Heuristic Retailer’s Day-Ahead Pricing Based on Online-Learning of Prosumer’s Optimal Energy Management Model

Abstract

Smart grids have introduced several key concepts, including demand response, prosumers—active consumers capable of producing, consuming, and storing both electrical and thermal energies—retail market, and local energy markets. Preserving data privacy in this emerging environment has raised concerns and challenges. The use of novel methods such as online learning is recommended to address these challenges through prediction of the behavior of market stakeholders. In particular, the challenge of predicting prosumers’ behavior in an interaction with retailers requires creating a dynamic environment for retailers to set their optimal pricing. An innovative model of retailer–prosumer interactions in a day-ahead market is presented in this paper. By forecasting the behavior of prosumers by using an online learning method, the retailer implements an optimal pricing scheme to maximize profits. Prosumers, however, seek to reduce energy costs to the greatest extent possible. It is possible for prosumers to participate in a price-based demand response program voluntarily and without the retailer’s interference, ensuring their privacy. A heuristic distributed approach is applied to solve the proposed problem in a fully distributed framework with minimum information exchange between retailers and prosumers. The case studies demonstrate that the proposed model effectively fulfills its objectives for both retailer and prosumer sides by adopting the distributed approach.

1. Introduction

1.1. Background and Motivation

With the rapid rise in global demand for energy consumption and ever-increasing human dependence on electrical energy, as well as growing environmental concerns over pollutant emission, and volatile energy prices, the exploitation of high-efficiency energy generation methods (such as combined heat and power (CHP)) and smart grid technologies are of paramount importance [1]. The spread of the smart grid and the employment of advanced metering infrastructures (AMI) have enabled a mutual connection between consumers and other entities, such as retailers [2]. The benefits of this capability have enabled the introduction of several key concepts, including demand response (DR), prosumers (active consumers capable of producing, consuming, and storing both electrical and thermal energies), retail markets, and local energy markets [3]. Unlike conventional power systems, consumers in such an environment can actively react to market price bids and sell their surplus generated energy, as well as minimize the costs of their needed energy [4]. Therefore, such consumer behavior can result in a more dynamic and competitive market. Likewise, as a profit-centered entity in a smart grid environment, the retailer seeks to maximize its profit by predicting the consumers’ behavior, optimal energy pricing, and meeting the consumers’ energy demand at minimum cost [5]. Based on these considerations, the computational burden of centralized optimization methods and the privacy of local (retail) market players are of particular concern [6]. The motive to avoid privacy breach of prosumers has directed the retailers toward new tools, including machine learning, which do not require all (private) information of players to ensure optimal pricing with minimum information exchange [7]. One of the less-studied issues in retail energy management is the prosumers’ participation in DR programs and cost minimization of energy needs from their (prosumers’) view. In studies related to short-term scheduling of retailer energy management, the prosumers help the retailer’s optimal pricing via participation in the DR program. The proposed problem is then solved in a centralized manner, assuming access to all players’ information [8]. In other words, these studies have violated the prosumers’ privacy from two aspects. First, although the DR program is considered for prosumers, it is implemented from the retailer’s view. In other words, the retailers are in control of the flexible loads and can shift the loads based on their goals without considering the prosumers’ privacy. In fact, they assume that the prosumers will certainly participate in the proposed DR program. From the second aspect, the retailer has access to all prosumers’ information. The proposed optimization problem is solved from the retailer’s viewpoint in a centralized manner, which further compromises the players’ information privacy.

This research presents a novel approach to explain the behavior of energy market participants in terms of balancing demand and supply, and ensuring privacy. Considering the abovementioned issues, presenting a decentralized approach to protect the prosumers’ privacy from both DR and data points of view seems to be an underaddressed issue. In the present paper, we aim to solve this problem by developing a learning-based approach.

1.2. Related Works

This section reviews the research conducted on the short-term scheduling of retailer’s energy management. The existing study gap is evaluated from three aspects: the voluntary participation of consumers in the DR program, privacy protection through minimal information exchange of consumers, and the use of online learning in retailer pricing. In the following, in three separate paragraphs, each of these research issues are examined in recent studies respectively.

