Probabilistic interactive fuzzy satisfying generation and transmission expansion planning using fuzzy adaptive chaotic binary PSO algorithm

Abstract

In this paper, an interactive fuzzy satisfying method based on fuzzy adaptive chaotic binary particle swarm optimization (FACBPSO) algorithm is proposed to investigate the multi-objective Generation and Transmission Expansion Planning (G-TEP). The objective functions of the G-TEP problem, which are modeled by fuzzy sets, present the total investment/operation cost and the total pollutant emission. In modern power systems, the necessity of considering the wind/solar energy resources in Generation Expansion Planning (GEP) studies is important. In addition, HVDC links, transmitting renewable resource power from remote sites, should be considered in TEP problem. The use of the wind/solar energy due to the uncertainty of their generation and the integration of HVDC links would consequently impose more complexity to solution of the G-TEP problem. In this paper, the proposed algorithm is tested on an IEEE test system based on economic and environmental considerations to generate an optimal expansion plan.

Keywords

Stochastic modeling Generation and Transmission Expansion Planning HVDC links Wind/solar energy resources fuzzy adaptive PSO interactive fuzzy satisfying method

1 Introduction

In power system expansion planning problem, determination of an optimal generation mix and transmission upgrades have a vital role in economic and secure operation of system [1 –6]. In order to keep the security of the power system, usually a large portion of the total capacity of generating units has been formed by steam and gas units [7], which is inconsistent with the Kyoto protocol commitments. There are many different policies to reduce emission effects in generation expansion planning (GEP) problem such as finance theory [8], emission trade and carbon tax [9], impact of renewable energy sources (RES) incentives (including feed-in tariffs, quota obligations, and tender schemes) [10] and considering CO₂ emission constraints [11]. However, the most favorable wind and solar sites are located far from load centers and their generation is stochastic. Hence, additional transmission lines are needed to access these sites. On the other hand, by transporting generated electricity to the load centers through HVAC transmission system, the line losses and subsequently emission are increased. As a consequence, there are different ways to reduce the emission:

Increasing the share of RES in particular wind and solar energy in generation sector,

Transforming the AC transmission lines into HVDC transmission lines.

The lower losses, lower investment costs, less land space, easier to control the active power are advantages of using HVDC systems over HVAC ones for transmission of bulk power over long distances [12, 13]. So, the main goal of this paper is to simultaneously develop renewable generation and AC/DC transmission expansion planning.

In the literature, there are numerous publications, considering GEP [14 –16] and TEP [17 –19] problems separately. A comprehensive review of GEP and TEP problems is given in [20]. In [7], a multi-objective model to obtain the power generation capacity of new generating units has been proposed. The conventional steam units, coal units, combined cycle modules, nuclear plants, gas turbines, wind farms, and geothermal and hydro units have included as candidate units. In [7], a framework for GEP in a pool market has been presented and a hybrid modified game theory has been proposed, which uses PSO as a useful technique for solving GEP optimization problem. In [21], the impact of RES incentives and mitigation policies is considered in the framework of GEP problem. In [21], a multi-objective GEP problem, considering relevant costs, fuel consumption, CO2 emission and energy price risks, has been presented. In [22], the TEP problem, considering both the network losses and construction cost of lines, has been solved by discrete particle swarm optimization (PSO) algorithm. In [23] and [24], the effect of the load demand and wind production uncertainties on network expansion planning has been investigated. In [25], the TEP problem under uncertainties of renewable generation and load has been considered. In [26], due to deregulation of the power system, new uncertainties such as market rules, bidding strategies, load forecast and energy at risk have been introduced and studied. The inherent uncertainties associated with estimated investment costs of candidate transmission lines and forecasted electricity demands during long term planning horizon has been investigated in [27]. Many researches have been carried out to simultaneously coordinate TEP and GEP [28 –30]. In [29], a co-optimization model for G-TEP including micro-grid effects has been used. The micro-grid may be a suitable alternative to reduce the total planning cost of the conventional G-TEP problem. Transmission switching applicability for the power system planning has been addressed in [30], to maximize the profit over the planning horizon. A new probabilistic model for G-TEP problem considering reliability criteria has been presented in [31].

