A multi-objective LMP pricing strategy in distribution networks based on MOGA algorithm

Abstract

This paper presents a new electricity pricing methodology in distribution networks by imploying Distributed Generations (DGs). As long as the Locational Marginal Price (LMP) is used in the pricing of short-term operations as an efficient method, it can be performed in distribution network consequently. The proposed pricing method is modeled as an optimization problem with the specific control variables and objectives. The variables are LMPs and DGs power factors, and objectives are total losses and emission. Also, profit earned from reduction of loss and emission was allocated between DGs. Reduction of loss and emission was compensated by DGs production. As a result, more production resulted to high price of DG buses rather than market price. This price should be provided by Distribution Company (DISCO), and DISCO earns this money form the profit. Because of the multi-objective nature of the problem, a Multi-Objective Genetic Algorithm Optimization (MOGA) is implemented to solve. In order to validate the proposed method, a comparison between MOGA and Multi-Objective Particle Swarm Optimization (MOPSO) is performed consequently. The proposed method allows the decision-makers to apply their preferences among losses/emission reduction and DISCO’s benefit. furthermore, the feasibility of the proposed method is investigated using the IEEE-32 bus test system.

Keywords

Electricity price multi-objective optimization locational marginal price loss reduction emission reduction

1. Introduction

Recently, electrical distribution systems become large in size and complex in configuration leading to higher system losses. Thus, more investments are needed inevitably. Studies indicate that almost 10-13% of the total generated power is lost as line losses at the distribution level [1]. These losses lead the distribution feeder to increase the cost of energy and make the voltage profile undesirable. Fossil fuels which are commonly burned in different ways to generate electricity, produce greenhouse gases (GHGs) especially CO₂. The result of this problem is environment pollution [2].

Environmental pollution and losses of distribution systems result to utilization of DGs in an effective way. In this view, the use of DGs can offer effective alternatives to provide a good solution to these problems. DG’s can employ to mitigate the loss of networks by changing transmission line flows [3]. This procedure can improve the reliability of system considering reducing losses [4]. Comparing to conventional thermal power, DGs use natural gas and other clean energies that can be more efficient, and also they reduce emission of greenhouse gases and dust. As a result, they have less detrimental effects on the surrounding environment. Therefore, DGs are considered environmentally preferable to the conventional source.

Although DGs could be beneficial in system management, there are some problems in their operation such as electricity pricing at DG connected buses [5]. In recent years, a lot of techniques have been developed for solving the electricity pricing problems at DG connected busses. The locational marginal pricing is discussed in [6].

LMP method is a well-known and popular strategy in electricity market transactions [7]. LMP can reflect the impact of power flows and contingency problem in specific lines. This approach provides individual nodal pricing, whereas the zonal method does not involve the monitoring of individual lines, assuming that all prices are the same in each zone. The LMP problem was solved based on demand response [8]. In [9], a method based on game theory was considered for LMP calculation in distribution networks. The energy balance, losses, and the network congestion and technical limits (bus voltage limits) were considered to determine the LMP value in each bus [10]. Fuzzy Q-learning was proposed for hour-head electricity marketing with renewable energy [11]. In [12] a new method was presented for optimal DG placement with the nodal pricing, by considering profit, losses reduction and the voltage improvement in distribution systems. In [13] LMP calculation was performed with distributed slack power flow formulation. In [14] the LMP calculation was obtained by a linear programming formulation with considering losses based on the information of network. All these methods suggested important features, but they have not included the following important features:

Considering the other technical aspects of network such as emission in the LMP formulation.

Emission reduction is considered as a technical issue. Therefore, conventional emission reduction techniques are often carried out with technical methods such as installing filters, switching from conventional to Combined Cycle, or replacing existing generation resources [15]. These methods usually involve the cost, and the replacement of equipment. However, in the proposed method, by employing an economic approach, this technical issue has been remedied.

On the other hand, because reducing emission results in economic profit directly, considering the pollution as a parameter in calculating the price (LMP) can lead the DGs owners to apply maximum changes in order to get more profit.

Encourage the DG unit’s owner to participate in the technical programs of network such as losses/emission reduction.

Efficient reduction in the losses/emission of the network.

Applying mandatory policies on DGs will reduce their participation in meeting the networks’ objectives in the long term operation [16]. In this regard, if a method can optionally create the opportunity for DGs owners to gain more profit in proportion to their participation rate, it will achieve the acceptable improvement in the network. The proposed method of this paper, by applying a voluntary approach, provides the maximum participation of the DGs. The ability of decision maker to deal with the system priority among the different objectives such as losses/emission reduction, DG unit’s profit, etc.

In spite of predetermined scheduling and predictions, there are sometimes problems occurring in the network that can lead to transmission congestion and loads being not supplied. In such a circumstance, the priority of the decision making must be to reduce the loss in comparison with the emission reduction.

On the other hand, in case of intensified emission and its adverse effects, this priority can be made to reduce further emission.

The proposed method provides this capability to the operators. In this paper, the LMP problem in the distribution networks with consideration of the above mentioned features is solved using a Multi-Objective optimization algorithm called MOGA. In this algorithm, control variables are LMPs at DG connected busses, and power factor of DGs. Additionally, objectives of the problem are losses and emission of network, which should be minimized [17].