The authors in [8] have comprehensively reviewed the literature regarding energy-management scheduling in long-term, medium-term, and short-term periods. These periods are listed in a table under the headings objective function, type of pricing, solving method, uncertainty, considering DR, and smart grid technologies. None of the scheduling periods included real-time pricing based on online learning from the reviewed papers in this reference. The authors of [9] have designed a short-term scheduling scheme for retailer’s energy management considering fixed, time of use, and real-time pricing strategies. In the proposed model, the consumers’ behavior is presented by the consumer to the retailer in the form of a price-quota curve, and the consumers can participate in the price-based DR program. In this type of modeling, the consumers are assumed to announce their behavior to the retailer by using the price-quota curve to maximize its profit. Such assumptions neglect the competitive market environment in which every entity pursues its benefit. In addition, the proposed problem is built on the assumption that the consumers will indeed participate in the developed DR program. An optimal bidding strategy for industrial consumers with cogeneration facilities, power-only and heat-only units for participation in the day-ahead electricity market is demonstrated in [10]. The proposed model is solved in a decentralized manner and from the retailer’s perspective. The authors have applied the information gap decision theory (IGDT) method to determine the optimal pricing strategies in the presence of market price uncertainty. The model obtains different expected profit and consumers’ price bid curves based on consumers’ risk-averse or risk-taker decision making. The authors of [11] have studied pricing and operation strategy along with the DR program for the microgrid retailer in an integrated energy system (IES). By simultaneous optimization of the retail price and microgrid distribution, the proposed model presents a dynamic pricing scheme that reflects the generation costs and promotes DR. Despite the comprehensive nature of the proposed algorithm, the problem is solved in a centralized manner under the inherent assumption that the retailer is responsible for DR and the loads have no role in determining their consumption schedule. The authors of [12] employed robust optimization (RO) in a centralized manner for price-based DR management of heat and electricity consumers under price uncertainty with the assumption of having all consumers’ information. The authors demonstrated a considerable increase in the daily cost of consumers without smart grid technology compared to those using the proposed model in real time. Comprehensive research from the perspective of considering a wide spectrum of electrical and thermal energy sources, including photovoltaic modules, wind turbines, CHPs, fuel cells, electric vehicles, electrical and thermal storages, power-only units, heat-only units, and DR program is presented in [13]. However, the retail price is assumed to be fixed, and the problem is solved from the microgrid operator’s view in a centralized manner.

In [14], the authors modeled a two-stage DR program for electricity retailers. Furthermore, they presented a two-layer optimal model for the purchase and sale of electricity retailers. In the upper layer, the retailers seek to minimize the power purchasing cost considering the uncertainty, whereas in the lower layer, the consumers aim to maximize their profit from power selling by implementing DR. In this study, DR is considered from the retailer’s view in both stages. Moreover, the players’ data privacy is violated due to utilizing the Karush–Kuhn–Tucker (KKT) method to convert the two-layer problem into a one-layer problem. The authors in [15] have proposed a novel pricing approach for a virtual energy station (VES) in a multicarrier energy system. In this model, the integrated DR is established considering the electrical and thermal loads. However, due to using KKT for model integration, this paper failed to address data privacy. An integrated retailer for energy management in the multicarrier energy market is considered in [16]. Authors in this study have particularly focused on uncertainty and risk. However, the data privacy was again not protected as the KKT method was applied. A bilevel Stackelberg-based model is developed in [17] for the interaction of retailers with consumers. The proposed problem was solved by using the KKT integration method, which ignores data privacy. Moreover, in this study, DR is addressed from the load’s perspective, considering different welfare-level scenarios. The authors of [18] evaluated the retailer’s dynamic pricing in the day-ahead market as a three-stage game, considering DR. The designed game was solved by integration by using the KKT method, and the consumers’ data privacy was not fulfilled. Authors in [19] have addressed the problem of retailers’ competition over prosumers as a multileader, multifollower game. DR is included in this study via considering the integrated energy hub. By relying on the KKT method, integrating the bilevel optimization problem and reaching the equilibrium point is achieved at the expense of a data privacy breach. The authors of [20] have modeled the retailer’s short-term decision-making as a two-stage bilevel problem considering the integrated DR. In this study, DR is undertaken by an independent entity, though it is thought that DR implementation should be driven privately by end users’ own desires. Furthermore, in solving the two-level problem by KKT, the lower-level optimization problem is transferred to the upper level, which does not protect consumers’ information privacy. In another study, the relation between retailer and distributed renewable energy (DER) market is considered as a bilevel game in which the problem is reduced to a one-level mixed-integer linear programming (MILP) problem by using the KKT method [21].

