Social welfare maximization in deregulated power market incorporating wind power plants using metaheuristic algorithm

Abstract

The increasing integration of renewable energy sources (RESs), particularly wind power plants (WPP), into deregulated power markets introduces complexities in optimizing social welfare (SW). This article proposes a recent metaheuristic algorithm to address this challenge and maximize SW while accounting for the presence of WPP and the inherent uncertainty associated with wind power forecasting. The proposed algorithm optimizes generation scheduling and demand-side bidding strategies in the deregulated power market to maximize SW while ensuring economic efficiency. To validate the effectiveness and robustness of the proposed algorithm, MATLAB simulations are conducted on IEEE 30 and IEEE 118-bus systems. The results demonstrate that the proposed algorithm provides promising solutions for maximizing SW, especially in the context of incorporating WPP. This research contributes to the advancement of power market optimization methods and promotes the seamless integration of RESs, fostering a more sustainable energy future.

Keywords

Wind power plants deregulated power sector renewable energy sources social welfare metaheuristic algorithms

Introduction

In today’s rapidly advancing world, the electricity demand is experiencing a remarkable surge due to technological progress and the quest for an improved quality of life (Sharma et al., 2023). In response to the pressing need to reduce the strain on finite raw materials and minimize the environmental impact associated with conventional energy sources, the utilization of renewable energy sources (RESs) has become increasingly vital (Maheshwari and Sood, 2022). RESs not only address environmental concerns but also cater to the escalating global energy requirements. Among these renewable sources, wind energy sources have gained significant prominence within the power sector, offering a cost-effective means to meet the growing energy demands of consumers (Saini and Rahi, 2023).

However, the integration of RESs also introduces uncertainties that can pose challenges to the security and stability of the power system. Inadequate handling or management of renewable-integrated systems may even lead to grid failures. Moreover, over the years, there has been a significant shift from the traditional regulated model to a deregulated one in the energy sector. This shift has revolutionized the power supply industry (Bouddou et al., 2020), moving away from government monopolies toward a more open and competitive electricity market. The deregulated approach involves separating generation, transmission, and distribution as independent entities, as shown in Figure 1, leading to lower costs, reduced rates, more consumer choices, and improved services. In this system, different market players, including generation companies (GENCOs), transmission companies (TRANSCOs), distribution companies (DISCOs), retailers, and independent system operators (ISOs), operate within the electricity market (Necoechea-Porras et al., 2021).

Figure 1.

Structure of regulated and deregulated power industry.

Optimizing bidding values from generators and consumers in the deregulated power sector becomes crucial since ISO oversees the entire electricity market and sets electricity prices. However, achieving optimal solutions in this complex environment is challenging. To address this, the implementation of metaheuristic algorithms proves beneficial. These algorithms enhance the efficiency and effectiveness of the optimization process, ensuring a fair and efficient distribution of resources among GENCOs and DISCOs.

Literature review

A comprehensive literature review was conducted to explore the integration of RESs into deregulated power systems, with a specific focus on critical objectives such as maximizing social welfare (SW) and minimizing system generation costs. Social welfare is difference between society’s willingness to pay for its energy demand and the actual cost of energy. This concept holds critical importance because it serves as a fundamental metric for evaluating overall societal well-being in electricity generation and distribution. Maximizing SW optimizes resource allocation, enhancing economic efficiency, ensuring affordable electricity, promoting equitable power access, and aligning with regulatory goals. Prioritizing SW within the deregulated power market is essential for achieving equilibrium and efficiency in the energy ecosystem, benefiting all stakeholders, and sustaining societal well-being. Several notable studies in this domain were examined. In Reddy and Bijwe (2016), the authors proposed an optimal methodology for integrating RESs into the day-ahead market to maximize system security and economic benefits. Ramesh et al. (2023) introduced an improved mayfly algorithm (IMA) to minimize the operating costs of thermal generators through optimal power flow (OPF) under varying load conditions within a deregulated environment. In Patil et al. (2022a), the authors introduced the artificial gorilla troops optimizer algorithm (AGTOA) to maximize system profit within deregulated power networks incorporating RESs. The effectiveness of AGTOA was evaluated on IEEE 14 and 30-bus test systems. Basu et al. (2022a) proposed the moth flame optimization (MFO) technique to investigate the impact of wind farms and pumped hydroelectric storage (PHS) systems on system economy, losses, voltage profiles, and operating costs in both regulated and deregulated environments. Sood and Singh (2010) utilized the interior point method (IPM) for OPF studies on the IEEE 30-bus system, aiming to maximize social benefit while considering RESs. Similarly, in Kumari Behera and Kant Mohanty (2021), the authors presented the grey wolf optimization (GWO) algorithm for a similar objective on IEEE 14 and 30-bus systems, with and without a thyristor-controlled series compensator (TCSC). In Basu et al. (2022b), the cuckoo search algorithm (CSA) was proposed to effectively maximize SW through the optimal placement and sizing of TCSC devices. Dawn et al. (2021) introduced the MFO algorithm to address the congested market-clearing power problem, integrating wind and pumped hydroelectric storage systems in a deregulated environment, showcasing improved SW post-integration. Mixed-integer programing (MIP) was introduced in Khaloie et al. (2020) to optimize profit and minimize emissions in RES-integrated power systems within a deregulated power sector, accounting for uncertainties associated with electricity market prices and RESs output through scenario characterization. Patil et al. (2022b) conducted OPF analysis using metaheuristic techniques to maximize SW in wind-integrated power systems within a deregulated market. They also examined the significance of deregulation concerning system generation costs, bus voltage profiles, and locational marginal pricing (LMP). In Vemula et al. (2023), the authors explored the dynamics of regulated and deregulated systems, emphasizing their impact on a power system. Their primary goal was to identify significant changes in system profit brought about by introducing solar power plants under regulated and deregulated conditions. The simulation used MATPOWER, with sequential quadratic programing (SQP) as the selected programing language. In Dawn and Tiwari (2014), authors proposed an optimization approach that aims to maximize SW while minimizing generation and investment costs associated with a unified power flow controller (UPFC) in a power system connected to wind generators considering a competitive power market with double auction mechanisms. Finally, in Das et al. (2022), the primary objective was to improve SW through the implementation of a market-clearing mechanism for a wind farm and compressed air energy storage hybrid system. The authors employed the artificial bee colony (ABC) algorithm and slime mold algorithm (SMA) to determine the optimal market clearing price (MCP) and market clearing volume (MCV) using a modified IEEE-30 bus test system.