The conventional methods used to improve distribution network performance often involve costing [18 –20], Sophisticated methods [21], adding/removing equipment [22, 23], or changing network configuration [24]. However, the proposed method, fulfills the network objectives by providing an economic-technical framework, only through the use of voluntary incentives resulting from the losses/emission reduction, without any additional cost, adding equipment, or making changes to the network configuration, satisfying the operators’ intended objectives.

In order to make the method more complete and practical, for future work, the following suggestions can be mentioned:

Considering renewable resources with non-quadratic cost function

Consider uncertainty in loads and prices

As the LMP calculation problem is a mixed integer nonlinear programming problem which cannot be solved by conventional methods properly, in this paper, an efficient evolutionary algorithm namely MOGA was used [25]. In addition, the considered problem refers to the simultaneous optimization of two conflicting objectives, i.e. losses and emission. Therefore, a Multi-Objective algorithm was used to obtain non-dominated (Pareto) solutions during the search process and store them in a repository. Since the objective functions are not the same, the size of the repository was controlled using a fuzzy clustering technique [26].

The resume of this paper is divided into sections: Section 2 presents the losses and emission calculation. Reduction of losses and emission for Multi-Objective problem and the proposed algorithm is presented in Section 3. In section 4, simulation results are provided to confirm the effectiveness of the proposed method. Finally, Section 5 presents the conclusion of the paper.

2. Formulation of loss and emission

In the first part of this section, the formulation of locational marginal pricing procedure is presented. Then, the formulation of emission and loss indices and the applied constraints in the problem are described.

2.1. Objective functions for pricing method

Due to the large number of feeders, connection and loads in the distribution network, one of the most important goals in the operation of this network is the loss reduction. On the other hand, the most environmental challenge about power generation is reducing emission from power plants due to undesirable environmental effects [27]. In this paper, LMPs are calculated at the DG buses based on the impact of corresponding DGs on the losses and emission reduction. Consequently, loss and emission are calculated as follows:

- Active Loss: $P_{loss} (X) = \sum_{br = 1}^{N_{br}} R_{br} | I_{br} |^{2}$ (1) where, P_loss is active power loss, N_br is the total number of branches, R_br and I_br are the resistance and current of brth branch consequently, and X is the control variable set at LMP calculation problem in DG busses. This variable is defined as follow: $\begin{matrix} X = [π_{1}, π_{2}, \dots π_{i}, \dots, π_{N_{DG}} \\ , {pf}_{1}, {pf}_{2}, \dots {pf}_{i}, \dots, {pf}_{N_{DG}}] \end{matrix}$ (2) where π_i is LMP at ith DG bus, pf_i is power factor of ith DG, and N_DG is the total number of DGs.

- Emission:

In the distribution network operation problem, the emission produced by conventional thermal power plants and DGs is calculated as follows: ${\begin{matrix} E_{DG, i} (X) = ({NOx}_{DG, i} + SO 2_{DG, i} + \\ CO 2_{DG, i} + {CO}_{DG, i}) \times P_{DG, i} \\ E_{grid} = ({NOx}_{grid} + SO 2_{grid} + CO 2_{grid} \\ + {CO}_{grid}) \times P_{grid} \\ E = \sum_{i = 1}^{N_{DG}} E_{DG, i} + E_{grid} \end{matrix}$ (3)

where E_DG,i is emission produced by ith DG and E_grid is the emission produced by the substation bus connected to the network. These values are obtained by emission coefficients at DG units which are presented in Table 1 [28].

Table 1

Emission coefficients of DG units and substation bus

Type	CO2	SO2	NOx	CO
Combined cycle gas turbines (CCGT)	695	1.25	2.13	2.8
Gas internal combustion engines (G-ICE)	477	0.024	0.015	0
Diesel internal combustion engines (D-ICE)	625	0.032	0.29	0.42
Substation bus	965	5.64	1.7	0.07

2.1.1 Objective functions for pricing method

Limitations and constraints of this LMP pricing problem are listed as follows:

- Distribution power flow equations: $\begin{matrix} P_{i} = \sum_{i = 1}^{N_{bus}} V_{i} V_{j} Y_{ij} cos (θ_{ij} - δ_{i} + δ_{j}) \\ Q_{i} = \sum_{i = 1}^{N_{bus}} V_{i} V_{j} Y_{ij} \sin (θ_{ij} - δ_{i} + δ_{j}) \end{matrix}$ (4)

P_i and Q_i are the net injected active and reactive power at ith bus, respectively. V_i and δ_i are the amplitude and angle of bus voltage, respectively. Y_ij and θ_ij are the amplitude and angle of admittance between ith and jth buses, respectively.

- Constraints of active and reactive power: $P_{min, DG, i} ⩽ P_{DG, i} ⩽ P_{max, DG, i}$ (5)

- P_min,DG,i, P_max,DG,i are the minimum and maximum active power of ith DG. $Q_{min, DG, i} ⩽ Q_{DG, i} ⩽ Q_{max, DG, i}$ (6) where Q_min,DG,i, Q_max,DG,i are the minimum and maximum reactive power of ithDG.

- Constraint of power factor: ${Pf}_{min, i} ⩽ {Pf}_{i} ⩽ {Pf}_{max, i}$ (7) where Pf_min,i, Pf_max,i are the minimum and maximum power factor of the ith DG.