The retailer’s profit maximization considering elastic demand in the presence of renewable energies is explored in [22]. Time of use pricing (TOU) has been implemented considering the uncertainty in wholesale prices and renewable energy generation. Assuming the aggregation of prosumers to form a retailer, the authors of [23] studied the relationship between the retailer and end users as the relationship between the aggregated prosumers with end users. The problem is regarded as a multileader–multifollower game with prosumers and end users taken as leaders and followers, respectively. The authors in [24] have developed a stochastic bilevel problem for integrating thermal energy and reserve scheduling into smart microgrids considering the maximum social welfare and price-based DR. Given the utilization of the particle swarm optimization (PSO) method and integrated problem solving, this study has not addressed data privacy. In a compelling study in [25], home energy management is considered based on reinforcement learning and artificial neural network (ANN). This paper assumes that the retail price, or in other words, the service provider, is fixed and predicted by the homes. The fundamental issue in this study is that the fully inactive retailer does not react to the variations in its environment with DR-enabled loads. Although DR is implemented from the load’s view by applying a learning approach and the load’s data privacy is preserved, the retail market and retailer reactions to the variations in its environment are not addressed. In [26], a combined DR program consisted of real-time and incentive-based DR is presented. The authors designed a trilevel Stackelberg game in which the power system operator seeks to minimize the incentive costs, while the retailers and users aim to maximize their profit and welfare, respectively. In an exciting work, a competitive market is designed consisting of prosumers and retailers such that all prosumers (as buyers and sellers) and retailers conduct peer-to-peer energy trading [27].

The studied articles are compared and classified in

Table 1. A review of papers related to retailer’s pricing in interaction with the end user.

Ref.	[14]	[23]	[17]	[16]	[24]	[15]	[18]	[19]	[20]	[25]	[26]	[21]	[22]	Current Paper
Retailer Pricing	-	✓	✓	-	✓	✓	✓	✓	-	-	✓	✓	✓	✓
DR (S/O)	O	-	S	-	O	O	O	O	O	S	S	-	-	S
Data Privacy	-	✓	-	-	-	-	-	-	-	✓	✓	-	-	✓
Learning	-	-	-	-	-	-	-	-	-	✓	-	-	-	✓

O: Operator DR. S: Self DR.

regarding the use of learning in solving methods, data privacy protection, DR implementer side, and retailer pricing.

From the performed literature review, a study gap exists in this area as follows.

• Neglecting data privacy (free access to private information of prosumers) in retailers’ decision-making model for optimal pricing.
• Applying centralized approaches such as integration methods to solve the optimization problem of retailer pricing interacting with prosumers, which is inherently a distributed problem.
• Not using forecasting and learning in retailer’s pricing as a powerful tool for solving this inherently distributed problem.

The conducted literature review on retail pricing suggests that DR is either neglected or performed by the operator in most studies. However, it seems that for prosumers’ privacy protection, DR should be implemented by the prosumers themselves. By using a distributed approach in this article, DR is considered from the perspective of the prosumers. In addition, many studies have integrated the optimization models of both beneficiaries to solve the problem of retailers’ interactions with prosumers. In general, the integration in optimization models means operator’s access to all information, which breaches the prosumers’ data privacy and thus is an unrealistic assumption. In reality, the retailer’s pricing in interaction with prosumers’ is solved as a distributed problem, and each prosumer as an intelligent agent can decide on optimal energy management based on its goal while exchanging minimum information. Based on the distributed approach presented in this article, the retailer and prosumers can be considered as completely separate agents that interact with each other by exchanging minimal information (i.e., the price signal from the retailer and the consumption signal from the prosumer). Furthermore, the proposed strategy for preserving data privacy is to forecast prosumers’ behavior in the market. Prediction becomes complicated when energy consumers have generation capability as well, meaning that they are prosumers. Learning is considered a powerful tool to deal with this issue, which is ignored in most studies. With the help of machine learning, this article has attempted to address this problem. Online learning allows retailers to predict prosumers’ behavior without having access to their private data.