Research gaps and contributions

The literature review has identified several critical research gaps that need to be addressed. Firstly, there is a noticeable lack of comprehensive studies that explore the potential benefits of employing metaheuristic algorithms to optimize social welfare (SW) within renewable-integrated power systems. Utilizing these advanced algorithms could lead to more effective SW maximization strategies. Secondly, existing studies often overlook the significance of addressing the inherent uncertainties and variations in renewable energy output. These uncertainties can profoundly impact the effectiveness of SW maximization strategies. Furthermore, there is a need to explore the relationship between integrating RESs and its overall impact on social welfare. Understanding how the increased penetration of RESs affects SW is essential for developing informed policies and strategies. Another critical area that requires further exploration is the changes in pricing mechanisms within the deregulated power market due to the integration of RESs. Investigating these changes can provide valuable insights into optimizing SW under such market conditions. Addressing these research gaps can significantly enhance our understanding of effectively maximizing SW in the deregulated power market while considering the integration of RESs and uncertainties associated with renewable energy generation. Additionally, investigating the promotion of RESs and their impact on pricing methods will provide valuable insights for policymakers, market operators, and stakeholders, aiding in the transition to a more sustainable and efficient electricity market. To bridge these research gaps, this study makes the following contributions:

Exploration of the application of state-of-the-art metaheuristic algorithm, particularly the grey wolf optimization algorithm, for maximizing social welfare in the context of the deregulated power market.

Utilization of probabilistic models to handle variations in RESs power output, resulting in more accurate social welfare maximization results.

Analysis of the direct impact of RESs penetration on social welfare, focusing on cost savings as a vital factor.

Investigation of the implications of incorporating RESs on pricing at different buses of the transmission system within the deregulated power market.

Other parts of the paper are as follows: Section 2 provides the mathematical models used and the optimization techniques employed in the study. In Section 3, a detailed discussion is provided to analyze and interpret the obtained results. The conclusions drawn from the study are presented in Section 4.

Problem formulation and optimization technique

The effectiveness of the proposed approach is evaluated using the following mathematical formulations and optimization techniques.

Wind power generation

The power output of a wind turbine (WT) can be determined using the equation (1) for a given wind speed $v$ (Maheshwari et al., 2023a):

P_{W} (v) = {\begin{matrix} 0 & v \leq v_{cutin} \\ \frac{v - v_{cutin}}{v_{rated} - v_{cutin}} & v_{cutin} \leq v \leq v_{rated} \\ P_{rated} & v_{rated} < v \leq v_{cutout} \\ 0 & v \geq v_{cutout} \end{matrix}

(1)

where, $P_{rated}$ represents the rated power output of the wind turbine. The cutin, rated and cutout wind speed is denoted by $v_{cutin}$ , $v_{rated}$ , and $v_{cutout}$ respectively.

The Weibull distribution (WD) is a valuable statistical tool used to characterize the random nature of wind speed at a specific location which is mathematically defined as (Maheshwari et al., 2023b):

f (v) = \frac{k}{C} {(\frac{v}{C})}^{k - 1} \times \exp {(- \frac{v}{C})}^{k}

(2)

where, C and k are the size and shape parameters of the WD, respectively.

The cumulative density function (CDF) for the WD is defined as:

F (v) = 1 - \exp {(\frac{v}{C})}^{k}

(3)

The inverse of CDF is considered to calculate the wind speed (v):

v = C \times {(- \ln (r))}^{1 / k}

(4)

where r be a uniformly distributed random integer, taking any value between 0 and 1. The estimated power output of the WT is calculated considering the probability of all potential states during the given time period by using the following equation (Maheshwari and Sood, 2022):

P_{W} = \frac{\sum_{g = 1}^{Nv} P_{W_{g}} . f_{v} ({v_{g}}^{t})}{\sum_{g = 1}^{Nv} f (v_{g}^{t})}

(5)

where ${v_{g}}^{t}$ represents $g^{th}$ state of the wind speed at $t^{th}$ time interval. $P_{W_{g}}$ is the active power output of the WT determined using the equation (1) for $v = v_{g}^{t}$ . N_v denotes the number of states. Furthermore, $f (v_{g}^{t})$ denoted probability of wind speed being in the $g^{th}$ state during the specified time interval t.

Objective function

The main objective of this study is to evaluate the impact of wind farm integration on social welfare (SW) in a deregulated environment and is mathematically defined as (Sood and Singh, 2010):

M i n F = (\sum_{i = 1}^{NG} C_{i} (P_{g, i}) - \sum_{j = 1}^{ND} B_{j} (P_{d, j}))

(6)

M a x SW = (\sum_{j = 1}^{ND} B_{j} (P_{d, j}) - \sum_{i = 1}^{NG} C_{i} (P_{g, i}))

(7)

The first term of equation (6) represents the generator bidding cost, and the second term shows the price consumer is willing to pay for its demand. $P_{g, i}$ and $P_{d, j}$ represents the active power generation at $i^{th}$ bus and load demand at $j^{th}$ bus, respectively. NG and ND represent the total number of generators and loads, respectively. As evident from equation (7), SW is the negative of equation (6); therefore, to maximize the SW, the objective function (F) defined in equation (6) needs to be minimized. It is to be noted that the generating cost of each GENCO is calculated using nodal pricing or LMP at different buses, mathematically defined as:

{\begin{matrix} C_{g} (P_{g i}) = [C_{i} (P_{Ti}) + C_{k} (P_{W, k})] \times L M P \\ where L M P = λ_{i} * Allowed Bid \end{matrix}

(8)

where

C_{i} (P_{Ti}) = \sum_{i = 1}^{N T G} (α_{i} + β_{i} P_{Ti} + χ_{i} P_{Ti}^{2})

(9)

C_{k} (P_{W, k}) = \sum_{k = 1}^{N W} (d_{k} P_{W, k})

(10)

where $C_{i} (P_{Ti})$ and $C_{k} (P_{W, k})$ represents the thermal and wind power generation cost, respectively (Maheshwari et al., 2023c). λ_i represents incremental cost at $i^{th}$ bus. $α_{i}, β_{i}$ and $χ_{i}$ represents the cost coefficients of thermal generators. $P_{Ti}$ , and $P_{W, k}$ represents the active power generation of $i^{th}$ thermal generating unit and $k^{th}$ wind power plant. NTG and NW denote the total number of thermal and wind power generators, respectively. The amount to be paid by each demand is also calculated based on LMP, mathematically defined as:

C_{d} (P_{d, j}) = \sum_{j = 1}^{ND} ((ε_{j} + φ_{j} P_{d, j} + γ_{j} P_{d, j}^{2}) \times L M P)

(11)

where $ε_{j}, φ_{j},$ and $γ_{j}$ represents the demand coefficients of the load or consumer.

Multiple constraints must be considered to solve the power flow optimization problem for maximizing social welfare. These constraints can be categorized into two parts: equality constraints and inequality constraints.

Equality constraints

Equality constraints are power balance equations that are mathematically represented as:

P_{g i} - P_{d i} - V_{i} \sum_{j = 1}^{N B} V_{j} (G_{ij} \cos ϕ_{ij} + B_{ij} \sin ϕ_{ij}) = 0

(12)

Q_{g i} - Q_{d i} - V_{i} \sum_{j = 1}^{N B} V_{j} (G_{ij} \sin ϕ_{ij} - B_{ij} \cos ϕ_{ij}) = 0

(13)

where $P_{g i}$ and $P_{d i}$ represents the active power generation and demand at $i^{th}$ bus; $Q_{g i}$ and $Q_{d i}$ are the reactive power generation and demand at $i^{th}$ bus. The transfer conductance and susceptance between $i^{th}$ and $j^{th}$ bus is denoted by $G_{ij}$ and $B_{ij}$ , respectively; $ϕ_{ij} = ϕ_{i} - ϕ_{j}$ is the voltage angle deviation between $i^{th}$ and $j^{th}$ bus. NB indicates the total number of buses.

