Multi-Stage Incentive-Based Demand Response Using a Novel Stackelberg–Particle Swarm Optimization

Abstract

Demand response programs can effectively handle the smart grid’s increasing energy demand and power imbalances. In this regard, price-based DR (PBDR) and incentive-based DR (IBDR) are two broad categories of demand response in which incentives for consumers are provided in IBDR to reduce their demand. This work aims to implement the IBDR strategy from the perspective of the service provider and consumers. The relationship between the different entities concerned is modelled. The incentives offered by the service provider (SP) to its consumers and the consumers’ reduced demand are optimized using Stackelberg–particle swarm optimization (SPSO) as a bi-level problem. Furthermore, the system with a grid operator, the industrial consumers of the grid operator, the service provider and its consumers are analyzed from the service provider’s viewpoint as a tri-level problem. The benefits offered by the service provider to its customers, the incentives provided by the grid operator to its industrial customers, the reduction of customer demand, and the average cost procured by the grid operator are optimized using SPSO and compared with the Stackelberg-distributed algorithm. The problem was analyzed for an hour and 24 h in the MATLAB environment. Besides this, sensitivity analysis and payment analysis were carried out in order to delve into the impact of the demand response program concerning the change in customer parameters.

1. Introduction

A smart grid (SG) epitomizes an unprecedented chance to motivate the energy sector into a new age of dependability, availability, and efficacy, contributing to future economic and environmental conditions. As per the strength, weakness, opportunities, and threat analysis report, the increasing per capita electricity consumption could be controlled by implementing the demand response programs (DRPs) that form a focal point of SG [1]. The evolution of the renewable energy systems, in addition to the altered consumption pattern of the consumers, aids in the effective implementation of DRPs. With the evolution of multi-energy systems (MES), demand response (DR) programming has been broadened into integrated DR [2]. DR is a sub-classification of a broader concept called demand side management, which is further classified into price-based DR (PBDR) and incentive-based DR (IBDR). PBDR relies on pricing in different ways, such as real-time and day-ahead pricing, etc. IBDR programs are based on contracts involving various market entities such as the grid operator (GO), retailers, and customers, etc. An IBDR scheme that provides coupon incentives to the customers was designed by several researchers [3,4]. It benefits both the customers and the load-serving entities (LSE). In [5,6], a novel IBDR scheme has been framed considering the various market operators that benefit the customers and the utilities. The SG technology advancement has made DR program implementation easier. The communication infrastructure of the SG technology has made it possible to implement DR programs for all kinds of customers. The modeling of the IBDR program for the residential customers, including the loads of the customer to reduce the power consumption, are carried out in the intelligent environment, thereby reducing the customer bill. The IBDR models for handling DR problems, with optimization techniques such as the Stackelberg game, GAMS, CPLEX, and others, are used in SG to obtain a better result of demand reduction and cost minimization [7,8,9,10]. In recent years, game-theory-based algorithms have been implemented to solve the IBDR strategy bidding model from the SP and GO ends [11,12,13]. IBDR problems with pricing schemes such as day-ahead pricing and intra-day renewable energy sources for procuring electricity at a minimal cost from the market were proposed in [14,15]. A multi-stage algorithm involving uncertain renewable sources, for serving the loads, considering entities such as GO and SP, was proposed in [16,17,18,19]. An integrated IBDR model considering the uncertainties coupled with the renewable sources [20,21] was proposed to improve the profit obtained by the customer [22]. An experimental setup based on flexible incentive-based DR to reduce critical demand and maximize consumer profits was proposed by Luo Zhe [23]. The IBDR was programmed using data mining for a virtual power plant. The authors in [24] proposed the IBDR problem for the Danish low-voltage (LV) grid with battery storage devices. Deepan et al. [25] proposed a novel self-reporting baseline estimation and outperformed the method for solving the IBDR program involving aggregators and consumers. The optimization techniques for effectively handling more variables maximization and minimization problems along with parameter selection and tuning are dealt with in [26,27,28,29].

Table 1. Highlights of the literature survey.

Reference	Entities/Participants	Pricing Schemes	Objective	Methodology and Simulation Tools
[3]	LSE and retail customers	Flat rate pricing	Optimizing social welfare	CPLEX
[4]	Service provider and customer	A day ahead of electricity pricing	Optimal incentives for SPs	Stackelberg game theory-GAMS tool
[6]	Utility and customers	Spot pricing	Maximizing benefits of retailers	MATLAB Yalmip toolbox
[7]	Grid operator, multi-service provider and customer	Incentive based pricing	Resource utilization in minimizing cost and maximizing profit of operators	Stackelberg game approach
[12]	Service provider and end-user	Real-time pricing	Peak demand and electricity bill reduction	MATLAB toolbox for optimization
[16]	Retailer and end-user	A day ahead of electricity pricing	Minimizing peak demand and finding hourly financial incentives for customers	NSGA II
[19]	LSE and ISO	Real-time market	LSE net revenue maximization	CPLEX

summarizes the highlights of the surveyed literature based on the objective of the DR problem, the entities considered, adapted pricing schemes, and optimization technology. The effect of inclusion and the interaction of the three market entities together has not been analyzed, although many algorithms were developed to solve IBDR. The influence of the customer parameters on the IBDR programs has also not been adequately shown. Furthermore, the literature has not concentrated on the IBDR work on 24-h timing, including all of the entities.

Accordingly, the proposed work compares the changes involved in the inclusion of various entities in the IBDR program using a novel SPSO algorithm. The IBDR problem designed concerning Case 1 (SP-customers) and Case 2 (GO-SP-industrial consumers (IC) customers) was optimized using SPSO algorithms, and was compared with the Stackelberg distributed algorithm for an hour and an entire day. A sensitivity analysis was also adapted by altering the customer parameters, namely the discomfort factor, the customer attitude to demand reduction, and the magnitude of demand change required to review the impact of IBDR. Furthermore, payment analysis was made to focus on the benefits obtained by the entities by varying the customer parameters.