In this paper, the main purpose of the G-TEP problem is the minimization of the investment and operation costs and also emission. As a consequence, it is important to determine the importance of different planning objectives which, should be solved in multi-objective G-TEP problem. The most common solving method for multi-objective problems is based on conversion to a single-objective case, e.g. using weighting method and ɛ-constraint [32]. In these methods, the multi-objective problem is converted into a scalar optimization problem. In the weighted sum method, a linear combination of different objectives is used. Low accuracy is the significant defect of this method. Also, to reach a set of desired optimal solutions, several runs are needed. In the ɛ-constraint method, the most appropriate objective will be optimized and the other objectives will be considered as constraints bounded by a pre-defined authorized level of ɛ. Therefore, in this paper, a fuzzy satisfaction decision approach, which utilizes the concept of Decision Making (DM), is applied to solve the multi-objective G-TEP problem [33, 34]. In this method, a membership function to associate a normalized value to each objective, is used. In addition, all optimum solutions may be found during the search process due to the adopted DM process. Generally, Evolutionary Algorithms (EAs) are suitable to be selected for solving multi-objective G-TEP problems. Different EA techniques such as genetic algorithm [35], firefly algorithm [36] and Teacher-Learning-Based Optimization algorithm [37] have been used for the G-TEP problem.

In this paper, due to its simplicity, robustness and effectiveness, Particle Swarm Optimization (PSO) is selected [38]. It is noted that the G-TEP problem is a binary integer problem, the binary form of the PSO should be used. The same as other EA methods, the original PSO have major drawbacks as follows:

Convergence to a local optimum solution.

Loss of population diversity after a few iterations.

Highly dependent on internal parameters.

To enhance the performance of PSO and improve the global searching capability, a chaotic local search (CLS) is proposed. Also, a fuzzy system is used to tune internal parameters such as inertia weight and learning factors.

Eventually, considering the uncertainty of wind/solar generations and load demand, the application of a probabilistic method is inevitable. An efficacious probabilistic method namely Point Estimation Method (PEM) is applied to the model uncertainties, considering its simple implementation, proper accuracy and low computational efforts needed.

2 Mathematical model of multi-objective generation and transmission expansion planning (G-TEP)

Generally, the interactive multi-objective optimization problem solving consists of three major steps: identify the objective functions, fuzzy- based clustering and finally solve a mini-max problem.

2.1 Objective functions identification

In this paper, two following conflicting objectives are considered, to generate a set of expansion alternatives: (a) Investment and operational costs minimization, and (b) minimization of losses and emission generated by power plants.

2.1.1 Total planning cost

The objective in the proposed problem is minimization of the total investment and operation costs for the generating units and AC/DC transmission systems, as follows: $\begin{matrix} min Total_Cost \\ = Investment_Cost + Operation_Cost \\ = \sum_{t = 1}^{Nt} \frac{1}{(1 + d)^{t - 1}} (\sum_{i = 1}^{CG} C_{G, i} . X_{i} + \sum_{j = 1}^{{CL}_{AC}} C_{AC, j} . Y_{j} \\ + \sum_{k = 1}^{{CL}_{DC}} C_{DC, k} . Z_{k}) + \sum_{h} \sum_{b} {DT}_{bht} {OC}_{i} P_{ibht} \\ s . t \\ C_{DC} = C_{line} + C_{stat} \\ C_{line} = a + b . V + S . (cN + d) $ / km \\ C_{stat} = A \times V^{B} \times P^{C} M $ \end{matrix}$ (1)

The coefficients of a, b, c, d, A, B and C are determined using data provided in [13] and by curve fitting of these data. The converter station cost depends on the line voltage as well as the power rating of the converter station. Additionally, the DC link configuration such as number of terminals and converter type may be affected. Also, the investment cost of AC transmission lines depends on the voltage level, length of line, line flow capacity, and financial issues which are considered as fixed and predetermined. For calculating the operation cost of existing and installed units, each planning year is divided into three load blocks representing off-peak, intermediate and peak loads. The planning is performed annually while the operation is carried out for each load block.

2.1.2 Emission Function

The total emission objective of the proposed problem is the minimization of the total fuel emission of generating units and emissions generated in AC/DC transmission systems, as follows:

$\begin{matrix} min Total_Emission \\ = {Emission}_{TU} + {Emission}_{GU} + {Emission}_{TL} \end{matrix}$ (2)

The total fuel emission of generating units depends on the type of the fuel and the amount of the power generated by that unit.