Inequality constraints

Generator constraints

\begin{matrix} {\begin{matrix} P_{Ti}^{\min} \leq P_{Ti} \leq P_{Ti}^{\max} \\ Q_{Ti}^{\min} \leq Q_{Ti} \leq Q_{Ti}^{\max} \\ V_{Ti}^{\min} \leq V_{Ti} \leq V_{Ti}^{\max} \end{matrix} i = 1 . \dots . N T G \\ {\begin{matrix} Q_{Wi}^{\min} \leq Q_{Wi} \leq Q_{Wi}^{\max} \\ V_{Wi}^{\min} \leq V_{Wi} \leq V_{Wi}^{\max} \end{matrix} i = 1 . . . . . N W \end{matrix}

(14)

where $Q_{Ti}$ and $Q_{Wi}$ represents the reactive power output of $i^{th}$ thermal and wind power generators, respectively. $V_{Ti}$ and $V_{Wi}$ denotes the $i^{th}$ bus voltages connected by thermal and wind power generators.

Transformer constraints

T_{i}^{\min} \leq T_{i} \leq T_{i}^{\max} i = 1 . . . . . XR

(15)

where $T_{i}$ represents the tap setting of $i^{th}$ transformers. XR indicates the total number of transformers.

Security constraints

{\begin{matrix} V_{Lj}^{\min} \leq V_{Lj} \leq V_{Lj}^{\max} j = 1 . . . . . NL \\ S_{i} \leq S_{i}^{\max} i = 1 . . . . . NTL \end{matrix}

(16)

where $V_{Lj}$ represents $j^{th}$ load bus voltage. $S_{i}$ represents the transmission line loading. NL and NTL denote the total number of load buses and the number of transmission lines.

Capacitor bank switching constraints

Q_{C i}^{\min} \leq Q_{C i} \leq Q_{C i}^{\max} i = 1 . . . . . NC

(17)

where $Q_{C i}$ denotes the shunt VAr compensation limit. NC represents the total number of shunt VAr compensating devices.

Solution methodology

The Grey Wolf Optimization (GWO) algorithm is a powerful nature-inspired optimization technique that draws inspiration from the social hierarchy and hunting behavior of grey wolves proposed by Mirjalili et al. (2014). Grey wolves work together as a pack to hunt, with different roles assigned to each member based on their position in the pack. These roles include alpha (α), beta (β), delta (δ), and omega (ω) wolves, representing different levels of dominance. It starts by randomly initializing a population of potential solutions (wolf pack) within the search space. Each wolf represents a solution, and its fitness is evaluated based on the objective function and problem constraints. The algorithm uses the social hierarchy of wolves, with the α wolf being the best solution found so far. The β, δ, and ω wolves follow the alpha’s lead during the hunting process, and their positions are iteratively updated.

Mathematical modelling of GWO algorithm

The mathematical model of the GWO algorithm assigns symbolic representations to different candidate solutions. In this representation, α denotes the most promising solution, while the β and δ represent the second and third-best solutions, respectively. All other candidate solutions are collectively designated as ω, symbolizing their subordinate status to the α, β, and δ, wolves. The hunting behavior, where wolves encircle their prey, is mathematically expressed through the following equations:

\vec{D} = | \vec{C} . \vec{X_{p}} (i t e r) - \vec{X} (i t e r) |

(18)

\vec{X} (i t e r + 1) = \vec{X_{p}} (i t e r) - \vec{A} . \vec{D}

(19)

where $i t e r$ represents the current iteration of the optimization process. $\vec{X_{p}}$ denotes the position vector of the prey, while $\vec{X}$ represents the position vector of a grey wolf. The vectors, $\vec{A}$ and $\vec{C}$ are calculated as follows:

\vec{A} = 2 \vec{a} . {\vec{r}}_{1} - {\vec{a}}_{1}

(20)

\vec{C} = 2 \vec{a} . {\vec{r}}_{2}

(21)

Throughout the iterations, the components of parameter $a$ are systematically reduced from 2 to 0, while ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$ represent random vectors within the range of [0, 1]. The hunting process involves the pack of grey wolves updating their positions, guided by the best agents α, β, and δ, along with the influence of random values ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$ for handling parameters $\vec{A}$ and $\vec{C}$ , as described earlier. The following equations mathematically represent this update process:

{\vec{D}}_{a l p h a} = | {\vec{C}}_{1} \cdot {\vec{X}}_{a l p h a} - \vec{X} |, {\vec{D}}_{b e t a} = | {\vec{C}}_{2} \cdot {\vec{X}}_{b e t a} - \vec{X} |, {\vec{D}}_{delta} = | {\vec{C}}_{3} \cdot {\vec{X}}_{delta} - \vec{X} |

(22)

{\vec{X}}_{1} = {\vec{X}}_{a l p h a} - {\vec{A}}_{1} \cdot {\vec{D}}_{a l p h a}, {\vec{X}}_{2} = {\vec{X}}_{b e t a} - {\vec{A}}_{2} \cdot {\vec{D}}_{b e t a}, {\vec{X}}_{3} = {\vec{X}}_{delta} - {\vec{A}}_{3} \cdot {\vec{D}}_{delta},

(23)

X (t + 1) = \frac{X_{1} + X_{2} + X_{3}}{3}

(24)

The hunting behavior of grey wolves culminates in attacking the prey when it halts its movement. To mathematically simulate this behavior, the value of the parameter $a$ gradually decreases from 2 to 0. Additionally, the range of fluctuation for $\vec{A}$ is also reduced by a factor of $a$ . In essence, $\vec{A}$ becomes a random value within the interval [−2 $a$ , 2 $a$ ]. Consequently, when random values fall within the range of [−1, 1], the next position of a search agent can potentially lie anywhere between its current position and the position of the prey.

Implementation of GWO algorithm for maximizing the social welfare

Maximizing the social welfare in a deregulated power sector with wind power plants (WPP) using the GWO algorithm involves the following step-by-step methods:

Step 1: Define the population size of grey wolves (agents) that will represent potential solutions in the optimization process. Randomly initialize the position of each wolf (representing the potential solution) within the search space. In this case, the search space corresponds to the dispatch schedules of generators and loads in the power system.

Step 2: For each grey wolf (potential solution), compute the SW using the LMP and additional power system constraints. The fitness function should incorporate the revenue received from the market and the expenditure on generation costs, considering constraints like power balance, transmission limits, and bidding characteristics and others other factors considered in this study.

Step 3: Identify the best three solutions (α, β, and δ,) among the grey wolves based on their fitness values. Update the positions of these α, β, and δ, wolves using the GWO equations to explore the search space efficiently.

Step 4: For the remaining ω grey wolves, update their positions using the GWO equations based on the positions of the α, β, and δ, wolves. This step encourages search space exploration while gradually converging toward the optimal solution.