The content of this paper is organized as follows: Section 2 elaborates on the problem formulation of IBDR and Case 1. Section 3 focuses on the problem formulation of Case 2 of the IBDR problem, followed by a briefing on the optimization technique adapted to solve the IBDR problem in Section 4. Section 5 discusses the results obtained for the two different cases. Finally, Section 6 presents the conclusions.

2. Problem Formulation of IBDR with One SP and Two Customers: Case 1

This section modelled and formulated an IBDR problem including GO, SP, IC and customers to maximize or minimize the utility function depending on the entities considered and the demand reduction.

An SP sells electricity to its retail customers and procures their capacity through demand reduction. The SP can even sell the procured capacity and gain profit. The SP provides incentives to those customers who agrees to reduce their demand when they are told to do so by the SP. In this work, a system with one SP and two consumers is considered, and the interaction between them is modelled using Stackelberg’s game theory.

a.

Customer model

Let $N$ be the total number of customers. Here, $N$ is taken to be 2. Every customer $i,$ when provided with incentives from the SP, tries to increase their demand reduction to gain more incentives. Here, time $t$ ranges from 1 to T, where T is taken as 24. The customer will aim to maximize the utility function by using their demand reduction, and the above can be framed as follows [4]:

$$\underset{D_i}{\max } U_i={\displaystyle \sum }_{t=1}^T{D}_{i,t}\times {\pi }_t-{\mu }_i\times {\displaystyle \sum }_{t=1}^T{\phi }_{i,t}\left(D_{i,t}\right)$$

(1)

which is subjected to

$$0\le D_{i,t}\le D_{i,t}^{tar}-D_{i,t}^{min}, \forall i\in N,\forall t\in T$$

(2)

Equation (1) represents the customer utility function, which has to be maximized using demand reduction as the variable. The constraint given by Equation (2) restricts the demand reduction from going beyond $D_{i,t}^{tar}-D_{i,t}^{min}$, which represents the available quantity of demand reduction. The first term of Equation (1) means the income gained by the customer by reducing the demand $D_{i,t}$ for the incentive ${\pi }_t$. The second term represents the discomfort of the customer. The weight factor ${\mu }_i$ decides the level of discomfort each customer can accept. The small value of ${\mu }_i$ represents less importance given to discomfort. The dissatisfaction cost (${\phi }_{i,t}$) represents the customer’s discomfort involved in demand reduction. The dissatisfaction cost is modelled as follows:

$${\phi }_{i,t}(D_{i,t})=\frac{{\theta }_i}{2}\left(D_{\left(i,t\right)}^2+{\lambda }_i\times D_{i,t}\right){\theta }_i>0, {\lambda }_i>0$$

(3)

${\theta }_i$ and ${\lambda }_i$ are parameters set based on the customer’s attitude towards demand reduction.

b.

Service provider model

The SP gains profit by selling the capacity procured from the customers, and can profit by procuring the capacity (i.e., the demand reduction) at a minimal incentive. Thus, the SP aims to minimize the utility function using the incentive offered as the variable. This can be modeled as follows [4]:

$$\underset{\pi }{\min }{U}_{SP}={\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^N{D}_{i,t}\times {\pi }_t-{\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^N{D}_{i,t}\times P_t$$

(4)

which is subjected to

$${\pi }_t^{min}\le {\pi }_t\le {\pi }_t^{max}, \forall t\in T$$

(5)

$${\displaystyle \sum }_{i=1}^N{D}_{i,t}\ge D_t^{req}, \forall t\in T$$

(6)

The first term of Equation (4) represents the incentive payments given by the SP to the consumers according to their demand reductions, and the second term represents the amount gained by the SP in selling the procured capacity to the market.

c.

Stackelberg game formulation and analysis

The interaction between the SP and the customers is framed as a Stackelberg game. Here, the SP acts as a leader and sets incentives for its customers; thus, a one-leader N-follower Stackelberg game is framed (here, one leader and two followers). The SP act provides incentives as a motivation for the customers to reduce their demand. The optimization equations are solved using PSO. The SP model is solved to obtain the optimal incentive set by the SP to its customers, and this is utilized in solving the customer model, thereby obtaining the corresponding optimal demand reduction; this iteration process tries to reach the global condition. The customer and SP models are reformulated using the Stackelberg theory. If the incentive value set by the leader (SP) is identified, then the customer’s optimal demand reduction can be obtained.

d.

Optimal solution for customers

By solving the first-order derivative of the Equation (1), concerning $D_{i,t}$, the optimal value of the demand reduction can be obtained as:

$$D_{i,t}=\frac{{\pi }_t-{\mu }_i\times {\lambda }_i}{{\mu }_i\times {\theta }_i}$$

(7)

The value of the second-order derivative of Equation (1) for $D_{i,t}$ gives $-{\mu }_i{\theta }_i<0$. This equation is always negative; as such, the customer utility function is strictly concave for the feasible values of $D_{i,t}$. The optimal value of ${\pi }_t$ obtained from the SP is substituted in Equation (7) to obtain the optimal demand reduction.

e.