In fossil-based generating units, the amount of pollutants is considered by the following Equations [39]:

$\begin{matrix} {Emission}_{TU} = \sum_{i = 1}^{Ng} E_{i} (P_{i}) \\ = \sum_{i = 1}^{Ng} (α_{i} + β_{i} P_{i} + γ_{i} P_{i}^{2} + ξ_{i} exp (λ_{i} P_{i})) \end{matrix}$ (3)

In addition, the emission of natural gas units such as NOx, SO2 and CO2, can be expressed by the following Equation [33]:

$\begin{matrix} {Emission}_{GU} \\ = \sum_{i = 1}^{N_{Gas}} ({NO}_{X}^{Gas, i} + {SO}_{2}^{Gas, i} + {CO}_{2}^{Gas, i}) \times P_{Gas, i} \end{matrix}$ (4)

The emissions generated in the power transmission technology are due to technical losses. The relation between emission and losses are derived as emission factors associated to transmission losses as follows:

$\begin{matrix} {Emission}_{TL} \\ = Active_Power_losses * Emissionfactor \end{matrix}$ (5)

2.2 Fuzzy- based clustering

In the proposed procedure, decision making is based on normalized fuzzy membership value as follows:

Firstly, the membership function of each objective function is defined for any individual. $μ_{fi} (X) = {\begin{matrix} 1, f_{i} (X) \leq f_{i}^{min} \\ 0, f_{i} (X) \geq f_{i}^{max} \\ \frac{f_{i}^{max} - f_{i} (X)}{f_{i}^{max} - f_{i}^{min}}, f_{i}^{min} \leq f_{i} (X) \leq f_{i}^{max} \end{matrix}$ (6)

Then, the normalized membership value is evaluated as follows: $N μ (j) = \frac{\sum_{k = 1}^{n} ω_{k} \times μ_{fk} (X_{j})}{\sum_{j = 1}^{m} \sum_{k = 1}^{n} ω_{k} \times μ_{fk} (X_{j})}$ (7)

This membership function shows a type of decision making criterion that is adaptive and changes with the available decision options. In the fuzzy-based clustering, the normalized membership values are sorted and the best individuals are selected as best compromising solution.

2.3 Mini-max problem

After fuzzy clustering process and computation of the membership function for all objectives, the following mini-max problem is solved [33, 34].

$\begin{matrix} F (X) & = & min {\max | (μ_{r, i} - μ_{f_{i}} (X) |} \\ X \in Ω; i = 1, 2, \dots, OF \end{matrix}$ (8)

In the paper, the mini-max problem is solved by fuzzy adaptive chaotic binary particle swarm optimization (FABPSO) algorithm. If the DM is not satisfied the new value of the mini-max problem is updated based on the change in the reference membership. This interactive updating process will continue till the satisfying solution for the DM is obtained.

3 Modeling of HVDC systems

A DC transmission system can be categorized based on converter technologies such as Current and Voltage Source Converter (CSCs or VSCs), converter configurations (mono-polar, bi-pole and tri-pole) and number of converter stations (back-to-back, two-terminal and multi-terminal). In this paper, VSC-based two-terminal (TTDC) and multi-terminal DC (MTDC) configurations are studied. Figure 1 shows the schematic of the TTDC and MTDC systems.

4 Probabilistic model of G-TEP

4.1 Modeling of load, wind and solar farms uncertainty

The growth of the load demand for every bus may affect the performance of different planning schemes and should be considered in the G-TEP. In order to determine the annual peak demand, the normal distribution is taken into account. $PDF (S_{i}^{L}) = \frac{1}{\sqrt{(2 π σ_{i}^{L})}} exp (- \frac{(S_{i}^{L} - μ_{i}^{L})^{2}}{2 σ^{2}})$ (9)

A wind farm is composed of a set of wind turbines (WTs). The total available power of a wind farm at each instant, is equal to the sum of the production of all WTs. Therefore, it is enough to model a WT. The power output characteristics of a WT are related to the wind speed. So, the modeling of the wind speed is inevitable. The Weibull probability distribution function (PDF) is commonly used to model the wind speed. The Weibull PDF can be described by two parameters: shape and scale factors. The first one is based on historical data and the other one depends on the forecasted wind. The relationship between power output of WT (P_WT) and wind speed (v) is given as [1]: $PDF (v) = (\frac{k}{c}) {(\frac{v}{c})}^{k - 1} exp (- {(\frac{v}{c})}^{k})$ $P_{WT, i} = {\begin{matrix} 0 \\ if v \leq v_{cut - in} OR v \geq v_{cut - out} \\ \frac{v - v_{cut - in}}{v_{rated} - v_{cut - in}} \times P_{rated, i} \\ if v_{cut - in} \leq v \leq v_{cut - out} \\ P_{rated, i} O . W \end{matrix}$ (10)