Step 5: Check and enforce any constraints related to the dispatch schedules of generators and loads.

Step 6: Define a termination criterion, such as a maximum number of iterations.

Step 7: Once the termination criterion is met, output the optimal dispatch schedules of generators, and loads that maximize the SW. The pseudo-code of GWO algorithm considered for this problem is shown as follows:

1.	Initialize the population of wolves randomly
2.	Set the maximum number of iterations.
3.	Set convergence criteria.
4.	While (iter < max_iter and not converged)
5.	Evaluate the fitness of each wolf
6.	Identify alpha, beta, delta, and omega wolves.
7.	For each wolf in the population
8.	Update wolf position based on alpha, beta, delta, and omega wolves.
9.	End For
10.	Decrease the value of “a” gradually from 2 to 0
11.	Check convergence criteria.
12.	Increment iterations.
13.	End While
14.	Select the best wolf as the optimized solution.

Result and discussion

This study examines the effectiveness of the proposed Grey Wolf Optimization (GWO) algorithm by analyzing both standard and modified IEEE 30 and 118-bus systems. The necessary bus data, line data, and cost coefficients of thermal generators for the standard IEEE 30 and 118-bus systems were acquired from Maheshwari et al. (2023a). The simulations were carried out using MATLAB software on a PC equipped with a 2.4 GHz processor and 8 GB of RAM.

The parameters controlling the proposed GWO algorithm were adjusted differently for the IEEE 30 and 118-bus systems. Specifically, for the IEEE 30-bus system, the maximum number of iterations was set to 200, and the population size was 40. In contrast, for the IEEE 118-bus system, the maximum number of iterations was raised to 500, with a population size of 100. It’s worth noting that each case was run for a maximum of 30 trials. To comprehensively evaluate the algorithm’s performance, two scenarios were considered in this study: one without the inclusion of wind power plants and another with their inclusion.

Scenario 1: System performance without wind power plants

Case 1: Standard IEEE 30-bus system

In this case, the proposed algorithm was employed to determine optimized values for both supply and demand bidding quantities for maximizing the social welfare, considering the standard IEEE 30-bus system. All constraints specified in equations (12)–(17) were considered during the process. Detailed information regarding the bidding coefficients of dispatchable loads, along with their maximum and allowed bids, was presented in Table 1.

Table 1.

Scheduling and revenue received from dispatchable loads in standard IEEE 30-bus system.

Bus no.	Active power demand (MW)		γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max. bid	Allowed bid	γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	LMP	Actual Bidding	Profit surplus ($/hr.)
3	30.00	13.19	0.0250	4.5	3.841	50.6628	63.703	13.0399
14	15.00	12.69	0.0676	5.8	4.084	51.8260	84.486	32.6605
15	20.00	1.06	0.0685	4.2	4.054	4.2972	4.529	0.2317
17	15.00	9.09	0.0283	4.5	3.985	36.2237	43.247	7.0230
18	20.00	13.71	0.0297	5	4.186	57.3901	74.129	16.7387
20	10.00	6.34	0.0294	4.5	4.127	26.1652	29.713	3.5474
Total	110.00	56.08				226.5649	299.806	73.2412

As observed from Table 1, GENCOs provided a total dispatchable load of 56.08 MW out of 110 MW. The surplus profit for dispatchable loads amounts to $73.2412 per hour, representing the disparity between the actual bidding cost and the locational marginal pricing (LMP).

Table 2 displays the revenue received from fixed loads determined based on the LMP method rather than considering fixed costs. Compared to the fixed cost method, the LMP-based approach proved to allocate costs more efficiently, accurately reflecting the actual network conditions and constraints. The total revenue generated from the fixed load sums up to $1130.5318 per hour.

Table 2.

Revenue received from fixed load in standard IEEE 30-bus system.

Bus no.	Fixed load (MW)	λ ($/MWhr.)	Revenue received ($/hr.)
Bus no.	Fixed load (MW)	λ ($/MWhr.)	LMP
2	21.7	3.745	81.2665
3	2.4	3.841	9.2184
4	7.6	3.898	29.6248
5	94.2	4.012	377.9304
7	22.8	3.998	91.1544
8	30	3.935	118.05
10	5.8	3.949	22.9042
12	11.2	3.905	43.736
14	6.2	4.084	25.3208
15	8.2	4.054	33.2428
16	3.5	3.958	13.853
17	9	3.985	35.865
18	3.2	4.186	13.3952
19	9.5	4.165	39.5675
20	2.2	4.127	9.0794
21	17.5	3.993	69.8775
23	3.2	4.074	13.0368
24	8.7	4.063	35.3481
26	3.5	4.09	14.315
29	2.4	4.065	9.756
30	10.6	4.15	43.99
Total	283.4	84.277	1130.5318

Furthermore, Table 3 provides the particulars of bidding coefficients of thermal generators with their minimum and maximum bidding quantities, allowed bid, and the expenditure associated with the generation. It has been noted that the profit surplus for all generators amounts to $277.550 per hour, denoting the difference between LMP and the actual bidding price.

Table 3.

Scheduling and expenditure on thermal generators in standard IEEE 30-bus system.

Bus no.	Active power demand (MW)			χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Min	Max	Allowed	χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
1	50	200	200	0.00372	2	3.512	702.400	548.800	153.600
3	20	100	57	0.0175	1.75	3.745	213.465	156.608	56.858
5	15	80	24.09	0.0625	1	4.012	96.649	60.361	36.289
8	10	60	35	0.00834	3.25	3.935	137.725	123.967	13.759
11	10	50	18.83	0.025	3	3.941	74.209	65.354	8.855
13	12	60	18.1	0.025	3	3.905	70.681	62.490	8.190
Total	117	550	353.02				1295.129	1017.579	277.550

Based on the information presented in Tables 1 to 3, it is evident that the LMP pricing method is advantageous for both loads and generators. Notably, loads are required to pay less than their actual bidding price, while generators receive a higher price than their actual bidding cost.

Case 2: Standard IEEE 118-bus system

A similar analysis has been executed on the standard IEEE 118-bus system to evaluate the effectiveness and scalability of the proposed algorithm. Table 4 summarizes the comprehensive information on dispatchable loads, including bidding coefficients, maximum bids, allowed bids, and profit surplus.

Table 4.

Scheduling and revenue received from dispatchable loads in standard IEEE 118-bus system.