Optimal solution for the service provider

Equation (7) is substituted in Equation (4), and the SP utility function becomes

$$\underset{\pi }{\min }{U}_{SP}={\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^N\left(\frac{{\pi }_t-{\mu }_i\times {\lambda }_i}{{\mu }_i\times {\theta }_i}\right)\times {\pi }_t-{\displaystyle \sum }_{t=1}^T{\displaystyle \sum }_{i=1}^N\left(\frac{{\pi }_t-{\mu }_i\times {\lambda }_i}{{\mu }_i\times {\theta }_i}\right)\times P_t$$

(8)

which is subjected to

$${\pi }_t^{min}\le {\pi }_t\le {\pi }_t^{max}, \forall t\in T$$

(9)

The constraints can be modified by arranging equation (7) as

$${\pi }_t=D_{i,t}\left({\mu }_i\times {\theta }_i\right)+{\mu }_i\times {\lambda }_i$$

(10)

Constraints (5) and (6) can be modified using the above-obtained equations. Thus, the value of ${\pi }_t^{min}$ and ${\pi }_t^{max},$ in (9), can be replaced by Equations (11) and (12).

$${\pi }_t^{max}=min\left({\pi }_t^{max},min \left\{\left(\left(D_{i,t}^{tar}-D_{i,t}^{min}\right)\times {\mu }_i\ast {\theta }_i+{\mu }_i\times {\lambda }_i\right)\forall i\in N\right\}\right)$$

(11)

$${\pi }_t^{min}=max\left({\pi }_t^{min},max\left\{{\mu }_i\times {\lambda }_i,\forall i\in N\right\}, \frac{D_t^{req}+{\sum }_{i=1}^N\frac{{\lambda }_i}{{\theta }_i}}{{\sum }_{i=1}^N\frac{1}{{\theta }_i}}\right)$$

(12)

The optimization equation in (8) is solved using PSO, and the optimal incentive is obtained.

3. Problem Formulation of IBDR with GO, ICs, SPs and Consumers: Case 2

The role of the GO in IBDR is re-studied in this section. Here, IBDR is implemented in a system consisting of GO, ICs, and SPs under the GO, and the customers under each SP. The GO sets the incentive for the ICs and the SPs. The GO procures the capacities from its SPs and ICs to meet the demand deficit. The SPs set the incentive for the customers under them. The SPs procure the capacity from the customers regarding demand reduction and sell it to the GO. The GO meets the required demand deficit of the day either through the generators or by procuring the capacity from the SPs and ICs (i.e., through their demand reductions), and the GO aims to reduce the cost of procuring these capacities. Figure 1 depicts the role of each domain in the smart grid. The mathematical models of the entities are framed and solved using the Stackelberg game theory, where the outcome of the leader is first obtained (i.e., the incentive of GO), and then this outcome is utilized to solve the model of the IC and the customers of the SP for demand reduction.

a.

Industrial consumer model

The IC aims to maximize its utility function, thereby maximizing the gain obtained as an incentive from the GO. The IC’s utility function involves the gain obtained by the energy consumed and the incentive profit obtained from the GO for demand reduction. Let L be the total number of ICs. For each IC_l, where $l\in L$, the utility function is given by [15].

$$\underset{D_{IC,l}}{\max }{U}_{IC,l}={\displaystyle \sum }_{t=1}^T{\Psi }_l\left(D_{l,t}^{ava}-D_{IC,l,t}\right)+D_{IC,l,t}\times {\pi }_{GOIC,t}$$

(13)

which is subjected to

$$0\le D_{IC,l,t}\le D_{l,t}^{ava}$$

(14)

The constraint in (14) restricts the value of the demand reduction within the range of the available amount of demand reduction, and ${\Psi }_l$ is the term used to describe the ICs’ profit; it is calculated by the following formula:

$${\Psi }_l=\left\{\begin{array}{c}{\omega }_l\times \left(D_l^{ava}-D_{IC,l}\right)-\frac{{\sigma }_l}{2}\times {\left(D_l^{ava}-D_{IC,l}\right)}^2 if 0\le \left(D_l^{ava}-D_{IC,l}\right)\le \frac{{\omega }_l}{{\sigma }_l}\\ \frac{{\omega }_l{}^2}{2{\sigma }_l} if \left(D_l^{ava}-D_{IC,l}\right)\ge \frac{{\omega }_l}{{\sigma }_l}\end{array}\right.$$

(15)

${\sigma }_l$ and ${\omega }_l$ denote the rate and magnitude of the profit change of the IC when the power consumption of the IC is varied. As ${\sigma }_l$ increases, the IC gains more by selling its resource to the GO.

b.

Service provider model

The SP procures capacity from its customers by giving incentives to them, and it sells the procured capacity to the GO at the incentive rate the GO gives. As such, the utility function of the SP involves these terms. The SP aims to maximize its profit obtained by trading with the GO and its customers. Here, if the SP model is compared with Case 1, the second term of the equation is modified into Equation (16). In Case 1, the second term describes the capacity sold by the SP to the market at the electricity pricing, whereas in (16), it describes the capacity sold by SP to the GO at the incentive rate given by GO. Let K be the total number of SPs, and for each SP, the utility function can be written as follows [15]:

$$\underset{{\pi }_{SP,k} D_{SP,k}}{\max } U_{SP,k}={\displaystyle \sum }_{t=1}^T{D}_{SP,k,t}\times {\pi }_{GOSP,t}-D_{SP,k,t}\times {\pi }_{SP,k,t}$$

(16)

which is subjected to

$$D_{SP,k,t}={\displaystyle \sum }_{i\in N_k}{D}_{i,k,t}$$

(17)

The customer receives incentives from SP in return for their demand reduction. The utility function of the customers will also involve the dissatisfaction cost. While comparing Equation (18) with Case 1 customer model, it could be seen that the number of SPs considered in Case 2 is more than that in Case 1, and the incentives the SPs give to their customers differ. As such, the customers belonging to different SPs receive different incentives. The utility function of the customers is written as follows [4,7]:

$$\underset{ D_{i,k}}{\max } U_{i,k,t}={\displaystyle \sum }_{t=1}^T{D}_{i,k,t}\times {\pi }_{SP,k,t}-{\mu }_{i,k,t}\times {\phi }_{i,k,t}$$

(18)

which is subjected to

$$0\le D_{i,k,t}\le D_{i,k,t}^{ava}$$

(19)

The dissatisfaction cost function ${\phi }_{i,k,t}$ is a function of $D_{i,k},$ and the level of discomfort that a customer might experience due to the reduction of demand is modeled by ${\phi }_{i,k}$.

$${\phi }_{i,k}=\frac{{\theta }_{i,k}}{2}\left(D_{\left(i,k\right)}^2+{\lambda }_{i,k}\times D_{i,k}\right),{\theta }_{i,k}>0, {\lambda }_{i,k}>0$$

(20)

Here, ${\theta }_{i,k}$ and ${\lambda }_{i,k}$ are the parameters set based on the customer’s attitude towards demand reduction. By increasing the value of ${\theta }_{i,k}$, the customer is more reluctant towards the demand reduction.

c.