- Constraint on Merchandising Surplus (MS):

As the losses and emission produced by DG units are minimized, the benefits of distribution company (DISCO) is increased consequently. This increased benefit is formulated as follows [28]: $MS = [\begin{matrix} λ ({loss}^{base} - loss) + \\ γ_{t} ω_{e} ({emission}^{base} - emission) - \\ \sum_{i = 1}^{N_{DG}} (π^{r})_{i} (Q_{DG, i}) - \sum_{i = 1}^{N_{DG}} ((π^{a})_{i} - λ) (P_{DG, i}) \end{matrix}]$ (8) where, λ is market price, loss^base and emission^base are loss and emission of network without DG units, respectively. γ_t is emission allowance price. ω_e determines the share of each DG units from mitigation of the emission allowance of the DISCO. π^a and π^r are active and reactive power price at DG busses.

In general, MS is greater than zero, and it must be minimized in a fair competition. In this regard, the MS constraint are considered as MS ⩽ ɛ, which ɛ is maximum deviation of the MS.

3. Multi-objective genetic algorithm (MOGA)

In this section, optimization algorithm has been introduced structurally. Then, the modeling of multi-objective optimization problem has been studied as a multi-objective approach. Furthermore, the best compromise solution for the aforementioned problem has been investigated. At the end of this section application of proposed algorithm for problem with interconnected bus has been studied.

In this paper, the decision maker is assumed to be honest since the pricing process in electrical distribution networks is carried out under the supervision of an independent entity called ISO (Independent System Operator) or IDSO (Independent electric Distribution System Operator) [29]. Therefore, none of the participants can behave in a dishonest way [30].

However, in the general process of decision making, several methods have been proposed by the authors. For example, the authors in [31] develop a mixed integer linear programming aimed to obtain the optimum solution to the proposed consistency improving model in certain real decision-making situations.

The personalized individual semantics (PIS) model is proposed to personalize individual semantics by means of an interval numerical scale model for supporting linguistic group decision making [32]. Also the similar method is applied to solve linguistic GDM problems with a consensus reaching process in [33]. Also, the decision makers may express their opinions dishonestly to obtain their own interests, which is referred to as strategic manipulation or non-cooperative behavior [33, 34]. This procedure is described in the following.

As a practical example in the distribution network, let X = {x₁, x₂, x₃} be the set of alternatives that each one represents one of the distribution network operation states. x₁ reflects the state of the network that the DGs do not contribute to improvement of the emission conditions and are exclusively used to reducing network losses. x₂ states the condition of the network that the DGs do not contribute to the loss and exclusively reduces emission. Finally, x₃ represents the normal state of the network, with DGs contributing both to reducing emission and reducing loss.

Also, let A = {a₁, a₂} the set of predefined attributes that a₁ represent network loss and a₂ represent the network emission, and w = {w₁, w₂} the associated weight vector of the attributes, such that w₁, w₂ ≥ 0 and w₁ + w₂ = 1.

In the event of an intensification of emission, corresponding to the reduction of emission, the amount of the w₂ will be greater than the w₁. On the other hand, In the event of a contingency on the network and the need to release the network capacity, the w₂ will be greater than the w₁, proportionally.

Let be the decision matrix given by the V = [v_nm] _3×2 decision maker that is shown in Table 2, where v_nm denotes the preference value for the alternative x_i ∈ X with respect to the attribute a₁ and a₂, representing how well alternative x_n verifies attribute a_m ∈ A. It should be noted that all predefined attributes are quantitative and measurable.

Table 2
Decision matrix of the network

Alternative\Attribute a ₁ a ₂

x₁ v ₁ v ₁₂

x₂ v ₂₁ v ₂₂

x₃ v ₃₁ v ₂₃

Alternative\Attribute	a ₁	a ₂
x₁	v ₁	v ₁₂
x₂	v ₂₁	v ₂₂
x₃	v ₃₁	v ₂₃

In order to normalize the decision making matrix into a corresponding standardized individual’s decision matrix $\bar{V} = {[v_{nm}]}_{3 \times 2}$ , since a_m ∈ A is a cost attribute, $\bar{v_{nm}}$ should be considered as follows: $\bar{v_{nm}} = \frac{max_{n} (v_{nm}) - v_{nm}}{max_{n} (v_{nm}) - min_{n} (v_{nm})}$ (9)

If weighted average (WA) operator is chosen as aggregation operator F with an associated vector w = {w₁, w₂}, the decision evaluation value of the alternative x_n is shown as follows: $D (x_{n}) = {WA}_{w} (\bar{v_{n 1}}, \bar{v_{n 2}}, \dots, \bar{v_{nm}}) = \sum_{m = 1}^{2} w_{m} \bar{v_{n 1}}$ (10) where $\bar{v_{n 1}}$ is the mth largest value in $(\bar{v_{n 1}}, \bar{v_{n 2}})$ .