The solar energy is produced when the sun is shining during the day. To predict the solar cell production, the model of solar irradiance should be determined. An individual solar cell is usually quite small, and they are connected in parallel and series to form larger units called arrays, to meet the required electricity demand. Therefore, it is enough to model a solar cell by using the beta PDF as follows: $PDF (s) = \frac{Γ (α + β)}{Γ (α) + Γ (β)} \times s^{α - 1} \times (1 - s)^{β - 1}$ (11)

The generated power of the solar cell is based on Equation (12) and related to short-circuit capacities (SCCs), current/voltage at maximum power point, ambient temperature of the installed site, and determined as follows [1]: $\begin{matrix} P_{solar, i} = N \times FF \times V (s) \times I (s) \\ s . t \\ FF = \frac{V_{MPP} \times I_{MPP}}{V_{OC} \times I_{SC}} \\ V (s) = V_{OC} - K_{v} \times T_{c} \\ I (s) = s_{a} \times (I_{SC} + K_{i} (T_{c} - 25)) \\ T_{c} = T_{A} + s_{a} \times \frac{N_{OT} - 20}{0.8} \end{matrix}$ (12)

4.2 Probabilistic Optimal Power Flow (OPF)

Several parameters of G-TEP problem can be considered as stochastic parameters and thus it should be solved by probabilistic methods. In a power system, if the load and generation distributions have been determined, the active power distribution and overload probability of each line could be determined with probabilistic power flow calculation. So, it is impossible to implement the conventional OPF. In the literature, several approximations for probabilistic OPF analysis such as Taylor Series Expansion Method, First-Order Second-Moment Method (FSOMM), Monte Carlo Simulation (MCS), Gram Charlier and Descritization Methods have been proposed [40, 41].

The most significant drawbacks of these methods, in particular MCS method, are their limited search space and time-consuming calculation procedure. The 2 m point estimate method (PEM), as an approximate method, is an efficient and reliable method to model the uncertainty in power systems [42].

4.2.1 2m-point estimation method

Mathematically, the PEM is applied to stochastically calculate the output variables using the solution set of the deterministic G-TEP problem for only few estimated values of input variables as follows: $\begin{matrix} Y = g (X) \\ X = [x_{1}, x_{2}, \dots, x_{m}] \\ Y = [P_{loss - line}^{AC}, P_{loss - line}^{DC}] \end{matrix}$ (13)

The input variables (X) are the load demand and the available output power of the WT and PV units. The function f is the set of AC/DC OPF equations connecting the uncertainty of input variables to output variables.

In 2m-PEM, the mean and standard deviation of the output variable μ_Y and σ_Y can be obtained through the following steps [42].

Step 1: Define m.

Step 2: Set E (Y) =0 and E (Y²) =0

Step 3: Calculate two locations of concentrations and probability of concentrations as follows: $\begin{matrix} ξ_{k, 1} = \frac{λ_{k, 3}}{2} + \sqrt{m + (\frac{λ_{k, 3}}{2})^{2}} \\ ξ_{k, 2} = \frac{λ_{k, 3}}{2} - \sqrt{m + (\frac{λ_{k, 3}}{2})^{2}} \end{matrix}$ (14) $λ_{k, 3} = \frac{E (z_{k} - μ_{Z_{k}})^{3}}{σ_{Z_{k}}^{3}}$ (15) $\begin{matrix} Pr {ob}_{k, 1} = \frac{- ξ_{k, 2}}{2 m . \sqrt{m + (\frac{λ_{k, 3}}{2})^{2}}} \\ Pr {ob}_{k, 2} = \frac{ξ_{k, 1}}{2 m . \sqrt{m + (\frac{λ_{k, 3}}{2})^{2}}} \end{matrix}$ (16)

Step 4: Compute two estimated locations. $\begin{matrix} x_{k, 1} = μ_{X, k} + ξ_{k, 1} . σ_{X . k} \\ x_{k, 2} = μ_{X, k} + ξ_{k, 2} . σ_{X . k} \end{matrix}$ (17)

