent power of 200 MVA, can be placed in buses 26, 30, 41, 68, and 85. High-capacity SCBs [71], i.e., 50 MVAr, can placed in buses 13, 22, 30, 33–35, 60, 70, 103, and 111. In this system, the peak load (active and reactive) in the first 2-year planning period is equal to 4242 MW and 1438 MVAr, respectively. In other periods, they increase at a rate of 6%. Other problem data, such as ΔF, the daily load factor curve, and the power generation rate of WFs are as in Section 4.1. The investment budget in this subsection is 100 times that of Section 4.1.

The results of planning, economics, operation, and convergence of the proposed GTEP scheme for different values of the flexibility tolerance (ΔF), are reported in

Table 5. Economic, technical, and environmental results in ΔF.

ΔF (p.u)		0.025	0.05	0.075
Installing time		h = 1
Installing TLs		TL4, TL11, TL17, TL39, TL50, TL96, TL117, TL132, TL143, TL156, TL175
Installing GUs		GU5, GU10, GU21, GU24, GU32, GU45, and all proposed WFs
Constructed SCBs		All proposed SCBs
Total expected planning cost (M USD)		1921	1864	1737
Total energy loss (TWh)	First case (I)	4.79
	Last case (V)	3.08	3.01	2.87
Max of voltage deviation (p.u)	First case (I)	0.055
	Last case (V)	0.040	0.039	0.038
Peak load carrying capability (MW)	First case (I)	5097
	Last case (V)	16322	16408	16502
Calculation time (min)		25.2	24.5	23.3

. Based on this table, for the different amounts of ΔF, 11 TLs, 11 GUs, and 10 new SCBs, they are added to the mentioned network in the year of the first planning period (h = 1). These choices are based on the cost minimization and the estimation of operation and flexibility constraints. As can be observed from this table, under conditions of a high flexibility (ΔF = 0.025 p.u.), the total expected planning cost is USD 1921 million, the TEL and MVD in the proposed scheme (Case V) are decreased by about 36%, and 29.8%, compared to Case I, but the PLCC increases by about 218%. In terms of the problem convergence, the proposed design reaches the convergence point in about 25.2 min. moreover, by increasing the flexibility tolerance, more favorable economics, operation, and convergence conditions are provided for the proposed GTEP scheme. This is because the increasing ΔF means releasing Constraint (25), and consequently the solution space is expected to increase. In this case, the convergence point is obtained in a shorter computational time with a lower planning cost and a more suitable operating condition than the conditions ΔF = 0.025 p.u.

Figure 6 shows the results of the optimal operation of the GUs, SCBs, and DRPs in the last year of planning. Figure 6a shows that in Case V, consumers participating in the DRP, increase their energy consumption during 1:00–13:00, compared to Case I, while they decrease their energy consumption at the peak load time. This has caused the changes in the GU’s active power, according to Figure 6a, to be negligible at most hours. Additionally, as the GUs do not need to produce high power during peak hours, inexpensive GUs are expected to supply power during these hours. Therefore, its operation costs will be reduced if the DRP is used for the network. According to Figure 6b, the SCBs always inject a constant reactive power equal to 500 MVAr (maximum capacity) to the network at different operating hours. This has caused the daily curve of the GU’s reactive power to have a trend, similar to the daily reactive power curve.

5. Conclusions

In this paper, the dynamic GTEP problem with the SCB location in a network with the DRP is presented. The scheme minimizes the total expected construction cost of the GUs, TLs, and SCBs and the operation costs of the GUs. It was also subject to the AC-PF constraints, the planning-operation formulation of the mentioned elements, the DRP operation model, and operation, and the flexibility limits. In this scheme, the load and renewable power parameters had uncertainties, and the SBSP was used to model these uncertainties. Then, a HEA, based on the hybrid process of the CSA and GWO, was adopted to solve the proposed problem. In the end, according to the simulation results, it was observed that the mentioned solver obtains a more optimal point than the non-hybrid algorithms in the high calculation speed. Furthermore, the standard deviation of the final response was about 0.92%, which corresponds to the approximate unique response conditions. The proposed scheme with the optimal planning of the GUs, TLs, and SCBs, along with their optimal operation and the DRP, has been able to reduce the energy losses and the voltage deviations in the conditions of high flexibility, in the range of 20–36% and 25–30%, compared to the network power flow studies. It also has the potential to gain more feed loads than the power flow studies.