Due to ranking of alternatives, if Q_k = {x_n|D (x_n) > D (x_k) , n = 1, 2, 3} is considered as the set of the alternatives whose decision evaluation value is greater than that of the alternative x_k, and |Q_k| be its cardinality, the alternative x_k such that D (x_k) = max {D (x₁) , D (x₂) , D (x₃)} might verify as well that |Q_k| = 0, while alternative x_m such that D (x_j) = min {D (x₁) , D (x₂) , D (x₃)} might as well have |Q_k| = n - 1, and therefore this alternative will be ranked in 1st and 2nd positions. Thus, the ranking position of an alternative is shown in the following [34, 35]. $r (x_{k}) = | {x_{n} | D (x_{n}) > D (x_{k}), n = 1, 2, 3} | + 1$ (11)

3.1. Multi-objective approach

A multi-objective decision problem is defined as follows: Consider an N-dimensional decision variable vector q ={ q₁, …, q_N } in the solution space Q, the problem is to find a vector q^* that minimizes a set of K objective functions r (q*) = { r₁ (q *) , …, r_K (q *) }. A series of constraints such as W_j (q *) = b_j for j = 1, …, m which limit the decision variables are confined the solution space (Q). The perfect multi-objective solution which simultaneously optimizes each objective function is almost impossible [36]. Then, the main goal of the multi-objective optimization problems is to obtain a set of Pareto optimal solution. Since Pareto optimal set is assemble of the solutions, genetic algorithm which is one of multi-point search methods is suitable to derive the Pareto optimal set [37]. Most of the other multi-objective algorithms are dependent on the user to set their parameters simultaneously. On the other hand, the multi-objective GA does not force the user to prioritize, scale or weigh objectives. Therefore, GA has been nominated as the most popular heuristic approach to the multi-objective design and optimization problems [22]. Multi-objective optimization problem is formulated as follows [38]: $min \vec{f} (\vec{q}) = \vec{h} (f_{1} (\vec{q}), f_{2} (\vec{q}), \dots, f_{k} (\vec{q}))^{T}$ (12)

Subjected to the following constraints: $g_{k} (\vec{q}) ⩽ 0 (j = 1, \dots, m)$ (13)

By solving the proposed multi-objectives problem, the pareto optimal solutions are obtained. Consequently, the best comprised solution (BSC) should be selected for proposed microgrid scheduling problem. In this regard, the max-min fuzzy satisfying criterion is utilized to select the BCS. In this method, firstly, the Fuzzy Membership Function (FMFs) are calculated [38, 39]. This procedure can be mathematically expressed as follows: $\begin{matrix} φ_{q, z} = {\begin{matrix} 1 & f_{q, z} < f_{q}^{min} \\ \frac{f_{q}^{max} - f_{q, z}}{f_{q}^{max} - f_{q}^{min}} & f_{q}^{min} < f_{q, z} < f_{q}^{max} \\ 0 & f_{q, z} > f_{q}^{max} \end{matrix}}, \dots \\ \dots, \forall q \in Φ, z \in S \end{matrix}$ (14) where z refers to the zth solution of the qth objective function. $f_{q}^{max}$ is the membership objective function with the maximum value of f_q calculated for the minimum value of f_q′, which is in the conflict with f_q, i.e., q′ ∈ Φ − q. This means that the maximum value of f_q is obtained when the f_q′ is optimally minimized. Generally speaking, the values of $f_{q}^{min}$ and $f_{q}^{max}$ are obtained for the optimization of qth objective function in a single objective problem, separately. Therefore, the maximum value of overall satisfactions is obtained as the BCS as follows: $Θ = max_{\forall z \in S} {min_{\forall q \in Φ} {φ_{q, z}}}$ (15)

3.2. Modeling multi-objective problem

In a Multi-Objective optimization problem, the concept of optimality is replaced by non-dominated solution (Pareto optimality) [26]. A general Multi-Objective optimization problem can be formulated as follows: $\begin{matrix} Minimize \\ F = [f_{1} (X), f_{2} (X), \dots, f_{N_{obj}} (X)] \end{matrix}$ (16) where, X is a feasible solution, f_i (X) is ith objective function, and N_obj is number of objective functions.

For any two solutions X₁ and X₂, the solution X₁ dominates X₂ if the following two conditions are satisfied: ${\begin{matrix} \forall m \in {1, 2, \dots, N_{obj}}, f_{m} (X_{1}) ⩽ f_{m} (X_{2}) \\ \exists n \in {1, 2, \dots, N_{obj}}, f_{n} (X_{1}) < f_{n} (X_{2}) \end{matrix}$ (17)

If any of the conditions are unacceptable, X₁ cannot dominate X₂ otherwise X₁ dominates X₂, and named non-dominated solution.

3.3. Best compromise solution

According to the proposed model for solving Multi-Objective problem, the algorithm’s output is a set of non-dominated solutions, and the decision maker wants to select one solution as the best compromise solution among the obtained non-dominated solutions. To reach this aim, the following section should be considered:

Section 1: Each objective function should be converted to a membership function by (18): $μ_{fi} (X) = {\begin{matrix} 1 & for : f_{i} (x) ⩽ f_{i}^{min} \\ 0 & for : f_{i} (x) \geq f_{i}^{max} \\ \frac{f_{i}^{max} - f_{i} (x)}{f_{i}^{max} - f_{i}^{min}} & for : f_{i}^{min} ⩽ f_{i} (x) ⩽ f_{i}^{max} \end{matrix}$ (18) where, $f_{i}^{min}$ and $f_{i}^{max}$ are the lower and upper bounds of each objective subject to the given constraints.

Section 2: The normalized membership function of kth non-dominated solution should be calculated as: $μ^{k} = \frac{\sum_{i = 1}^{N_{obj}} μ_{fi} (X_{k})}{\sum_{j = 1}^{M} \sum_{i = 1}^{N_{obj}} μ_{fi} (X_{j})}$ (19) where, M is the number of non-dominated solutions.

Section 3: The solution which has maximum μ^k is selected as the best compromise solution.