Step 5: Calculate the deterministic PF for the k-th estimated location using the following matrix: $X = [μ_{k, 1}, μ_{k, 2}, \dots, x_{k, i}, \dots, μ_{k, m}]$

Step 6: Update the first and second moment of the output random variable. $\begin{matrix} E (Y)^{k + 1} = E (Y)^{k} + \sum_{i = 1}^{2} Pr {ob}_{k, i} . f (X) \\ E (Y)^{k + 1} = E (Y)^{k} + \sum_{i = 1}^{2} Pr {ob}_{k, i} . f^{2} (X) \end{matrix}$ (18)

Step 7: Compute the mean and standard deviation of the output random variable. $\begin{matrix} μ_{Y} = E (Y) \\ σ_{Y} = \sqrt{E (Y^{2}) - μ_{Y}^{2}} \end{matrix}$ (19)

It is clear that the 2m-PEM algorithm is essentially very simple and easy to implement. It must be emphasized that the deterministic G-TEP should be simulated 2m times in the PEM.

5 Proposed FACBPSO algorithm

5.1 Binary Particle Swarm Optimization (BPSO) algorithm

The particle Swarm Optimization (PSO) is a population-based evolutionary algorithm, which was originally proposed in 1995 by Kennedy and Eberhart [38]. In PSO, each solution can be considered as an individual particle in a given search space, which characterized by its position and velocity. Each particle adjusts its position according to the movement rule by changing its velocity based on its own experience (G_best), as well as the experience of its neighbors (P_best,i). $\begin{matrix} V_{i}^{(t + 1)} = ω \cdot V_{i}^{(t)} + c_{1} \cdot {rand}_{1} (\cdot) \cdot (P_{best, i} - X_{i}^{(t)}) \\ + c_{2} \cdot {rand}_{2} (\cdot) \cdot (G_{best} - X_{i}^{(t)}) \\ X_{i}^{(t + 1)} = X_{i}^{(t)} + V_{i}^{(t + 1)} \end{matrix}$ (20)

Where, i is the index of each particle, t is the current iteration number, rand₁ (.) and rand₂ (.) are two random functions in the range [0,1].

The entire algorithm of the binary version of PSO is almost the same as that of the basic continuous version, except for the state equations listed below. The position of each particle is a vector in the d-dimensional binary solution space and the velocity is a vector in the continuous space. A new particle is generated according to the velocity, as follows: $X_{i}^{t + 1} = {\begin{matrix} 1 if rand (.) < sig (V_{i}^{t + 1}) \\ 0 O . W \end{matrix}$ (21) ${V^{'}}_{i}^{t} = sig (V_{i}^{t}) = \frac{1}{1 + exp (- V_{i}^{t})}$ (22)

5.2 Chaotic BPSO algorithm

The BPSO often converges to local optima. In order to avoid this shortcoming, a chaotic BPSO algorithm is proposed in this paper combines BPSO with the Chaotic Local Search (CLS) presented in [43].

To apply CLS on the BPSO algorithm, the initial binary sequences are generated by the followingequation: $\begin{matrix} {cx}_{i + 1}^{j} = 4 \times {cx}_{i}^{j} \times (1 - {cx}_{i}^{j}), i = 0, 1, 2, \dots, \\ N_{choas}; j = 1, 2, \dots, Ng \\ {cx}_{0}^{j} = rand (.) \\ b () = ⌊ 2 \times {cx}_{i + 1}^{j} ⌋ \end{matrix}$ (23)

For all individuals of CLS, the fitness function is evaluated and the best solution among them with one particle selected randomly, is replaced.

5.3 Fuzzy adaptive CBPSO

The inertia weight (ω), the learning factors c₁ and c₂, are adjustable parameters of BPSO that greatly influence the algorithm performance. The ω is used to control the influence of the particle velocity. A fine-tune of the inertia weight can provide a balance between global and local optimal solution. The learning factors c₁ and c₂ determine the influence of the personal best P_best,j and global best G_best. Overall, it is not quite clear how to vary the mentioned parameters during a run to get the best tradeoff between exploration and exploitation. So, in this paper, the inertia weight and learning factors of the BPSO is dynamically adjusted using fuzzy IF/THEN rules. The Mamdani-type fuzzy rule is used to formulate the conditional statements that comprise fuzzy logic. The fuzzy inference system maps crisp set of input variables into a fuzzy set using membership functions. According to the predefined logic, the output is assigned based on these fuzzy input sets. The AND operator is used to combine the membership values and method of centroid (center-of-sums) is used to obtain a deterministic control action. The fuzzy setting of the inertia weight and learning factors are described in the following subsections.