1.3. Novelties and Contributions

This paper presents a novel model for retailer’s pricing in the smart grid environment. In the proposed model, the prosumers, as active consumers capable of generating, consuming, and storing energy, are regarded as new players in interaction with retailers. These prosumers can participate in the DR program voluntarily and without the retailer’s interference. In turn, the retailer benefits from online learning to predict the prosumer’s behavior in optimal pricing. Even though prosumers can exchange information with retailers regarding their energy consumption, it is important for them as a stakeholders in the market to keep more detailed information private, including the type and number of energy generators, the mechanism for responding to demand, and the type and quantity of storage devices. In this model, the distributed solving algorithm is applied to protect the prosumer’s information privacy. To do so, the retailer exchanges its proposed prices, whereas the prosumer only exchanges its demand quantity based on the proposed price. Each player (prosumer and retailer) individually solves its model in a distributed manner, needless of further information exchange.

This article makes the following contributions:

• designing and modeling the optimal pricing problem of the retailer in interaction with prosumer by presenting a distributed approach based on online learning;
• participation of prosumer in price-based DR without operator interference;
• minimum information exchange between market beneficiaries and the retailer’s lack of free access to private information of the prosumer’s decision-making model; and
• the retailer’s optimal pricing is independent of the prosumer model due to the lack of necessity for an initial database in the presented algorithm.

2. Problem Definition

According to Figure 1, the wholesale prices in the day-ahead market are submitted to the retailer as a price signal. This paper aims to balance between two opposing objectives of minimizing the prosumer’s cost and maximizing the retailer’s profit. The retailer decides on sending the final optimal price by using the iterative learning method. Notably, this learning is online, meaning that the database becomes complete while executing the algorithm. Furthermore, given the retailer’s learning of the prosumer consuming behavior, the presented model fully protects the information privacy of the prosumer. The reason for this is that, unlike the conventional bilevel optimization models, the retailer has no access to information related to decision making and internal equipment of prosumers and decides on the optimal retail price only by relying on the received consuming signals from these loads in response to its submitted prices. Learning the consuming behavior model of prosumers in response to the retailer’s submitted prices is performed by creating a database based on both the retailer’s submitted price and the prosumer’s consumption after sufficient iterations. In the retailer’s decision-making model, the regulatory rules for submitting a fair price to prosumers are considered. In addition, thanks to smart meters, which enable mutual connection with the retailer, the prosumers can submit their optimal consumption for the day-ahead scheduling period. The prosumers’ decision-making model considers a variety of electricity and heat generation and storage units. Strikingly, in the proposed model, the prosumers’ load pattern is optimized by DR through load shifting by the prosumers, and the retailer plays no role in decision making, which protects the prosumers’ privacy. Furthermore, because the retailer and prosumers are modeled as two separate entities or agents connected through changes they make in the environment (sending signals to each other), the scalability and data privacy protection are addressed.

2.1. Modeling of Prosumer Energy Management

In light of the benefits of CHP systems, like pollution reduction and increased energy efficiency, CHP units are considered in the prosumer model in this section. The optimal pricing by a retailer is taken into account through modeling of the relationship between retailer’s pricing and prosumer’s consumed power, by using multivariate linear regression (MVLR) by the retailer. In the following, the prosumer energy management problem and prediction of its behavior by the retailer is presented.