Bus no.	Active power generation (MW)		γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max	Allowed	γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
93	25	22.22	0.0045	42	41.801	928.818	935.447	6.629
94	30	30	0.0067	45	42.24	1267.200	1356.039	88.839
95	140	140	0.0048	50	43.399	6075.860	7094.864	1019.004
96	35	35	0.0097	55	42.091	1473.185	1936.895	463.710
97	45	45	0.0345	50	41.221	1854.945	2319.863	464.918
98	35	35	0.00273	55	40.814	1428.490	1928.344	499.854
101	350	189.36	0.02096	52	44.062	8343.580	10,598.287	2254.707
102	50	50	0.0076	55	42.081	2104.050	2769.000	664.950
106	40	40	0.00853	50	42.014	1680.560	2013.648	333.088
108	30	30	0.00377	50	42.595	1277.850	1503.393	225.543
109	80	80	0.00742	48	42.764	3421.120	3887.488	466.368
114	80	80	0.00679	50	41.768	3341.440	4043.456	702.016
115	25	25	0.01481	55	41.736	1043.400	1384.256	340.856
117	40	40	0.00348	52	42.226	1689.040	2085.568	396.528
118	35	35	0.00871	55	41.384	1448.440	1935.670	487.230
Total	1040	876.58				37,377.979	45,792.218	8414.239

During the study, thermal generators supplied a cumulative dispatchable load of 876.58 MW out of the total capacity of 1040 MW, resulting in a profit surplus of $8414.239 per hour attributed to the dispatchable loads. Additionally, Table 5 unveils that the ISO garnered revenue amounting to $172,113.11 per hour from fixed loads.

Table 5.

Revenue received from fixed load in standard IEEE 118-bus system.

Bus no.	Fixed load (MW)	λ	RR	Bus no.	Fixed load (MW)	λ	RR	Bus no.	Fixed load (MW)	λ	RR	Bus no.	Fixed load (MW)	λ	RR
Bus no.	Fixed load (MW)	λ	LMP	Bus no.	Fixed load (MW)	λ	LMP	Bus no.	Fixed load (MW)	λ	LMP	Bus no.	Fixed load (MW)	λ	LMP
1	51	40.77	2079.17	32	59	41.02	2420.24	59	277	39.88	11,047.04	94	30	42.24	1267.20
2	20	40.86	817.26	33	23	40.77	937.80	60	78	39.38	3071.33	95	42	43.40	1822.76
3	39	40.60	1583.44	34	59	40.36	2381.12	62	77	39.39	3032.65	96	38	42.09	1599.46
4	39	39.67	1547.01	35	33	40.45	1334.69	66	39	38.45	1499.43	97	15	41.22	618.32
6	52	40.29	2095.29	36	31	40.44	1253.67	67	28	39.11	1095.16	98	34	40.81	1387.68
7	19	40.46	768.68	39	27	41.04	1108.03	70	66	40.18	2651.62	99	42	40.65	1707.09
8	28	39.56	1107.57	40	66	41.15	2715.83	72	12	40.10	481.16	100	37	40.97	1516.00
11	70	40.53	2837.31	41	37	41.42	1532.54	73	6	40.15	240.88	101	22	44.06	969.36
12	47	40.64	1909.99	42	96	41.02	3937.54	74	68	40.82	2775.90	102	5	42.08	210.41
13	34	41.01	1394.24	43	18	41.17	741.01	75	47	40.76	1915.86	103	23	41.05	944.20
14	14	40.81	571.27	44	16	41.54	664.69	76	68	41.17	2799.63	104	38	41.25	1567.50
15	90	40.65	3658.32	45	53	41.40	2194.20	77	61	40.28	2457.26	105	31	41.53	1287.49
16	25	40.69	1017.33	46	28	40.53	1134.73	78	71	40.41	2869.25	106	43	42.01	1806.60
17	11	39.98	439.74	47	34	39.88	1355.78	79	39	40.29	1571.31	107	50	41.39	2069.40
18	60	40.47	2428.20	48	20	39.89	797.78	80	130	39.50	5135.00	108	2	42.60	85.19
19	45	40.63	1828.35	49	87	39.44	3431.45	82	54	41.67	2249.96	109	8	42.76	342.11
20	18	40.81	734.58	50	17	40.14	682.41	83	20	41.40	827.90	110	39	41.29	1610.27
21	14	40.69	569.63	51	17	41.06	697.95	84	11	40.70	447.68	112	68	41.32	2809.96
22	10	40.32	403.20	52	18	41.40	745.16	85	24	40.12	962.78	113	6	40.11	240.67
23	7	39.48	276.33	53	23	41.45	953.26	86	21	40.56	851.82	114	8	41.77	334.14
24	13	39.78	517.11	54	113	40.98	4630.29	88	48	39.63	1902.43	115	22	41.74	918.19
27	71	40.91	2904.33	55	63	40.94	2579.16	90	163	40.46	6594.17	116	184	38.86	7150.24
28	17	41.28	701.76	56	84	40.97	3441.14	91	10	40.39	403.93	117	20	42.23	844.52
29	24	41.44	994.56	57	12	40.80	489.61	92	65	40.58	2637.77	118	33	41.38	1365.67
31	43	41.32	1776.55	58	12	41.16	493.87	93	12	41.80	501.61	Total		4039.99	172,113.11

λ: ($/MWhr.); RR: revenue received ($/hr.).

Table 6 provides an illustrative depiction of bidding coefficient data for thermal generators, including their minimum and maximum bidding quantities, allowed bids, and associated generation costs. The profit surplus of the GENCOs stood at39,885.605 $/hr.

Table 6.

Scheduling and expenditure on thermal generators in standard IEEE 118-bus system.