Grid operator model

The GO tries to minimize the cost of capacity procurement. Let the expected demand deficit be taken as $D^{req}$. The GO compensates for the demand deficit by acquiring capacities from SP and IC. The utility function of the GO can be framed as follows [15]:

$$min C_{GO}={\displaystyle \sum }_{t=1}^T{C}_{gen}\left(G\right)+{\pi }_{GOIC,t}\times {\displaystyle \sum }_{i\in L}{D}_{IC,l,t}+{\pi }_{GOSP,t}\times {\displaystyle \sum }_{i\in L}{D}_{SP,k,t}$$

(21)

which is subjected to

$${\pi }_{GO,t}^{min}\le {\pi }_{GOSP,t}\le {\pi }_{GOIC,t}\le {\pi }_{GO}^{max}$$

(22)

$$G=D^{req}-{\sum }_{i\in L}{D}_{IC,l}-{\sum }_{i\in L}{D}_{SP,k}$$

(23)

The value of the cost of generating the required quantity $G$ is calculated as follows:

$$C_{gen}\left(G\right)=a\left(G^2\right)+b\left(G\right)+c$$

(24)

In Equation (21), the first term represents the cost required for generation, and the second and the third terms represent the cost of procuring the capacities from the IC and SP.

d.

Stackelberg game formulation and analysis

The interaction between the various entities involved in IBDR is modelled using Stackelberg’s game theory. Including GO and IC in the problem makes it a leader–multi-follower game, increasing the complexity compared to Case 1. The GO model is solved, and the optimal incentive set by the GO for IC and SP is found, and then those optimal incentive values are utilized for the calculation of the optimal demand reductions. The GO fixes the incentives for the IC and SP using the following equations:

$${\pi }_{GOSP}={\pi }_{GO}$$

(25)

$${\pi }_{GOIC}=\rho \times {\pi }_{GO}, 0\le \rho \le 1$$

(26)

e. Optimal solution for the industrial consumer

Using the first-order derivative of the utility function in (13) with respect to $D_{IC,l}$ when equated to zero, we can obtain the equation for $D_{IC,l}$, as follows:

$$D_{IC,l}=D_l^{ava}-\frac{{\omega }_l}{{\sigma }_l}+\frac{\rho \times {\pi }_{GO}}{{\sigma }_l}$$

(27)

The total demand reduction of all of the ICs could be summed as

$${\displaystyle \sum }_{i\in L}{D}_{IC,l}={\displaystyle \sum }_{l\in L}{D}_l^{ava}-\frac{{\omega }_l}{{\sigma }_l}+{\displaystyle \sum }_{l\in L}\frac{\rho \times {\pi }_{GO}}{{\sigma }_l}$$

(28)

For convenience, the constant terms in the equation are taken as follows:

$$\eta ={\displaystyle \sum }_{l\in L}{D}_l^{ava}-\frac{{\omega }_l}{{\sigma }_l},\gamma ={\displaystyle \sum }_{l\in L}\frac{1}{{\sigma }_l}>0$$

(29)

For Equation (29), when it is substituted into (28), we obtain

$${\displaystyle \sum }_{i\in L}{D}_{IC,l}=\eta +\gamma \times \rho \times {\pi }_{GO}$$

(30)

The constraint in (14) can be regulated to make the IC contribute the minimum load by replacing the zero to $D^{min}$.

$$D^{min}=max\left(0,min\left(D_l^{ava}-\frac{{\omega }_l}{{\sigma }_l}\right)\right)$$

(31)

The optimal demand reduction of IC can be found by substituting the optimal GO incentive using (30).

f.

Optimal solution for the service provider and its customers

The optimal demand reduction of the SP can be calculated as follows:

$$D_{SP,k}=\frac{1}{2}{\pi }_{GO}\times {{\displaystyle \sum }}_{i\in N_k}\frac{1}{{\mu }_{i,k}\times {\theta }_{i,k}}-\frac{1}{2}{{\displaystyle \sum }}_{i\in N_k}\frac{{\lambda }_{i,k}}{{\theta }_{i,k}}$$

(32)

The total demand reduction of all SPs could be found as follows:

$${\displaystyle \sum }_{k\in K}{D}_{SP,k}=\frac{1}{2}{\pi }_{GO}\times {\displaystyle \sum }_{k\in K}{\displaystyle \sum }_{i\in N_k}\frac{1}{{\mu }_{i,k}\times {\theta }_{i,k}}-\frac{1}{2}{\displaystyle \sum }_{k\in K}{\displaystyle \sum }_{i\in N_k}\frac{{\lambda }_{i,k}}{{\theta }_{i,k}}$$

(33)

For convenience, the constant terms in the equation are taken as follows:

$$\alpha ={{\displaystyle \sum }}_{k\in K}{{\displaystyle \sum }}_{i\in N_k}\frac{1}{{\mu }_{i,k}\times {\theta }_{i,k}}>0,\beta ={{\displaystyle \sum }}_{k\in K}{{\displaystyle \sum }}_{i\in N_k}\frac{{\lambda }_{i,k}}{{\theta }_{i,k}}>0$$

(34)