2.1.1. CHP Model

CHP units are utilized for cogeneration of heat and power. The thermal and electrical output power of CHP is interdependent, and this relationship is shown by a curve known as a feasible operational region (FOR) [28]. In this paper, two types of curve are considered in the modeling. A detailed description of the modeling of these units is provided in the following [13]. Equations (1) and (2) determine the CHPs operating points in the FOR region [13], where $P_{i,t}^{chp}$ is the electrical power generated by the i-th CHP at time t, $V_{i,t}^{chp}$ is a binary variable showing the on or off state of the i-th CHP at time t, and $H_{i,t}^{chp}$ is the heat power generated by i-th CHP at time t. Equations (1) and (2) can be expanded to (3)–(16) separately for the first and second types of CHP. As the first type of CHP scheme in Figure 2 shows, Equations (3)–(7) describe feasible regions of first type CHP [13]. According to first type of CHP illustrated in Figure 2, Equation (3) shows the area below the AB line, whereas Equations (4) and (5) model the upper areas of the BC and CD lines, respectively. Equations (6) and (7) also describe the generation limits of electrical and thermal output power. As is evident from the second type of CHP scheme in Figure 2, Equations (8)–(16) describe feasible regions of second type of CHP [13]. According to Figure 2, the feasible region of second type CHP is nonconvex, and divided into two convex regions by decision variables $X_{1,t}$ and $X_{2,t}$. The position of the CHP’s operating point is determined by two binary variables $X_{1,t}$ and $X_{2,t}$. Hence, the operating point could be placed in either of the two areas dependent upon whether the CHP is on or off in that area, and this condition is expressed in Equation (14). The cost of CHP systems can be calculated from Equation (17). In this equation, ${\lambda }^{NG}$ is the price of natural gas and ${\eta }^{chp}$ is the efficiency of CHP units. We have

$$P_i^{{chp}_{min}}\left(H_{i,t}^{chp}\right)\times V_{i,t}^{chp}\le P_{i,t}^{chp}\le P_i^{{chp}_{max}}\left(H_{i,t}^{chp}\right)\times V_{i,t}^{chp}fori=1,\cdots ,N^{chp},t=1,\cdots ,24$$

(1)

$$H_i^{{chp}_{min}}\left(P_{i,t}^{chp}\right)\times V_{i,t}^{chp}\le H_{j,t}^{chp}\le H_i^{{chp}_{max}}\left(P_{i,t}^{chp}\right)\times V_{i,t}^{chp}fori=1,\cdots ,N^{chp},t=1,\cdots ,24$$

(2)

$${\mathrm{P}}_{i,t}^{chp}-P_{i,A}^{chp}-\frac{P_{i,A}^{chp}-P_{i,B}^{chp}}{H_{i,A}^{chp}-H_{i,B}^{chp}}\left(H_{i,t}^{chp}-H_{i,A}^{chp}\right)\le 0$$

(3)

$${\mathrm{P}}_{i,t}^{chp}-P_{i,B}^{chp}-\frac{P_{i,B}^{chp}-P_{i,C}^{chp}}{H_{i,B}^{chp}-H_{i,C}^{chp}}\left(H_{i,t}^{chp}-H_{i,B}^{chp}\right)\ge -(1-V_{i,t}^{chp})\times M$$

(4)

$${\mathrm{P}}_{i,t}^{chp}-P_{i,C}^{chp}-\frac{P_{i,C}^{chp}-P_{i,D}^{chp}}{H_{i,C}^{chp}-H_{i,D}^{chp}}\left(H_{i,t}^{chp}-H_{i,C}^{chp}\right)\ge -(1-V_{i,t}^{chp})\times M$$

(5)

$$0\le {\mathrm{P}}_{i,t}^{chp}\le P_{i,A}^{chp}\times V_{i,t}^{chp}$$

(6)

$$0\le {\mathrm{H}}_{i,t}^{chp}\le H_{i,B}^{chp}\times V_{i,t}^{chp}$$

(7)

$${\mathrm{P}}_{i,t}^{chp}-P_{i,B}^{chp}-\frac{P_{i,B}^{chp}-P_{i,C}^{chp}}{H_{i,B}^{chp}-H_{i,C}^{chp}}\left(H_{i,t}^{chp}-H_{i,B}^{chp}\right)\le 0$$

(8)

$${\mathrm{P}}_{i,t}^{chp}-P_{i,C}^{chp}-\frac{P_{i,C}^{chp}-P_{i,D}^{chp}}{H_{i,C}^{chp}-H_{i,D}^{chp}}\left(H_{i,t}^{chp}-H_{i,C}^{chp}\right)\ge 0$$

(9)

$${\mathrm{P}}_{i,t}^{chp}-P_{i,D}^{chp}-\frac{P_{i,D}^{chp}-P_{i,E}^{chp}}{H_{i,D}^{chp}-H_{i,E}^{chp}}\left(H_{i,t}^{chp}-H_{i,D}^{chp}\right)\ge -\left(1-X_{2,t}\right)\times M$$