Bus no.	Active power generation (MW)		χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max	Allowed	χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
1	100	38.38	0.01	40	40.768	1564.676	1549.930	14.746
4	100	0	0.01	40	39.667	0.000	0.000	0.000
6	100	14.71	0.01	40	40.294	592.725	590.564	2.161
8	100	0	0.01	40	39.556	0.000	0.000	0.000
10	550	408.44	0.02	20	38.153	15,583.211	11,875.983	3707.228
12	185	87.71	0.1176471	20	40.638	3564.359	2659.264	905.095
15	100	32.41	0.01	40	40.648	1317.402	1306.904	10.498
18	100	23.51	0.01	40	40.47	951.450	945.927	5.522
19	100	31.48	0.01	40	40.63	1279.032	1269.110	9.922
24	100	0	0.01	40	39.778	0.000	0.000	0.000
25	320	198.08	0.0454545	20	38.007	7528.427	5745.040	1783.386
26	414	285.7	0.0318471	20	38.197	10,912.883	8313.506	2599.377
27	100	45.32	0.01	40	40.906	1853.860	1833.339	20.521
31	107	7.46	1.4285714	20	41.315	308.210	228.702	79.508
32	100	51.06	0.01	40	41.021	2094.532	2068.471	26.061
34	100	17.9	0.01	40	40.358	722.408	719.204	3.204
36	100	22.07	0.01	40	40.441	892.533	887.671	4.862
40	100	57.45	0.01	40	41.149	2364.010	2331.005	33.005
42	100	50.8	0.01	40	41.016	2083.613	2057.806	25.806
46	119	19.5	0.5263158	20	40.526	790.257	590.132	200.125
49	304	198.31	0.0490196	20	39.442	7821.743	5893.987	1927.756
54	148	50.34	0.2083333	20	40.976	2062.732	1534.741	527.991
55	100	46.94	0.01	40	40.939	1921.677	1899.634	22.043
56	100	48.31	0.01	40	40.966	1979.067	1955.739	23.329
59	255	154.08	0.0645161	20	39.881	6144.864	4613.255	1531.610
61	260	153.48	0.0625	20	39.185	6014.114	4541.857	1472.257
62	100	0	0.01	40	39.385	0.000	0.000	0.000
65	491	365.91	0.0255754	20	38.716	14,166.572	10,742.500	3424.072
66	492	361.56	0.0255102	20	38.447	13,900.897	10,566.038	3334.860
69	805.2	472.5	0.0193648	20	38.3	18,096.750	13,773.320	4323.430
70	100	8.82	0.01	40	40.176	354.352	353.578	0.774
72	100	4.85	0.01	40	40.097	194.470	194.235	0.235
73	100	7.36	0.01	40	40.147	295.482	294.942	0.540
74	100	41.09	0.01	40	40.822	1677.376	1660.484	16.892
76	100	58.56	0.01	40	41.171	2410.974	2376.693	34.281
77	100	14.14	0.01	40	40.283	569.602	567.599	2.002
80	577	465.07	0.0209644	20	39.5	18,370.265	13,835.784	4534.481
85	100	5.78	0.01	40	40.116	231.870	231.534	0.336
87	104	4.1	2.5	20	40.477	165.956	124.025	41.931
89	707	571.34	0.0164745	20	38.825	22,182.276	16,804.550	5377.726
90	100	22.75	0.01	40	40.455	920.351	915.176	5.176
91	100	19.67	0.01	40	40.393	794.530	790.669	3.861
92	100	29.04	0.01	40	40.581	1178.472	1170.033	8.439
99	100	32.27	0.01	40	40.645	1311.614	1301.214	10.401
100	352	264.26	0.0396825	20	40.973	10,827.525	8056.365	2771.160
103	140	42.1	0.25	20	41.052	1728.289	1285.103	443.187
104	100	62.51	0.01	40	41.25	2578.538	2539.475	39.062
105	100	76.6	0.01	40	41.532	3181.351	3122.676	58.676
107	100	69.39	0.01	40	41.388	2871.913	2823.750	48.164
110	100	64.47	0.01	40	41.289	2661.902	2620.364	41.538
111	136	37.2	0.2777778	20	40.664	1512.701	1128.400	384.301
112	100	66.16	0.01	40	41.323	2733.930	2690.171	43.758
113	100	5.55	0.01	40	40.111	222.616	222.308	0.308
116	100	0	0.01	40	38.86	0.000	0.000	0.000
Total	9966.200	5216.490				205,488.359	165,602.754	39,885.605

Derived from the obtained outcomes (Tables 4 –6), it can be deduced that the LMP method consistently confers surplus advantages upon both loads and generators, thereby substantiating its viability and effectiveness within the assessed scenario.

Scenario 2: System performance with wind power plants

Case 3: Modified IEEE 30-bus system

To evaluate the impact of wind power plants (WPPs) on social welfare, modified IEEE 30-bus system has been considered, which included the integration of two wind farms. These wind farms are connected to bus 15 and bus 30, with rated capacities of 75 MW and 50 MW, respectively. The minimum and maximum reactive power limit of these wind farms is considered in between $- 0.4 \times P_{W}^{\max}$ to $0.5 \times P_{W}^{\max}$ . It is important to emphasize that the active power generation from wind farms is inherently non-schedulable, meaning it cannot be precisely planned or controlled in advance. Thus, system operators must rely on forecasted values to effectively manage and utilize the generated power. Conversely, the voltage of the wind farms is introduced as a control variable in the optimization problem formulation. This provision empowers the system operator to make well-informed decisions and efficiently regulate power generation.

Table 7 provides an overview of dispatchable loads, encapsulating bidding coefficients, maximum and permissible bids, and the resulting profit surplus. It is noteworthy that the analysis has demonstrated a significant rise in the penetration of dispatchable loads, ascending from 56.08 MW (Case 1) to 66.79 MW (Case 3) due to the integration of WPPs.

Table 7.

Scheduling and revenue received from dispatchable loads in modified IEEE 30-bus system.

Bus no.	Active Power demand (MW)		γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max bid	Allowed bid	γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
3	30.000	21.11	0.0250	4.5	3.445	72.7240	106.131	33.4074
14	15.000	15	0.0676	5.8	3.349	50.2350	102.208	51.9728
15	20.000	7.84	0.0685	4.2	3.126	24.5078	37.140	12.6318
17	15.000	13.87	0.0283	4.5	3.714	51.5132	67.867	16.3538
18	20.000	5.13	0.0297	5	4.696	24.0905	26.431	2.3406
20	10.000	3.84	0.0294	4.5	4.274	16.4122	17.714	1.3017
Total	110.000	66.79				239.4826	357.491	118.0080

As a significant outcome, the incorporation of WPPs resulted in a reduction of locational marginal prices (LMP) compared to Case 1, thereby leading to increased profits for both dispatchable and fixed loads and generating units. Consequently, the profit surplus for dispatchable loads substantially rose from $73.2412 per hour (Case 1) to $118.008 per hour (Case 3).

Furthermore, Table 8 discloses that fixed loads experience a reduction in costs, settling at 1013.9905 $/hr, marking an 11.49% decrease compared to Case 1.

Table 8.

Revenue received from fixed load in modified IEEE 30-bus system.

Bus no.	Fixed load (MW)	λ ($/MWhr.)	Revenue received ($/hr.)
Bus no.	Fixed load (MW)	λ ($/MWhr.)	LMP
2	21.7	3.376	73.2592
3	2.4	3.445	8.268
4	7.6	3.471	26.3796
5	94.2	3.615	340.533
7	22.8	3.585	81.738
8	30	3.524	105.72
10	5.8	3.731	21.6398
12	11.2	3.304	37.0048
14	6.2	3.349	20.7638
15	8.2	3.126	25.6332
16	3.5	3.527	12.3445
17	9	3.714	33.426
18	3.2	4.696	15.0272
19	9.5	4.441	42.1895
20	2.2	4.274	9.4028
21	17.5	3.738	65.415
23	3.2	3.336	10.6752
24	8.7	3.591	31.2417
26	3.5	3.636	12.726
29	2.4	3.244	7.7856
30	10.6	3.096	32.8176
Total	213.6	75.819	1013.9905

Table 9 illustrates bidding coefficients for thermal and WPPs, including their minimum and maximum bidding quantities, allowable bids, and associated generation expenditures. Notably, the inclusion of WPPs results in a remarkable increase of approximately 20.88% in the profit surplus of GENCOs when compared to Case 1.

Table 9.

Scheduling and expenditure on thermal and wind power generators in modified IEEE 30-bus system.

Bus no.	Active Power generation (MW)		χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max	Allowed	χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
1	200	166.99	0.00372	2	3.242	541.382	437.715	103.667
3	100	46.45	0.0175	1.75	3.376	156.815	119.046	37.770
5	80	20.92	0.0625	1	3.615	75.626	48.273	27.353
8	60	16.42	0.00834	3.25	3.524	57.864	55.614	2.250
11	50	13.07	0.025	3	3.654	47.758	43.481	4.277
13	60	34.77	0.025	3	3.304	114.880	134.534	−19.654
15	75	50.27	0	1	3.126	157.144	50.270	106.874
30	50	34.77	0	1	3.096	107.648	34.770	72.878
Total	675	383.66				1259.116	923.701	335.415

These findings underscore the positive economic implications of integrating wind power into the system. This integration not only enhances the profitability of load and generating units but also fosters the overall efficiency and sustainability of the energy network.