For Equation (34), when substituted in (33), we obtain

$${\displaystyle \sum }_{k\in K}{D}_{SP,k}=\frac{\alpha }{2}\left({\pi }_{GO}\right)-\frac{\beta }{2}$$

(35)

In order to solve the customer equation, the second derivative of Equation (18) with reference to $D_{i,k}$ is taken and equated to zero. The demand reduction of each customer under each SP can be found as follows:

$$D_{i,k}=\frac{{\pi }_{SP,k}-{\mu }_{i,k}\times {\lambda }_{i,k}}{{\mu }_{i,k}\times {\theta }_{i,k}}$$

(36)

g. Optimal solution for the grid operator

The cost function of the GO in (21) can be rewritten by substituting (24) and (23) into it, as follows:

$$\begin{gathered} min C_{GO}=a\left({(D^{req}-{\displaystyle \sum }_{i\in L}{D}_{IC,l}-{\displaystyle \sum }_{i\in L}{D}_{SP,k})}^2\right)+b\left(D^{req}-{\displaystyle \sum }_{i\in L}{D}_{IC,l}-{\displaystyle \sum }_{i\in L}{D}_{SP,k}\right) \\ +c+{\pi }_{GOIC}\times {\displaystyle \sum }_{i\in L}{D}_{IC,l}+{\pi }_{GOSP}\times {\displaystyle \sum }_{i\in L}{D}_{SP,k} \end{gathered}$$

(37)

By substituting (29) and (34) into (37) and equating the first derivative of the obtained equation to zero, we can realize the optimal incentive of the GO.

$${\pi }_{GO}^{*}=\frac{2a \left(D^{req}+0.5\beta -\eta \right)\times \left(0.5\alpha +\gamma \rho \right)+b\left(0.5\alpha +\gamma \rho \right)+0.5\beta -\rho \eta }{2a{\left(0.5\alpha +\gamma \rho \right)}^2+\alpha +2\gamma {\rho }^2}$$

(38)

The obtained optimal incentive value of the GO is utilized to obtain the optimal demand reduction of ICs and SPs.

h.

Stackelberg distributed algorithm

The parameters of the IC and SP customers must be disclosed to the GO, which is difficult in practice. As such, the distributed algorithm is used [15]. Here, the incentive of the GO is used as the variable, and SP and IC demand reductions are found.

The initial value of the GO incentive ${\pi }_{GO}^{*}$ is initialized to ${\pi }_{GO}^{min}$. A value of $\rho$ is chosen, and the value of ${\pi }_{GOIC} and {\pi }_{GOSP}$ are calculated. Then, the initial values of $D_{IC,l}^{*}$ and $D_{SP,k}^{*}$ are calculated using Equations (27) and (32), respectively.

The initial value of the procurement cost of the GO is calculated as

$$\begin{gathered} C_{GO}^{*}=(a{(D^{req}-{\displaystyle \sum }_{i\in L}{D}_{IC,l}+{\displaystyle \sum }_{i\in L}{D}_{SP,k})}^2 +b\left(D^{req}-{\displaystyle \sum }_{i\in L}{D}_{IC,l}+{\displaystyle \sum }_{i\in L}{D}_{SP,k}\right) \\ +c)+{\pi }_{GOIC}\times {\displaystyle \sum }_{i\in L}{D}_{IC,l}+{\pi }_{GOSP}\times {\displaystyle \sum }_{i\in L}{D}_{SP,k} \end{gathered}$$

(39)

These initial values are then used to calculate the optimal value of $C_{GO}$ with ${\pi }_{GO}$ as the variable which is updated for every iteration, as shown below:

• Step 1: for the iteration m = m + 1;
• Step 2: update the value of ${\pi }_{GO}^m$ using

  $${\pi }_{GO}^{m+1}={\pi }_{GO}^m+\Delta {\pi }_{GO}$$

  (40)

  where

  $$\Delta {\pi }_{GO}=\delta \left(e^{\varpi \left|C_{GO}^m-C_{GO}^{m-1}\right|}-1\right)$$

  (41)
• Step 3: the value of $D_{IC,l}^{m+1}$ and $D_{SP,k}^{m+1}$ are updated from Equations (27) and (32), respectively, and the ${\pi }_{GO}^{m+1}$ value;
• Step 4: with the values found in step 3, the value of $C_{GO}^{m+1}$ is calculated as follows:

  $$\begin{array}{ll}{C}_{GO}^{m+1}=\left(a\right(D^{req}& -{\displaystyle {\displaystyle \sum }_{i\in L}} D_{IC,l}^{m+1}+{\displaystyle {\displaystyle \sum }_{i\in L}}{D}_{SP,k}^{m+1}){}^2\\ & +b\left(D^{req}-{\displaystyle {\displaystyle \sum }_{i\in L}} D_{IC,l}^{m+1}+{\displaystyle {\displaystyle \sum }_{i\in L}}{D}_{SP,k}^{m+1}\right)+c)\\ & +{\pi }_{GOIC}\times {\displaystyle {\displaystyle \sum }_{i\in L}} D_{IC,l}^{m+1}+{\pi }_{GOSP}\times {\displaystyle {\displaystyle \sum }_{i\in L}}{D}_{SP,k}^{m+1}\end{array}$$

  (42)
• Step 5: f $C_{GO}^{m+1}\le C_{GO}^{*}$ and ${\pi }_{GO}^{m+1}\le {\pi }_{GO}^{*}$, then update the values of GO as ${\pi }_{GO}^{*}={\pi }_{GO}^{m+1}$ and $C_{GO}^{*}=C_{GO}^{m+1}$;
• Step 6: end if;
• Step 7: end for;
• Step 8: these steps are repeated until the condition shown in (43)

  $$\left|C_{GO}^{m+1}-C_{GO}^m\right|\le \epsilon$$

  (43)
• Step 9: the equilibrium is reached when the value of $C_{GO}$ does not decrease further.