5.3.1 Inertia weight (ω)

To determine a suitable inertia weight under the fuzzy environment the set of input variables are normalized fitness value (NFV) and current inertia weight (ω), whereas output variable is the change in the inertia weight (Δω). The normalized fitness value is used to show the effects of objective function as a binary value as follows: $NFV = (FV - {FV}_{min}) / ({FV}_{max} - {FV}_{min})$ (24)

The FV_max is a very large value, which is greater than any acceptable feasible solution. The new value of the inertia weight is calculated based on the change in the inertia weight as follows: $ω^{k + 1} = ω^{k} + Δ ω$ (25)

Both positive and negative corrections are required for the inertia weight. Therefore, a range of –0.1 to 0.1 has been chosen for the inertia weight correction. For fuzzification of input and output, the membership functions are used. Here, all the membership functions are triangular in shape for simplicity and all the memberships of the input are presented in three linguistic values; S, M and L for Small, Medium and Large, respectively. The output variable is presented in three fuzzy sets of linguistic values; NE (Negative), ZE (Zero), and PE (Positive) with associated membership functions, as shown in Fig. 2 [44]. The fuzzy rules are used to present a mapping from the input space to the output space for calculating the inertia weight correction as listed in Table 1. Each rule is an “IF-THEN” statement.

5.3.2 Learning factors c1 and c2

In a similar way, a fuzzy system is developed to adjust the learning factors with best fitness (BF) and number of generations for unchanged best fitness (NU) as the input variables, and the learning factors (c₁ and c₂) as the output variables. All the memberships of the input and output are presented in four linguistic values; PS (positive small), PM (positive medium), PB (positive big) and PR (positive bigger). For fuzzification of input and output, the membership functions shown in Fig. 3, are used. The fuzzy rules in Table 2 are used to adjust the learning factors [44].

6 Planning methodology

The iterative planning methodologies rely on bi-level programming models as shown in Fig. 4 whose upper level model represents the amount of the investment in generation, transmission and operating cost of new generating units based on new topologies. Also, the lower level problems represent the total fuel emission of generating units and emissions generated in the AC/DC transmission systems. New transmission lines and generating units can accommodate a cheaper generation dispatch, thereby alleviating congestion and lowering emission. In the mentioned levels, the feasibility and optimality check are considered. The feasibility check is related to the transmission network constraints and the other one finds the optimal system economic operation in the proposed plan. To handle the constraints, proper cuts are generated and added to the next iteration. The constraints handling procedure will continue until an optimal expansion planning solution is achieved.

7 Implementation of proposed FACBPSO on G-TEP problem

To apply the FACBPSO algorithm to the G-TEP problem, the following steps should be taken and repeated:

Step 1: Define the input data.

Step 2: Generate the initial binary sequences based on candidate list.

The initial binary sequences for each particle, which must meet constraints, are randomly generated based on (23).

Step 3: Probabilistic multi-objective evaluation.

Step 3-0: Set i = 1.

Step 3-1: Select the ith individual.

The values of the objective functions are evaluated for the ith individual using Equations (6) and (7).

Step 3-2: Run 2m-PEM algorithm.

To calculate the mean and standard deviation of the objective function based on the uncertainty of the load demand, the generated power of the WT and PV units, the 2m-PEM algorithm is used.

Step 3-3: Calculate the membership function.

Step 3-4: Solve the mini-max problem.

Step 4: Update the FACBPSO parameters.

Step 5: Update velocity and positive vectors for the next iteration according to Equation (20).

Step 6: Time horizon.

For a time horizon, in each year, the incremental rate of the load and generation capacity is considered.

Step 7: Optimal plan presentation.