3.4. Application of the MOGA in multi-objective LMP calculation in DG connected bus

To apply the intended algorithm to Multi-Objective LMP calculation the following steps should be performed:

Step 1. Define the input data.

Step 2. Randomly generate the first population P₀ of size N.

At the first, all buses have identical price equal to market price.

Step 3. Then, calculation of DG generations and cost of each DG are performed as follows respectively by (20) and (21): $P_{DG, i} = \frac{π_{i} - b_{i}}{2 a_{i}}$ (20) ${Cost}_{DG, i} = a_{i} P_{DG, i}^{2} + b_{i} P_{DG, i} + c_{i}$ (21) where, Cost_DG,i is the cost of ith DG. a_DG,i and b_DG,i are cost coefficients of ith DG. After calculating $P_{DG, i}^{t}$ , these terms should be checked by (steps: 4, 5 and 6).

Step 4. Evaluate the values of the objective functions individually. To estimate these values, first the load flow should be applied and then should be calculated by (1) and (2).

Step 5. Check the constraints (5–7). If the constraints aren’t satisfied, a big value must be considered for the objective functions.

Step 6. Extract non-dominated solutions of initial population using (16) and (17).

Step 7. Calculate the normalized membership function of non-dominated solutions using (18) and (19).

Step 8. The MOGA algorithm is described based on the outline is summarized as follows [22]:

Repair the first population to make it feasible

Fitness evaluation of P₀.

Apply the sorting process for each solution based on the fitness evaluation

i_t = i_t + 1

Crossover: generate the set R_i of offspring of size N

If p > p_m, then activate the Mutation process

Apply the sorting process for each solution in R_i

Fitness evaluation for P_i

Update the non-dominated list Q_i

Evaluate and select best N solution from P_i

While stopping criteria

Step 9. Update the variables of step 8.

Step 10. Evaluate the values of the objective functions for updated variables.

Step 11. Check the constraints (5–7). If the constraints aren’t satisfied, a big value must be considered for objective functions.

Step 12. Extract non-dominated solutions of initial population using (18) and (19).

Step 13. Calculate the normalized membership function of non-dominated solutions using (20) and (21).

Step 14. Check the termination criteria. If it has been satisfied, then stop algorithm; else go to the step 8.

Also, the flowchart of the proposed algorithm is shown in Fig. 1.

Fig. 1

Flowchart of the proposed MOGA algorithm for LMP calculation problem.

4. Simulation results

In this section, the Multi-Objective LMP calculation in distribution networks is tested on a 32-bus distribution test system with four DGs. The proposed test system is a hypothetical 12.66 kV system including a substation, two feeders, and 32 buses. The system data is given in [28] and the single line diagram of this system is shown in Fig. 2. The initial loss of system is 202.47 kW and the initial emission is 3617.365 kg. The DG units are placed in 5,11,25,30 buses. Emission coefficients for these DGs are shown in Table 1. The of DG units in the term of coefficients of equation a, b, c are presented in Table 3.

Fig. 2

Single line diagram of 32-bus distribution test system.

Table 3

Characteristics of DGs

No. DG	Type	Capacity of DG (kW)	a($/MW²)	b($/MW)	c($)
DG1	Combined cycle gas turbines (CCGT)	1000	5.8 × 10^-6	0.021	0
DG4
DG2	Gas internal combustion engines (G-ICE)	1000	5.3 × 10^-6	0.020	0
DG3	Diesel internal combustion engines (D-ICE)	1000	5 × 10^-6	0.020	0

First, to evaluate the superiority of the MOGA algorithm rather than MOPSO algorithm, the loss and emission were optimized by these two mentioned algorithms separately. In this comparison the market price is equal to 28 ($/MW). Table 4 presents the results of both algorithms in the case that loss is considered as the objective function. Also, in Table 5, the results of algorithms are presented considering that emission is the objective function. As shown in Tables 4 and 5, the results of MOGA algorithm are much reliable than the MOPSO algorithm. This superiority may result to more reduction of loss and emission.

Table 4

Results of optimizing the loss for MOPSO and MOGA Algorithms in 500 trials

Market price	Algorithm	Average	Worst solution	Best Solution	Standard deviation
($/MW)		(kW)	(kW)	(kW)	(kW)
28	MOPSO	26.058	32.313	22.643	3.833
	MOGA	24.223	29.105	22.643	2.108

Table 5

Results of optimizing the loss for MOPSO and MOGA Algorithms in 500 trials

Market price)	Algorithm	Average	Worst solution	Best Solution	Standard deviation
($/MW)		(kg)	(kg)	(kg)	(kg)
28	MOPSO	2024.55	2198.37	1917.00	51.39
	MOGA	1996.41	2189.54	1917.00	24.52

The results of the proposed MOGA algorithm in confronting with LMP calculation are presented in Tables 6–8. In these tables, the proposed method is compared with other two conventional LMP calculation methods namely uniform price method and marginal losses method [6]. In Tables 6 and 7, DG units’ nodal price and their generation for various market prices are presented, and the results of these three methods is compared. As the market price increases, the generation of DGs and nodal price is increases using all three methods consequently. In fact, the generation of DGs is adjustable with new nodal price, but generation and price of DGs for the proposed method are more than marginal losses and uniform price methods. More active power of DGs and high price of DG connected buses emphasize on the efficiency of this method in comparison with other methods. Due to the fact that more generation of DGs causes lower loss and emission, we can approach to our aim i.e. decreasing mentioned objectives. For starting the of DGs, coefficient ‘b’ in (15) must be lower than market price. As shown in Table 6, for market price equal to 20($/MW) all DGs are turned off and have not any generation. When the DG are turned off and does not generate active power, the price of DG buses are equal to market price equal to 20 ($/MW). The results of this problem are shown in the first row of Table 7.