(10)

$${\mathrm{P}}_{i,t}^{chp}-P_{i,E}^{chp}-\frac{P_{i,E}^{chp}-P_{i,F}^{chp}}{H_{i,E}^{chp}-H_{i,F}^{chp}}\left(H_{i,t}^{chp}-H_{i,E}^{chp}\right)\ge -\left(1-X_{1,t}\right)\times M$$

(11)

$$0\le {\mathrm{P}}_{i,t}^{chp}\le P_{i,A}^{chp}\times V_{i,t}^{chp}$$

(12)

$$0\le {\mathrm{H}}_{i,t}^{chp}\le H_{i,C}^{chp}\times V_{i,t}^{chp}$$

(13)

$$X_{1,t}+X_{2,t}=V_{i,t}^{chp}$$

(14)

$$H_{i,t}^{chp}-H_{i,E}^{chp}\le (1-X_{1,t})\times M$$

(15)

$$H_{i,t}^{chp}-H_{i,E}^{chp}\ge -(1-X_{2,t})\times M$$

(16)

$${COST}_t^{chp}=\sum _{i=1}^{N^{chp}}\left({\lambda }^{NG}\times \frac{P_{i,t}^{chp}+H_{i,t}^{chp}}{{\eta }^{chp}}\right).$$

(17)

2.1.2. Model of Power-Only Unit

In this paper, microturbines are considered as power-only units. The equations related to the operation and cost of these units are given below [13]. Equations (18) and (19) express the power generated by power-only units. In these equations, $P_{t,q,\xi }^{po}$ is the output power of power-only unit q at time t and block $\xi$, $P_{q,\xi }^{MAX}$ is the nominal power of block $\xi$ of power-only unit q and $N^b$ is the number of blocks in unit q[13]. Equations (20) and (21) exhibit the limitations of generation ramp-up and ramp-down, respectively [13]. In these equations, $U_{t,q}^{po}$ is the spinning status of unit q at time t, $R_q^{up}$ is the ramp-up rate, and $R_q^{down}$ is the ramp-down rate. The minimum up and down times are modeled in Equations (24) and (25) by using the auxiliary binary variables ${Up}_{q,m}$ and ${Dn}_{q,m}$ [13]. In addition, Equations (22) and (23) express the minimum up- and downtime constraints. The cost function of the power-only units is presented as piecewise linear, as shown in Figure 3 and Equation (26). We have

$$0\le P_{t,q,\xi }^{po}\le P_{q,\xi }^{MAX}-P_{q,\xi -1}^{MAX}\forall q,t,\xi =2,\cdots ,N^b$$

(18)

$$0\le P_{t,q,1}^{po}\le P_{q,1}^{MAX}\forall q,t$$

(19)

$$\sum _{\xi =1}^{N^b}{P}_{t,q,\xi }^{po}-\sum _{\xi =1}^{N^b}{P}_{t-1,q,\xi }^{po}\le R_q^{up}\times U_{t,q}^{po}\forall q,t$$

(20)

$$\sum _{\xi =1}^{N^b}{P}_{t-1,q,\xi }^{po}-\sum _{\xi =1}^{N^b}{P}_{t,q,\xi }^{po}\le R_q^{down}\times U_{t-1,q}^{po}\forall q,t$$

(21)

$$U_{t,q}^{po}-U_{t-1,q}^{po}\le U_{t+{Up}_{q,m},q}^{po}\forall q,t,m$$

(22)

$$U_{t-1,q}^{po}-U_{t,q}^{po}\le {1-U}_{t+{Dn}_{q,m},q}^{po}\forall q,t,m$$

(23)

$${Up}_{q,m}=\left\{\begin{array}{c}mm\le {MUT}_q\\ 0m\ge {MUT}_q\end{array}\right.$$

(24)

$${Dn}_{q,m}=\left\{\begin{array}{c}mm\le {MDT}_q\\ 0m\ge {MDT}_q\end{array}\right.$$

(25)

$${COST}_t^{po}=\sum _{q=1}^{N^{po}}\sum _{\xi =1}^{N^b}{\lambda }_{q,\xi }^{po}\times P_{t,q,\xi }^{po}\forall t.$$

(26)