8 Simulation results and discussion

In order to study the proposed algorithm in the G-TEP problem, IEEE-24 bus is considered. The single line diagram of the IEEE-24 bus is shown in Fig. 5. The base case of the test system consists of 24 buses, with 11 generators. The technical data of this test system have been presented in [42]. The total generation and load data are changed to 3900 MW and 3850 MW, respectively. The required data such as planning horizon year, discount rate, curve fitting parameters, the Weibull and beta parameters of the wind speed and solar radiation are tabulated in Table 3. The forecasted load in each load block is given in Table 4. The total planning horizon is considered to be 10 years divided into 30 periods. The list of candidate DC lines as two-terminal DC (TTDC) and multi-terminal DC (MTDC) lines are presented in Table 5. It is important to note that up to three AC transmission lines could be added per right of way. There are 8 generating units that they include nonrenewable sources such as fossil (steam and gas power plants) and renewable energy sources such as wind/solar farms that their data are presented in Table 6. It is noted that a set of wind turbines and set of solar arrays are considered as wind and solar farms with total generation of 200 MW. Also, it is assumed that wind/solar farms are located far from the network. The electrical energy produced by wind/solar farms, are injected to the nearest buses of the network via the integrated AC/DC system using candidate transmission lines. According to Fig. 5 it is assumed that the nearest buses to wind and solar farms are buses 3 and 9, respectively.

The optimum solution of the G-TEP problem using the proposed method, is tabulated in Tables 7 and 8. In years 1 and 2, the load cannot be served based on the existing generation and transmission network capacity (due to modification of total generation and load). Therefore, the system would need to install new generating units and lines to supply the power network. According to results, installation of HVDC link especially MTDC link would eliminate the installation of AC candidate transmission lines.

In Table 7, due to the congestion of some lines, generation units should increase its capacity for GEP section. Therefore, the generating units have been installed. In addition, it can be said that the wind/solar farms need no additional fuel and are a suitable choice for emission reduction.

The choices of the test system are only intended as 1) a way to identify a regulatory environment concerning RES support and emission reduction and 2) HVDC links

Therefore, to validate and compare the performance of the proposed method, the following three scenarios are studied:

Scenario 1: the G-TEP problem without considering HVDC links and wind/solar farms.

Scenario 2: the G-TEP problem without any wind/solar farms, but HVDC links are considered as candidate lines.

Scenario 3: the G-TEP problem without using any HVDC links, however the wind/solar farms are considered.

The obtained results using the proposed algorithm for above mentioned scenarios, are listed in Table 9.It should be noted that the initial reference membership values (η_r,1, η_r,2) are set to 1.

8.1 Comparison of proposed FACBPSO to other binary PSO algorithms

Since the proposed FACBPSO includes the binary PSO, chaotic local search and fuzzy adaptive parameters control, it is necessary to find the relative strength of each constituent. So, the following strategies are considered:

S1: The proposed FACBPSO.

S2: BPSO without considering fuzzy system. This strategy is selected to show the influence of the fuzzy system on parameter control in original BPSO. In this strategy, constant values are considered for the BPSO parameters.

S3: FABPSO without considering chaotic local search (CLS).

In order to compare the mentioned strategies, Table 10 is provided, which shows that by making BPSO parameters fuzzy rule base, the global search ability is improved and the probability of the premature convergence to local minima is decreased. In addition, by comparing the results obtained from strategies, it can be inferred that the CLS used in the proposed algorithm is a powerful solution to generate binary sequences to escape from local minima. Therefore, the proposed algorithm not only improves the global search ability but also prevents the local optimum. It is noted that because of the stochastic nature of Evolution Algorithms (EAs), a series of 20 independent runs have been carried out. In all of the strategies, the initial reference membership values are set to 1.

9 Conclusion

A fuzzy adaptive chaotic binary PSO (FACBPSO) algorithm has been presented in this paper, to solve the G-TEP problem by considering the effect of the wind/solar farms and HVDC links. To improve the performance of the PSO algorithm, the fuzzy adaptive PSO has been proposed to design a fuzzy system, which dynamically adapts the inertia weight and learning factors. Also, it should be noted that to escape from local minima, the application of the fuzzy system is not enough. To overcome this problem, the chaotic local search has been used as a powerful strategy to diversify the population in preventing the premature convergence.

The investment/operation cost is one of the objective functions in the G-TEP problem. Also, considering the features of the HVDC links and renewable energy resources in reduction of emission, the emission function has been also added as the other objective function. For stochastic modeling of the uncertainty and the integration of HVDC links in the multi-objective G-TEP problem, an efficient fuzzy set based optimization technique has been used. Finally, it is evident from the simulation results that the proposed G-TEP method is able to find the feasible and most economical/environmental plan.

10 Nomenclature

The list of symbols has been presented in Table 11.

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