Table 6

DG units’ nodal price for various market prices

Market	DG1 Nodal Price ($)			DG2 Nodal Price ($)			DG3 Nodal Price ($)			DG4 Nodal Price ($)
price	New	M.L	Uniform	New	M.L	Uniform	New	M.L	Uniform	New	M.L	Uniform
($)	Method	Method	price	Method	Method	price	Method	Method	price	Method	Method	price
20	20	20	20	20	20	20	20	20	20	20	20	20
22	22.364	22.011	22	25.845	22	22	22.398	22.29	22	23.532	22.81	22
24	26.155	25.13	24	27.155	24	24	25.346	24.20	24	27.545	24.32	24
26	29.182	26.700	26	30.174	26.750	26	29.028	26.36	26	27.902	26.08	26
28	29.743	28.57	28	31.035	28.56	28	30.663	28.25	28	31.148	28.03	28
30	30.161	30.026	30	31.969	30.73	30	32.507	30.10	30	32.700	30.10	30

Table 7

DG Units’ Generation for Various Market Prices (Deterministic Framework)

Market	DG1 Generation (kW)			DG2 Generation (kW)			DG3 Generation (kW)			DG4 Generation (kW)
price	New	M.L	Uniform	New	M.L	Uniform	New	M.L	Uniform	New	M.L	Uniform
($)	Method	Method	price	Method	Method	price	Method	Method	price	Method	Method	price
20	0	0	0	0	0	0	0	0	0	0	0	0
22	102.321	93.586	86.206	469.502	191.879	188.679	254.284	201.570	200	280.102	90.38	86.2069
24	472.209	263.751	258.620	676.746	381.563	377.358	523.878	403.950	400	618.682	260.400	258.620
26	588.595	435.125	431.034	929.327	569.078	566.037	698.319	604	600	740.907	435.785	431.034
28	753.740	607.450	603.448	1000	757.726	754.717	1000	802	800	874.881	604.568	603.448
30	833.429	779.874	775.862	1000	946.396	943.396	1000	1000	1000	1000	776.862	775.862

Table 8

Loss and emission of network for Various Market prices

Market	New method		M.L method		Uniform price
price($)	Loss(kW)	Emission (kg)	Loss(kW)	Emission (kg)	Loss(kW)	Emission (kg)
20	202.47	3617.365	202.47	3617.365	202.47	3617.365
22	108.184	2920.098	177.709	3164.097	178.572	3167.970
24	45.238	2527.618	110.890	2884.38	111.484	2888.260
26	33.535	2160.474	66.399	2604.676	66.750	2608.549
28	29.379	1995.075	41.920	2324.965	42.051	2328.839
30	29.3437	1973.575	35.509	2045.254	35.438	2049.128

Furthermore, as shown in Table 6, as the market price increases, the DG’s power also increases. For example, in the case that the market price is equal to 28 ($/MW), DG1 generates 753.74 kW, and in the other case that the market price is equal to 26 ($/MW), it generates 588.595 kW. When the market price is equal to 30 ($/MW), DG2, DG3 and DG4 generate 1000 kW active power. It’s due to the constraint of DG’s capacity, they can produce up to 1000 kW, but by (20) the generation of DGs calculated more than 1000 kW, and DGs are forced to generate maximum capacity of active power. If the capacity of DG increased, in the intended market price, more power can be reached.

Table 8 shows the optimized value of loss and emission for various market prices. As shown in this table, as the market price is increases, the loss and emission in all pricing methods increases. So, the proposed method makes more reduction in comparison of marginal losses and uniform price methods. Decreasing of these objectives causes extra benefit for DISCO. These benefit is equal to 1 MW power costs 1 $ and 1 kg emission costs 5.94 $. These values are λ₁ and λ₂ in simulation respectively. Based on allowance price of emission and market price, decreasing these objectives are equivalent with identical cost that delivered to DISCO. Also, DISCO allocates the benefit of reduction of loss and emission to DGs in order to generate more active power, and aims to reduce loss and emission of distribution network. DGs need more money to generate extra power and DISCO supplies more money to generate more active power of DGs. As the generation of DGs active power increases, the allocated cost to DGs increases consequently.

In Fig. 3, final solutions of LMP problem by MOGA algorithm are presented. As shown in Fig. 3, behavior of emission index is in contrast with the loss index. This means that increase in loss results to decrease in emission. Non-dominated solutions are extracted from many answers of optimization, and finally no answer dominates other answer, and it is named non-dominated solutions. Figure 3 demonstrates that non-dominated solutions are reasonable. The solution with larger membership function value is chosen the best solution for Multi-Objective optimization problem during the search process of MOGA algorithm.

Fig. 3

Obtained non-dominated solution using MOGA algorithm for various market prices when objectives are power loss and emission.

Also, in Tables 5–7 the results corresponding to the solutions with minimum emission and maximum loss in Fig. 3 are considered. This selection is based on the allowance price of 1 kg emission and market price. The price of 1 kg emission is higher than 1 MW loss, and the reduction of emission causes more benefit rather than reduction of loss and brought more money to DISCO.