2.1.3. Model of Heat-Only Unit

In addition to CHP and heat storage systems, the heat-only unit is used to generate heat power. Equations (27) and (28) demonstrate the constraint and cost of heat production in this unit, respectively [24]. We have

$$0\le H_t^{ho}\le {TH}^{ho,max}\forall t$$

(27)

$${COST}_t^{ho}={\lambda }^{ho}\times H_t^{ho}\forall t.$$

(28)

2.1.4. Model of the Electrical Storage System

The constraints of the electrical storage system are given in Equations (29)–(34). Equation (29) expresses the energy and initial conditions of energy-storage systems. Equations (30) and (31) describe the limitations of charge and discharge, respectively. Equation (32) shows the upper and lower limits of stored energy in the energy-storage system. Concerning the infeasibility of simultaneous charge and discharge, the corresponding constraint is shown in Equation (33). The dynamic energy model of energy storage is also given in Equation (34) [29]. We have

$$X_{t_0}^b=X_0^b$$

(29)

$$P_t^{charge}\le P_{charge}^{max}\times U_t^{charge}$$

(30)

$$P_t^{disc}\le P_{disc}^{max}\times U_t^{disc}$$

(31)

$$X_b^{min}\le X_t^b\le X_b^{max}$$

(32)

$$U_t^{charge}+U_t^{disc}\le 1$$

(33)

$$X_t^b=X_{t-1}^b+\gamma {\times P}_t^{charge}-\frac{P_t^{disc}}{\eta }.$$

(34)

2.1.5. Model of the Heat Storage System

Equations (35)–(38) represent the heat storage model. Equation (35) describes the storage constraints. Equations (36) and (37) express the limitations related to charging and discharging, respectively, and the energy balance associated with thermal energy storage is shown in (38) [29]. We have

$$B_{min}\le B_t\le B_{max}$$

(35)

$$0\le H_t^{b,c}\le H^{b,c,max}$$

(36)

$$0\le H_t^{b,d}\le H^{b,d,max}$$

(37)

$$B_t=B_{t-1}+{\eta }^{b,c}\times H_t^{b,c}-\frac{H_t^{b,d}}{{\eta }^{b,d}}.$$

(38)

2.1.6. Model of Demand Response

In this paper, the demand response model is assumed as load shifting at times of higher market prices. Equation (39) describes the power demand following a demand response. Equation (40) describes load shifting, Equation (41) expresses that the sum of loads in 24 h cannot change, and Equation (42) demonstrates the limitation of demand response per hour [30]. We have

$$P_t^{DR}=P_t^D+{ldr}_t$$

(39)

$${ldr}_t={DR}_t\times P_t^D$$

(40)

$$\sum _{t=1}^{24}{ldr}_t=0$$

(41)

$${DR}_t^{min}\le {DR}_t\le {DR}_t^{max}.$$

(42)

2.1.7. Heat and Power Balance Constraints

Equation (43) indicates the limitation of the electrical power balance in the considered problem. In this relation, $P_t$ is the purchased power from the retailer. Similarly, Equation (44) shows the limitation of the thermal power balance in this problem. WE have

$$\sum _{i=1}^{N_{chp}}{P}_{i,t}^{chp}+\sum _{q=1}^{N_{po}}\sum _{\xi =1}^{N_b}{P}_{t,q,\zeta }^{po}+P_t^{disc}-P_t^{charge}+P_t=P_t^{DR}$$

(43)

$$\sum _{i=1}^{N_{chp}}{H}_{i,t}^{chp}+H_t^{ho}+H_t^{b,d}-H_t^{b,c}=H_t^{demand}.$$

(44)

2.1.8. Objective Function in Prosumer Energy Management Model

The objective of this problem is to minimize the costs associated with the prosumer. The considered objective function is given in Equation (45). The first term in this relation is CHP costs, the second and third terms show the power-only and heat-only unit costs, respectively, and the last term is related to purchasing load from the retailer. We have

$$ObjectiveFunction=Min\left\{\sum _{t=1}^{24}({COST}_t^{chp}+{COST}_t^{po}+{COST}_t^{ho}+({\lambda }_t\ast P_t)\right\}.$$

(45)