4. Optimization Technique

Optimization techniques are utilized to find the best results from the set of feasible solutions. In this work, the interaction between the market entities is modelled using Stackelberg game theory, and the minimization and maximization equations of the entities are solved using PSO [30]. In the IBDR problem considered, there is a need for a repeated iteration with either maximization or minimization equations for the leader (GO) and the followers (IC, SP and customers) with two to three variables. As the Stackelberg game theory-based IBDR problem considered for optimization is a complicated iterative problem, the inertia-based PSO is considered for optimization in order to reduce its complexity.

In Case 1, the maximization equation of the customer is solved by setting the demand reduction as a variable, and the minimization equation of the SP is solved by setting the incentive as a variable. In Case 2, the industrial customer’s maximization equations are solved using the demand reduction as the variable; the SP is maximized by compromising the GO and customer by varying the demand reduction from customers and the incentives. Furthermore, the GO tries to minimize the cost of capacity procurement with the demand reduction obtained from IC and SP by varying the incentives.

The PSO procedure to solve the formulated Stackelberg-based problem for all of the entities is given below:

• Step 1: initialize a set of particles in the search space;
• Step 2: each particle will have a position and velocity. Initially, we generate the positions randomly based on the minimum and maximum limits of the variables shown in

  Table 2. Variables and their bounds for Case 2’s optimization.

  Entities	Variables	Lower and Upper Bounds	Number of Variables
  Grid operator	Incentive (πigo)	(3, 10)	1
  Industrial consumer	Demand reduction of IC1 (DIC1) Demand reduction of IC2 (DIC2) Demand reduction of IC3 (DIC3)	(0, 45.4) (0, 36.2) (0, 56.5)	3
  Service provider	Incentive of SP1 (πSP1) Incentive of SP2 (πSP2)	(3, 10) (3, 10)	2
  Consumer	Demand reduction of customer 1 (Dk,1,t) Demand reduction of customer 2 (Dk,2,t) Demand reduction of customer 3 (Dk,3,t)	(0, 11.35) (0, 16.55) (0, 12.77)	3

  ;
• Step 3: for each particle, evaluate the incentives and utility function (objective) as per Equations (13), (16), (18) and (21), and in the entities, each particle will have a position and velocity;
• Step 4: store the local and global best values (pbest and gbest);
• Step 5: if the utility function for the new particle changes, based on maximization or minimization of the problem, update the pbest and gbest values;
• Step 6: update the inertia weight factor by using w = w × wdamp;
• Step 7: update the position and velocity of the particle. The velocity update is given in (44);

  Velocity = w × velocity + C₁ × rand () × Pbest + C₂ × rand () × gbest

  (44)
• Step 8: velocity clamping is performed to maintain the velocity of the particle within the limit;
• Step 9: check the termination condition; if satisfied, stop; else, go to Step 3.

The optimization is carried out by considering the maximum iteration count as 1000, the number of particles as 100, and the acceleration coefficients C₁ and C₂ as 1.5 and 2. In this work, the inertia weight is dynamic, with a weight-damping ratio of 0.99.

5. Results and Discussions

The IBDR problem for Cases 1 and 2 is analyzed and optimized with the load data and the customer parameters considered.

a.

Results of Case 1 with one SP and two customers

In this case, one SP and two customers are considered. The minimum and target demands of customers 1 and 2 for each hour are illustrated in Figure 2 and Figure 3. The customer-related parameters (${\mu }_i$,${\theta }_i$,${\lambda }_i$) for the two customers are shown in

Table 3. Parameters of Case 1, for customers 1 and 2.

Parameters	Customer 1	Customer 2
$\mu$	(0.8, 1)	(0.8, 1)
$\theta$	3.0	4.5
$\lambda$	10.0	10.0

.

The hourly electricity market pricing required for the calculation of the SP’s utility function is represented in Figure 4.

The value of $D_t^{req},$ i.e., the required demand reduction of the SP and the available demand reduction, are given in Figure 5.

The optimal incentive of the SP is obtained using PSO for two different values of $\mu$, namely 0.8 and 1, for both customers. The optimal SP incentive and the optimal mean demand reductions of customers 1 and 2 are taken and plotted. The obtained optimal solutions for the corresponding incentives are given in Figure 6 and Figure 7. In order to ensure that the Stackelberg game is implemented correctly, Case 1 is tested with an existing system [4], and the results are obtained. The comparisons of the results for $\mu$ = 1.0 and 0.8 are listed in

Table 4. Comparison with the existing system for $\mu$ = 1.0.

Incentive by SP ($)		Demand Reduction (MW)		Incentive by SP ($)		Demand Reduction (MW)
For one hour (16th hour)				For 24 h
Stackelberg [4]	SPSO	Stackelberg [4]	SPSO	Stackelberg [4]	SPSO	Stackelberg [4]	SPSO
38	39	18.2	18.3	486.8	425.2	195.4	240.59

and

Table 5. Comparison with the existing system for $\mu$ = 0.8.

Incentive by SP ($)		Demand Reduction (MW)		Incentive by SP ($)		Demand Reduction (MW)
For one hour (16th hour)				For 24 h
Stackelberg [4]	SPSO	Stackelberg [4]	SPSO	Stackelberg [4]	SPSO	Stackelberg [4]	SPSO
32	32	18.5	19	439	394.8	230.02	253.44

.

The tables show that the results obtained are better than the existing system results when optimization is performed using SPSO to solve the IBDR program.

The value of the weight factor represents the importance given by the customer to the discomfort involved in reducing the demand. A small value of ${\mu }_i$ represents less importance given to discomfort. This can be seen in Figure 7. The demand reduction curve for $\mu$ = 0.8 is high compared to that for $\mu$ = 1.0, which shows that the customers with a smaller $\mu$ value impose less weightage on the dissatisfaction cost, i.e., they can afford to have more demand reduction, without giving too much consideration to the discomfort caused. However, the customers with a higher $\mu$ value impose more weightage on dissatisfaction, such that they reduce their demand less compared to those with lower values of $\mu$.

b.