5. Conclusion

A Multi-Objective optimization based on MOGA is considered to optimize the losses and emission in presence of DGs for various market prices in this paper. the DG generation is directly related to the price of its bus. As the price increase, the generation of DG would be increased. One of the most important advantages of the Multi-Objective formulation is that it has several non-dominated solutions allowing the decision-makers to select the best solution based on the performance of network. Furthermore, the suggested method which is based on non-dominated solution is much efficient rather than the other multi-objective methods. One of the most important advantages of the Multi-Objective formulation is that it has several non-dominated solutions allowing the decision-makers to select the best solution based on the performance of network. Therefore, in the process of reducing losses/emission, depending on the circumstances in the decision making procedure, the share of each objective in proportion to the allocated benefit, can be reduced or increased.

Also, the simulation results confirmed that the proposed method is a reliable answer to the distribution pricing problem.

References

Fazliana Abdul Kadir ,

Mohamed ,

Shareef ,

Asrul Ibrahim ,

Khatib and

Elmenreich , An improved gravitational search algorithm for optimal placement and sizing of renewable distributed generation units in a distribution system for power quality enhancement, Journal of Renewable and Sustainable Energy 6(3) (2014), 033112.

Doagou-Mojarrad ,

Rastegar and

G.B.

Gharehpetian , Probabilistic interactive fuzzy satisfying generation and transmission expansion planning using fuzzy adaptive chaotic binary PSO algorithm, Journal of Intelligent & Fuzzy Systems 30(3) (2016), 1629–1641.

M.K.

Narayanan ,

J.E.

Abdu and

T.P.I.

Ahamed , Loss minimization using optimal allocation of photovoltaic units in unbalanced radial distribution feeders: A case study, Journal of Renewable and Sustainable Energy 8(5) (2016), (055503.

Rahiminejad ,

Aranizadeh and

Vahidi , Simultaneous distributed generation and capacitor placement and sizing in radial distribution system considering reactive power market, Journal of Renewable and Sustainable Energy 6(4) (2014), (043124.

Sadeghi Mobarakeh ,

Rajabi-Ghahnavieh and

Haghighat , A bi-level approach for optimal contract pricing of independent dispatchable DG units in distribution networks, International Transactions on Electrical Energy Systems 26(8) (2016), 1685–1704.

P.M.

Sotkiewicz and

J.M.

Vignolo , Nodal pricing for distribution networks: Efficient pricing for efficiency enhancing DG, IEEE Transactions on Power Systems 21(2) (2006), 1013–1014.

Orfanogianni and

Gross , A general formulation for LMP evaluation, IEEE Transactions on Power Systems 22(3) (2007), 1163–1173.

Aazami ,

Aflaki and

M.R.

Haghifam , A demand response based solution for LMP management in power markets, International Journal of Electrical Power & Energy Systems 33(5) (2011), 1125–1132.

Azad-Farsani ,

S.M.M.

Agah ,

Askarian-Abyaneh ,

Abedi and

S.H.

Hosseinian , Stochastic LMP (Locational marginal price) calculation method in distribution systems to minimize loss and emission based on Shapley value and two-point estimate method, Energy 107 (2016), 396–408.

10.

De Jonghe ,

B.F.

Hobbs and

Belmans , Optimal generation mix with short-term demand response and wind penetration, IEEE Transactions on Power Systems 27(2) (2012), 830–839.

11.

M.R.

Narimani ,

Azizipanah-Abarghooee ,

Zoghdar-Moghadam-Shahrekohne and

Gholami , A novel approach to multi-objective optimal power flow by a new hybrid optimization algorithm considering generator constraints and multi-fuel type, Energy 49 (2013), 119–136.

12.

Shaloudegi ,

Madinehi ,

S.H.

Hosseinian and

H.A.

Abyaneh , A novel policy for locational marginal price calculation in distribution systems based on loss reduction allocation using game theory, IEEE Transactions on Power Systems 27(2) (2012), 811–820.

13.

M.R.

Salehizadeh and

Soltaniyan , Application of fuzzy Q-learning for electricity market modeling by considering renewable power penetration, Renewable and Sustainable Energy Reviews 56 (2016), 1172–1181.

14.

R.K.

Singh and

S.K.

Goswami , Optimum allocation of distributed generations based on nodal pricing for profit, loss reduction, and voltage improvement including voltage rise issue, International Journal of Electrical Power & Energy Systems 32(6) (2010), 637–644.

15.

J.A.

De Gouw ,

D.D.

Parrish ,

G.J.

Frost and

Trainer , Reduced emissions of CO 2, NOx and SO 2 from U.S. power plants due to the switch from coal to natural gas with combined cycle technology, Earth’s Future 2(2) (2014), 75–82.

16.

D.S.

Kirschen and

Strbac , Fundamentals of power system economics, John Wiley & Sons, 2004.

17.

Hu ,

Cheng ,

Yan and

Li , An iterative LMP calculation method considering loss distributions, IEEE Transactions on Power Systems 25(3) (2010), 1469–1477.

18.

Usman ,

Coppo ,

Bignucolo and

Turri , Losses management strategies in active distribution networks: A review, Electric Power Systems Research 163 (2018), 116–132.

19.

Lorestani and

M.M.