It is worth noting that advancements in wind power technology and the expanded scale of installations have bolstered its competitiveness within the energy market. As renewable energy garners greater traction, wind power plants are poised to gain cost advantages over traditional thermal power plants in deregulated sectors, attributed to lower cost coefficients, as demonstrated in Table 9.

Case 4: Modified IEEE 118-bus system

In this case, modified IEEE 118-bus system that incorporates wind power plants has been considered. This system consists of 8 wind farms, each having a rated capacity of 100 MW. These wind farms are interconnected with buses 28, 29, 33, 35, 37, 38, 39, and 41. The reactive power limits of these wind farms are set between $- 0.4 \times P_{W}^{\max}$ to $0.5 \times P_{W}^{\max}$ .

Detailed data pertaining to dispatchable loads is furnished in Table 10, encompassing bidding coefficients, maximum and permissible bids, as well as the revenue generated. It has been observed that the inclusion of wind power plants led to a rise in the penetration of dispatchable loads, escalating from 876.58 MW (Case 2) to 880.31 MW (Case 4). This results in a 2.54% increase in profits compared to Case 2. Fixed loads, as delineated in Table 11, yielded an amount of $169976.98 per hour, signifying a reduction of $2136.13 per hour as compared to Case 2.

Table 10.

Scheduling and revenue received from dispatchable loads for modified IEEE 118-bus system.

Bus no.	Active power demand (MW)		γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max	Allowed	γ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
93	25	24.99	0.0045	42	41.763	1043.657	1052.372	8.714
94	30	30	0.0067	45	42.144	1264.320	1356.039	91.719
95	140	140	0.0048	50	43.267	6057.380	7094.864	1037.484
96	35	35	0.0097	55	41.918	1467.130	1936.895	469.765
97	45	45	0.0345	50	40.992	1844.640	2319.863	475.223
98	35	35	0.00273	55	40.61	1421.350	1928.344	506.994
101	350	190.32	0.02096	52	44.022	8378.267	10,655.847	2277.580
102	50	50	0.0076	55	42.046	2102.300	2769.000	666.700
106	40	40	0.00853	50	41.986	1679.440	2013.648	334.208
108	30	30	0.00377	50	42.576	1277.280	1503.393	226.113
109	80	80	0.00742	48	42.747	3419.760	3887.488	467.728
114	80	80	0.00679	50	40.756	3260.480	4043.456	782.976
115	25	25	0.01481	55	40.712	1017.800	1384.256	366.456
117	40	40	0.00348	52	41.539	1661.560	2085.568	424.008
118	35	35	0.00871	55	41.23	1443.050	1935.670	492.620
Total	1040	880.31				37,338.414	45,966.702	8628.287

Table 11.

Revenue received from fixed load in modified IEEE 118-bus system.

Bus no.	Fixed load (MW)	λ	RR	Bus no.	Fixed load	λ	RR	Bus no.	Fixed load	λ	RR	Bus no.	Fixed load (MW)	λ	RR
Bus no.	Fixed load (MW)	λ	LMP	Bus no.	Fixed load	λ	LMP	Bus no.	Fixed load	λ	LMP	Bus no.	Fixed load (MW)	λ	LMP
1	51	40.36	2058.41	32	59	40.10	2366.14	59	277	39.50	10,942.61	94	30	42.14	1264.32
2	20	40.29	805.86	33	23	38.55	886.56	60	78	38.96	3038.57	95	42	43.27	1817.21
3	39	40.06	1562.38	34	59	38.72	2284.24	62	77	38.96	2999.61	96	38	41.92	1592.88
4	39	38.95	1518.93	35	33	38.59	1273.60	66	39	38.00	1481.84	97	15	40.99	614.88
6	52	39.66	2062.48	36	31	38.66	1198.37	67	28	38.67	1082.76	98	34	40.61	1380.74
7	19	39.81	756.47	39	27	38.86	1049.33	70	66	39.95	2636.37	99	42	40.56	1703.44
8	28	38.78	1085.95	40	66	39.56	2610.83	72	12	39.84	478.09	100	37	40.90	1513.45
11	70	39.84	2788.45	41	37	39.50	1461.46	73	6	40.00	239.98	101	22	44.02	968.48
12	47	39.97	1878.68	42	96	40.21	3860.26	74	68	40.68	2765.97	102	5	42.05	210.23
13	34	40.26	1368.87	43	18	39.79	716.17	75	47	40.57	1906.93	103	23	41.01	943.21
14	14	40.08	561.12	44	16	40.58	649.33	76	68	41.06	2791.81	104	38	41.23	1566.55
15	90	39.77	3579.39	45	53	40.60	2152.01	77	61	40.06	2443.72	105	31	41.51	1286.81
16	25	39.94	998.50	46	28	39.92	1117.79	78	71	40.18	2852.85	106	43	41.99	1805.40
17	11	39.05	429.56	47	34	39.38	1338.75	79	39	40.05	1561.79	107	50	41.38	2068.75
18	60	39.74	2384.58	48	20	39.38	787.64	80	130	39.22	5098.47	108	2	42.58	85.15
19	45	39.90	1795.59	49	87	38.97	3390.13	82	54	41.51	2241.32	109	8	42.75	341.98
20	18	40.09	721.58	50	17	39.72	675.22	83	20	41.27	825.34	110	39	41.28	1609.76
21	14	39.97	559.64	51	17	40.71	692.00	84	11	40.63	446.92	112	68	41.32	2809.49
22	10	39.62	396.18	52	18	41.07	739.22	85	24	40.07	961.78	113	6	39.22	235.31
23	7	38.80	271.58	53	23	41.16	946.61	86	21	40.52	850.96	114	8	40.76	326.05
24	13	39.28	510.67	54	113	40.71	4600.57	88	48	39.60	1900.85	115	22	40.71	895.66
27	71	39.82	2827.01	55	63	40.71	2564.86	90	163	40.44	6591.56	116	184	38.47	7078.11
28	17	39.16	665.65	56	84	40.72	3420.48	91	10	40.38	403.79	117	20	41.54	830.78
29	24	39.20	940.82	57	12	40.48	485.76	92	65	40.55	2635.69	118	33	41.23	1360.59
31	43	39.56	1701.12	58	12	40.85	490.21	93	12	41.76	501.16	Total		3985.28	169,976.98

λ: ($/MWhr.); RR: revenue received ($/hr.).

Bidding coefficients, maximum bidding quantities, allowed bids, and generation-related expenses for both thermal power plants and wind power plants are comprehensively presented in Tables 12 and 13, respectively. The analysis highlights that the net profit accrued by GENCOs utilizing the LMP method reached $53,466.696 per hour which is a notable increase of $13,581.091 per hour in contrast to Case 2. Mirroring the findings of Case 3, the incorporation of WPPs leads to a decrease in LMP when compared to Case 2, consequently fostering heightened profits for both loads and generating units.