Sensitivity analysis for Case 1

Here, two conditions for which the value of $\mu$ is different for both customers are considered. At first, the values of ${\mu }_1=$ 0.8 and ${\mu }_2=$ 1.0 were taken. It could be inferred from the output graphs in Figure 8 and Figure 9 that for the same optimal incentive curve obtained by the SP, the customer with a $\mu$ = 0.8 value shows more interest in demand reduction than the customer with $\mu$ = 1.0. This also happens when the condition is vice-versa; i.e., when ${\mu }_1=$ 1.0 and ${\mu }_2=$ 0.8, the demand reduction is more for customer 2, as depicted in Figure 10 and Figure 11.

c.

Case 2 with GO, ICS and SPS optimized using Stackelberg-distributed and SPSO algorithms

This case considers GO, IC, SP and its customers. The number of ICs is three and the number of SPs is two, with each SP being connected to three customers. The generator coefficients used are a = 0.2, b = 0, c = 0, and the value of $\rho$ is chosen as 0.6. The value of $\delta$ is taken as 2.5, $\varpi \mathrm{is} \mathrm{taken}$ as 0.13, $\epsilon$ is taken as 10⁻⁴, and μ is taken as 1. Initially, the results are tested for SPSO and the Stackelberg-distributed algorithm for an hour using the test data and parameters specified in

Table 6. Test data used in Case 2 for the ICs [15].

ICs	IC1	IC2	IC3
Load (kW)	45.4	36.2	56.5
$\sigma$	0.1	0.12	0.13
$\omega$	8	8	8

and

Table 7. Test data used in Case 2 for the SPs.

SPs	SP1			SP2
End Users	User 1	User 2	User 3	User 1	User 2	User 3
Load (kW)	11.4	7.5	14.4	5.5	13.7	9.2
$\theta$	3.0	4.5	5.0	4.0	5.5	6.0
$\lambda$	2.0	2.0	2.0	3.0	3.0	3.0
$\mu$	1.0	1.0	1.0	1.0	1.0	1.0

.

Figure 12 and Figure 13 depict the optimal incentive and total cost of GO executed using the Stackelberg PSO. The results show that the SPSO can obtain the optimal converged output obtained by the Stackelberg distributed algorithm [31]. As per [15], the optimal GO incentive should be 7.27 cents/kWh, and that obtained by the distributed algorithm was 7.3, whereas with SPSO, the obtained global optimal solution was exactly 7.273 cents/kWh, as shown in Figure 12. From the results, it can be observed that the optimal incentive for the GO is 7.27 cents/kWh [4]. With the distributed algorithm, the result obtained was 7.29, with an error percentage of 0.275 [15], whereas with SPSO, the result was 7.273, with an error percentage of 0.041.

Furthermore, the optimal incentive of the SPs obtained is shown in Figure 14, and the demand reduction of the three customers under SP₁ and SP₂ is given in Figure 15 and Figure 16, respectively. The comparison of the obtained optimal incentive for GO, IC, SP, and the total cost of GO with SPSO and the distributed algorithm are tabulated in

Table 8. Result comparison of Case 2’s Stackelberg-PSO and distributed algorithm.

Parameters	Stackelberg
	PSO Algorithm		Distributed Algorithm
Grid operator incentive (cents/kWh)	7.273		7.290
Total cost ($)	5.028		5.060
IC Incentive (cents/kWh)	4.363		4.38
SP incentive to users (cents/kWh)	SP1	SP2	SP1	SP2
	4.665	5.0	4.7	5.1

. Compared with the Stackelberg distributed algorithm, the SPSO can reach the exact optimal solution. Similarly, Case 2 with GO, IC, SP and its customers is considered for 24 h, and optimization is carried out using both Stackelberg-distributed and SPSO algorithms. Figure 17, Figure 18 and Figure 19 illustrate the evaluated load data for Case 2 for 24 h of the ICs and the customers under SP₁ and SP₂.

The customer and IC parameters are the same as those in Table 6 and Table 7. The load data was taken for the entire day. These parameters remain the same throughout the day. The value of $\rho$ is taken as 0.6. The required demand reduction optimized for the whole day is obtained, as illustrated in Figure 20. This demand reduction was used in the modeling of GO required for the calculation of the quantity of power generated [32].

The optimal incentive of GO is found using the Stackelberg equation which was derived, and the optimized incentives are utilized in the evaluation of the total cost. The total cost is further minimized using SPSO. The optimized incentive and the total cost incurred by the GO in procuring the capacities are depicted in Figure 21. The entire study is conducted for a day; therefore, the incentives and demand reductions vary daily.

The demand reductions of each ICs in response to the optimal GO incentive are optimized using PSO, and the results are given in Figure 22a. The customers with a larger value of $\sigma$ are the ones contributing to more load reduction. It could be observed that customer 3, with a higher $\sigma$ value, contributes to a greater demand reduction. The optimized GO incentives are utilized, and the incentives for SP₁ and SP₂ are found. The optimized values of the incentives of the SPs are given in Figure 22b. It is seen that SP₂ has a higher incentive compared to SP₁. The corresponding demand reduction of each customer under SP₁ and SP₂ is depicted in Figure 22c,d. These figures show that the demand reduction of customer 1 is high compared to customers 2 and 3.