Ardehali , Optimal integration of renewable energy sources for autonomous tri-generation combined cooling, heating and power system based on evolutionary particle swarm optimization algorithm, Energy 145 (2018), 839–855.

20.

Ahmadi ,

M.H.

Nazari and

S.H.

Hosseinian , Optimal resources planning of residential complex energy system in a day-ahead market based on invasive weed optimization algorithm, Engineering, Technology & Applied Science Research 7(5) (2017), 1934–1939.

21.

Khodadadi ,

Hasanpor Divshali ,

M.H.

Nazari and

S.H.

Hosseinian , Small-signal stability improvement of an islanded microgrid with electronically-interfaced distributed energy resources in the presence of parametric uncertainties, Electric Power Systems Research 160 (2018), 151–162.

22.

M.H.

Nazari ,

Khodadadi ,

Lorestani ,

S.H.

Hosseinian and

G.B.

Gharehpetian , Optimal multi-objective D-STATCOM placement using MOGA for THD mitigation and cost minimization, Journal of Intelligent & Fuzzy Systems 35(2) (2018), 2339–2348.

23.

Rahiminejad ,

Faramarzi ,

S.H.

Hosseinian and

Vahidi , An effective approach for optimal placement of non-dispatchable renewable distributed generation, Journal of Renewable and Sustainable Energy 9(1) (2017), (015303.

24.

Sultana ,

M.W.

Mustafa ,

Sultana and

A.R.

Bhatti , Review on reliability improvement and power loss reduction in distribution system via network reconfiguration, Renewable and Sustainable Energy Reviews 66 (2016), 297–310.

25.

Javidtash ,

Jabbari ,

Niknam and

Nafar , A novel mixture of non-dominated sorting genetic algorithm and fuzzy method to multi-objective placement of distributed generations in Microgrids, Journal of Intelligent & Fuzzy Systems 33(4) (2017), 2577–2584.

26.

Niknam ,

Kavousi Fard and

Baziar , Multi-objective stochastic distribution feeder reconfiguration problem considering hydrogen and thermal energy production by fuel cell power plants, Energy 42(1) (2012), 563–573.

27.

Aghaei ,

M.I.

Alizadeh ,

Abdollahi and

Barani , Allocation of demand response resources: Toward an effective contribution to power system voltage stability, IET Generation, Transmission & Distribution 10(16) (2016), 4169–4177.

28.

E.A.

Farsani ,

H.A.

Abyaneh ,

Abedi and

S.H.

Hosseinian , A novel policy for LMP calculation in distribution networks based on loss and emission reduction allocation using nucleolus theory, IEEE Transactions on Power Systems 31(1) (2016), 143–152.

29.

Independent Electric Distribution System Operator • GridPolicy. [Online]. Available: http://gridpolicy.com/content/articles/idso/. [Accessed: 30-Dec-2018].

30.

De Martini ,

Kristov and

Schwartz , Distribution systems in a high distributed energy resources future Planning, Market Design, Operation and Oversight, 2015.

31.

Dong ,

C.-C.

Li and

Herrera , An optimization-based approach to adjusting unbalanced linguistic preference relations to obtain a required consistency level, Information Sciences 292(C) (2015), 27–38.

32.

C.-C.

Li ,

Dong ,

Herrera ,

Herrera-Viedma and

Martínez , Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching, Information Fusion 33 (2017), 29–40.

33.

C.-C.

Li ,

R.M.

Rodríguez ,

Martínez ,

Dong and

Herrera , Personalized individual semantics based on consistency in hesitant linguistic group decision making with comparative linguistic expressions, Knowledge-Based Systems 145 (2018), 156–165.

34.

Dong ,

Liu ,

Liang ,

Chiclana and

Herrera-Viedma , Strategic weight manipulation in multiple attribute decision making, Omega 75 (2018), 154–164.

35.

Liu ,

Dong ,

Liang ,

Chiclana and

Herrera-Viedma , Multiple Attribute Strategic Weight Manipulation With Minimum Cost in a Group Decision Making Context With Interval Attribute Weights Information, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2018, pp. 1–12.

36.

Deb , Multi-objective optimization using evolutionary algorithms, John Wiley & Sons, 2001.

37.

Konak ,

D.W.

Coit and

A.E.

Smith , Multi-objective optimization using genetic algorithms: A tutorial, Reliability Engineering & System Safety 91(9) (2006), 992–1007.

38.

Jones ,

Mirrazavi and

Tamiz , Multi-objective meta-heuristics: An overview of the current state-of-the-art, European Journal of Operational Research 137(1) (2002), 1–9.

39.

Ongsakul ,

J.G.

Singh and

S.R.

Tuladhar , Multi-objective approach for distribution network reconfiguration with optimal DG power factor using NSPSO, IET Generation, Transmission & Distribution 10(12) (2016), 2842–2851.

A multi-objective LMP pricing strategy in distribution networks based on MOGA algorithm

Abstract

Keywords

1. Introduction

2. Formulation of loss and emission

2.1. Objective functions for pricing method

Table 2 Decision matrix of the network Alternative\Attribute a 1 a 2 x1 v 1 v 12 x2 v 21 v 22 x3 v 31 v 23

References

Table 2
Decision matrix of the network

Alternative\Attribute a ₁ a ₂

x₁ v ₁ v ₁₂

x₂ v ₂₁ v ₂₂

x₃ v ₃₁ v ₂₃