Table 12.

Scheduling and expenditure on thermal generators in modified IEEE 118-bus system.

Bus no.	Active Power generation (MW)		χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max	Allowed	χ ($/MWhr.)	β ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
1	100	18.05	0.01	40	40.361	728.516	725.258	3.258
4	100	0	0.01	40	38.947	0.000	0.000	0.000
6	100	0	0.01	40	39.663	0.000	0.000	0.000
8	100	0	0.01	40	38.784	0.000	0.000	0.000
10	550	392.85	0.02	20	37.460	14,716.161	11,286.580	3429.581
12	185	84.88	0.1176471	20	39.972	3392.823	2545.202	847.622
15	100	0	0.01	40	39.771	0.000	0.000	0.000
18	100	0	0.01	40	39.743	0.000	0.000	0.000
19	100	0	0.01	40	39.902	0.000	0.000	0.000
24	100	0	0.01	40	39.282	0.000	0.000	0.000
25	320	189.57	0.0454545	20	37.233	7058.260	5424.890	1633.370
26	414	273.22	0.0318471	20	37.403	10,219.248	7841.762	2377.486
27	100	0	0.01	40	39.817	0.000	0.000	0.000
31	107	6.85	1.4285714	20	39.561	270.993	204.032	66.961
32	100	5.18	0.01	40	40.104	207.739	207.468	0.270
34	100	0	0.01	40	38.716	0.000	0.000	0.000
36	100	0	0.01	40	38.657	0.000	0.000	0.000
40	100	0	0.01	40	39.558	0.000	0.000	0.000
42	100	10.54	0.01	40	40.211	423.824	422.711	1.113
46	119	18.93	0.5263158	20	39.921	755.705	567.203	188.502
49	304	193.46	0.0490196	20	38.967	7538.556	5703.846	1834.710
54	148	49.71	0.2083333	20	40.713	2023.843	1509.009	514.834
55	100	35.59	0.01	40	40.712	1448.940	1436.266	12.674
56	100	36.02	0.01	40	40.720	1466.734	1453.774	12.960
59	255	151.16	0.0645161	20	39.504	5971.425	4497.351	1474.073
61	260	150.11	0.0625	20	38.764	5818.864	4410.513	1408.351
62	100	0	0.01	40	38.956	0.000	0.000	0.000
65	491	356.57	0.0255754	20	38.239	13,634.880	10,383.118	3251.762
66	492	352.73	0.0255102	20	37.996	13,402.329	10,228.540	3173.789
69	805.2	463.13	0.0193648	20	37.937	17,569.763	13,416.151	4153.611
70	100	0	0.01	40	39.945	0.000	0.000	0.000
72	100	0	0.01	40	39.841	0.000	0.000	0.000
73	100	0.13	0.01	40	39.997	5.200	5.200	−0.001
74	100	33.82	0.01	40	40.676	1375.662	1364.238	11.424
76	100	52.78	0.01	40	41.056	2166.936	2139.057	27.878
77	100	3.05	0.01	40	40.061	122.186	122.093	0.093
80	577	458.37	0.0209644	20	39.219	17,976.813	13,572.076	4404.737
85	100	3.72	0.01	40	40.074	149.075	148.938	0.137
87	104	4.09	2.5	20	40.436	165.383	123.620	41.763
89	707	570.55	0.0164745	20	38.799	22,136.769	16,773.888	5362.881
90	100	21.96	0.01	40	40.439	888.040	883.222	4.818
91	100	18.97	0.01	40	40.379	765.990	762.399	3.591
92	100	27.45	0.01	40	40.549	1113.070	1105.535	7.535
99	100	27.89	0.01	40	40.558	1131.163	1123.379	7.784
100	352	263.39	0.0396825	20	40.904	10,773.705	8020.748	2752.956
103	140	42.02	0.25	20	41.009	1723.198	1281.820	441.378
104	100	61.23	0.01	40	41.225	2524.207	2486.691	37.516
105	100	75.5	0.01	40	41.510	3134.005	3077.003	57.002
107	100	68.76	0.01	40	41.375	2844.945	2797.679	47.266
110	100	63.79	0.01	40	41.276	2632.996	2592.292	40.704
111	136	37.17	0.2777778	20	40.652	1511.035	1127.180	383.855
112	100	65.8	0.01	40	41.316	2718.593	2675.296	43.296
113	100	0	0.01	40	39.218	0.000	0.000	0.000
116	100	0	0.01	40	38.468	0.000	0.000	0.000
Total	9966.200	4688.990				182,507.573	144,446.032	38,061.541

Table 13.

Scheduling and expenditure on wind power generators in modified IEEE 118-bus system.

Bus no.	Active power generation (MW)		χ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	Revenue received ($/hr.)		Profit surplus ($/hr.)
Bus no.	Max bid	Allowed bid	χ ($/MWhr.)	φ ($/MWhr.)	λ ($/MWhr.)	LMP	Actual bidding	Profit surplus ($/hr.)
28	100	67.28	0	10	39.156	2634.416	672.8	1961.616
29	100	66.56	0	10	39.201	2609.219	665.6	1943.619
33	100	67.42	0	10	38.546	2598.771	674.2	1924.571
35	100	65.48	0	10	38.594	2527.135	654.8	1872.335
37	100	66.84	0	10	38.528	2575.212	668.4	1906.812
38	100	68	0	10	38.6	2624.800	680	1944.800
39	100	65.48	0	10	38.864	2544.815	654.8	1890.015
41	100	66.49	0	10	39.499	2626.289	664.9	1961.389
Total	800	533.550				20,740.655	5335.5	15,405.155

Summary of results

The findings of each case are summarized in Figure 2. It has been observed that the incorporation of WPPs led to an 8.48% increase in social welfare for the IEEE 30-bus system. Similarly, there was a 26.49% rise in social welfare for the IEEE 118-bus system. These results demonstrated a significant enhancement in social welfare through the integration of WPPs. The above findings strongly advocate for the integration of RESs in the deregulated power sector.

Figure 2.

Summary of results for all cases.

Conclusion

This paper has effectively utilized the grey wolf optimization algorithm to maximize social welfare in wind-integrated power systems. The analysis was performed on both standard and modified IEEE 30-bus and IEEE 118-bus systems, and the following key findings were observed:

Integrating wind power plants (WPPs) significantly enhances social welfare within the power systems.

The presence of WPPs leads to a substantial reduction in locational marginal prices (LMP), resulting in increased profits for electricity consumers and power generators.

The LMP pricing method proves to be advantageous as it provides surplus benefits to both loads and generators. Consumers benefit by paying less than their actual bidding price, while power generators receive higher payments than their initial bidding cost.

Future research directions should focus on exploring the implementation of advanced technologies, such as energy storage systems and plug-in electric vehicles, in conjunction with wind power plants. This integration can further optimize social welfare and enhance the efficiency of wind-integrated power systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ankur Maheshwari

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