The effect of various parameters on the results is studied using sensitivity analysis. Before starting with the altering of the parameters, from the results obtained from Case 2, it could be inferred that the demand reduction of customer 1 of SP₁ is higher compared to that of customer 1 of SP₂, which is visible in Figure 22c,d. This result could be attributed to the fact of the $\theta$ value. The increase in the $\theta$ value means that the customer’s interest in demand reduction is low. From the input data of Table 6 and Table 7, it is visible that customer 1 of SP₁ has a lower θ value (i.e., 3.0) compared to customer 1 of SP₂ (i.e., 4.0); therefore, the demand reduction is high for customer 1 of SP1. Furthermore, from Figure 22a, it can be said that the ICs with a larger value of $\sigma$ will reduce more load. It could be seen that customer 3, with a higher $\sigma$ value, contributes to a greater demand reduction. $\mu$ for the SP customers and $\omega$ for the ICs are changed to investigate the changes caused by the parameters, and one hour’s (hour 16) output was observed and analyzed.

Table 9. Load data of the IC for the hour considered for the sensitivity analysis in Case 2.

ICs	IC1	IC2	IC3
Available load (kW)	45.4	36.2	56.5

and

Table 10. Load data of SP for the hour considered for the sensitivity analysis in Case 2.

SPs	SP1			SP2
Users	User 1	User 2	User 3	User 1	User 2	User 3
Load (kW)	11.54	7.47	9.29	11.35	16.55	12.77

present the input data of the performed analysis. The required demand deficit of the hour considered is 96 kW.

The values of $\omega$ of the ICs considered are 8 and 5 for the three ICs. The values of $\mu$ for the SP customers considered are 1 and 0.8. These changes in the outputs are also shown in

Table 11. Optimized output with changes in the $\omega$ value.

Parameters	$\mathit{\omega }=8$			$\mathit{\omega }=5$
Incentive of GO (cents/kWh)	7.3593			3.3383
Total cost of GO ($)	4.0397			2.0634
IC demand reduction (kW)	13.6714	4.9452	40.5349	8.2034	6.3111	34.7609

and

Table 12. Optimized output for changes in the $\mu$ value.

Parameters	$\mathit{\mu }=1.0$			$\mathit{\mu }=0.8$
Incentive of GO (cents/kWh)	7.3593			9.9254
Total cost of GO ($)	4.0397			5.2703
SP incentive (cents/kWh)	3.0415   3.9544			3.1819   3.8468
Demand reduction of SP1 customers (kW)	0.18	0.12	0.11	0.52	0.35	0.31
Demand reduction of SP2 customers (kW)	0.11	0.08	0.07	0.42	0.31	0.28

for IC and SP, respectively. The demand reduction of IC, the total cost of GO, and the incentive of GO are reduced when the value of $\omega$ is diminished.

The demand reduction of SP, and the total cost and the incentive of GO increase when the value of $\mu$ is reduced. The customer output of both of the SPs shows that as the value of $\mu$ is small, the customer shows more interest in demand reduction, and when observed closely, it is noted that the $\theta$ value also affects the demand reduction.

As $\theta$ increases, the value of the demand reduction decreases, which highlights that the customers show less interest in demand reduction as the value of $\theta$ increases.

A payment analysis is used to evaluate the benefits gained by each entity, and the results are tabulated. In Case 1, the demand reductions obtained by this approach for the entire day are much higher—the SP gains by selling these demand reductions at market prices. The incentive payment to be given to the customer is calculated. The incentive given by the SP to its customers is 5263 dollars for $\mu =$ 1.0. If the IBDR implementation was not performed, the SP must purchase the required demand reduction at the market price as tabulated in

Table 13. Payment analysis in Case 1.

Entity Considered	Incentive Benefits ($)
Incentives given by SP to customers in dollars	5263
SP purchasing $D_t^{req}$ at market price (with no IBDR) in dollars	6357.7

.

The gain obtained by SP₁ and SP₂ can be found in

Table 14. Payment analysis in Case 2.

Entity Considered	Incentive Benefits (Cents)
Incentives obtained by IC from GO (cents)	3900
Total incentives obtained by SP1 and SP2 from GO (cents)	237.4676
Total incentives given by SP1 and SP2 to its customers (cents)	110.69

as USD 126.77 cents. The total cost incurred by the GO from Figure 22 is USD 60.1813, which is USD 6018.13 cents; lesser than the cost obtained using generators to serve the entire demand. Therefore, all of the entities involved in the IBDR program is benefited.

6. Conclusions

A novel IBDR is implemented considering the viewpoint of the SP and the GO. It could be said that the introduction of the GO into the system brings many changes in how incentives are provided, and in the demand reductions being evaluated. The interactions between the various entities are modeled using the Stackelberg game, and the contradicting parameters are brought to equilibrium. In order to solve the optimization problems, PSO is used, which provides the best-optimized results. The interaction between the SP and its two customers is a one-leader, two-follower game, but due to the presence of multiple entities with the introduction of GO, the game becomes a little complex as a one-leader multi-follower game. For the first IBDR case implemented (considering only the SP and its two customers), the incentive of the SP and the corresponding demand reductions of its customers were optimized. For the second case, with the involvement of the GO, the IC, and the SP, the incentives of the GO, the total cost of the GO, the corresponding demand reduction of the ICs under GO, the incentives set by the SP to its customers, and the demand reductions of the customers under SP were optimized. A sensitivity analysis was conducted to study the influence of varying customer parameters on the proposed IBDR program. The results proved that various changes in the outcomes are obtained due to the effect of the customer parameters, and a change made to a parameter of one entity affects the results of other entities. In the first case, the discomfort weight factor and the customer’s attitude towards demand reduction are modified, and the outputs are monitored. In the second case, the discomfort weight factor and the customer’s attitude towards the demand reduction of the SP and the IC parameters are varied, and the results are evaluated. The results show that the IBDR proves itself as a valuable tool to help the SP and GO procure resource capacities, thereby enabling them to solve the demand deficit issue. This work can be extended in the future by increasing the number of customers, SPs and ICs, and by considering the effect of integrating renewable energy sources on the results.