2.2. Retailer Prediction by Using Multivariate Linear Regression

The multivariate linear regression model is defined in (46). Response variables are variables that we aim to predict. In addition, predictor variables are variables by which the response variables are predicted. In (46), for $j=1,\cdots ,n$, we have $m\ge 2$ response variables as $Y_1,\cdots ,Y_m$ and p predictor variables as $x_1,\cdots ,x_p$. B (coefficients matrix) and ${ϵ}_j$ (Bias matrix) also should be estimated. For this purpose, in the proposed model, we regard $y_j$ and $x_j$ as prosumer-consumed powers, and retailer’s suggested prices, respectively. The absolute values of error are minimized in matrix laboratory (MATLAB) to estimate the proposed model values. Equation (46) is rewritten in the form of (47) to apply it to the considered retail problem. In this equation, $P_j^h$ is the load-consumed power at time h and dataset j, and ${\lambda }_j^h$ is the retailer’s pricing at time h and dataset j. With the learning and estimation of coefficients, the retailer can predict the model of prosumer’s behaviors toward its suggested prices and, eventually, offer optimal prices according to the learned model. The reason for using this method is that the obtained values by this method can easily be entered into the general algebraic modeling system (GAMS) environment and perform the retailer profit maximization. The ANN method can also be employed to predict the prosumer’s behavior. In this case, the retailer profit optimization problem is required to be addressed by using optimization heuristic methods, such as PSO, which takes a substantial amount of time to converge [2]. We have

$$y_j=B^T{x}_j+{ϵ}_j$$

(46)

$$\left[\begin{array}{c}{P}_1^j\\ .\\ .\\ .\\ P_{24}^j\end{array}\right]=\left[\begin{array}{c}{b}_{1,1}\cdots b_{1,24}\\ .\cdots \\ .\cdots \\ .\cdots \\ b_{24,1}\cdots b_{24,24}\end{array}\right]\left[\begin{array}{c}{\lambda }_1^j\\ .\\ .\\ .\\ {\lambda }_{24}^j\end{array}\right]+\left[\begin{array}{c}{P}_1^{j0}\\ .\\ .\\ .\\ P_{24}^{j0}\end{array}\right].$$

(47)

The retailer profit maximization problem with prosumer’s behavior prediction is obtained from (47), in the form of Equation (48) along with Equation (49), which shows the constraint for controlling the retailer’s pricing. We have

$$\underset{{\lambda }_1,\cdots ,{\lambda }_t}{\max }{\left(P_t+P_{ft}\right)}^T\left({\lambda }_t-{\lambda }_{gt}\right)$$

(48)

s.t.

$$\left\{\begin{array}{c}{\lambda }_{min}\le {\lambda }_t\le {\lambda }_{max}\\ \frac{1}{t}\sum _{i=1}^t{\lambda }_t=\frac{1}{t}\sum _{i=1}^t{\lambda }_{gt}\end{array}\right..$$

(49)

The second part of Equation (49) assumes that the sum of prices declared by the retailer to the load at 24 h equals the sum of prices that it purchases from the wholesale during this period as a regulatory contract. It should be mentioned that GAMS software solves this nonlinear programming (NLP)-type optimization problem.

2.3. Putting the Pieces Together

Due to expensive computational overhead, the GAMS software coupled with MATLAB is used to solve the proposed problem, as shown in Figure 4. From this figure, the retailer first makes its initial proposal to the prosumer in the GAMS environment; then the prosumer announces its estimated power consumption to the retailer considering cost minimization. The retailer creates a database of the prosumer’s estimated power demand and its announced prices in MATLAB and learns the prosumer behavior model by using the multivariate linear regression method. Ultimately, the retailer maximizes its profit by determining the retail price based on both the learned model and the amount of fixed power to supply, and then announces the new price to the prosumer. This process will continue until the stopping criterion is reached. In this method, the retailer’s database becomes more extensive as the number of iterations increases, and the retailer can achieve a more accurate estimation of the prosumer’s behavior which ensures the convergence of the retailer’s profit and prosumer’s costs.

3. Simulation Results

To further illustrate the efficacy of the proposed approach, two case studies are presented and compared with each other for demand response in this section. We assume 30% and 10% demand response for the first and second case studies, respectively, which reflects the extent of demand response to the retailer’s suggested prices.

3.1. Data

In the prosumer energy management problem, two CHP types are assumed based on data from

Table 2. Parameters of CHP units.

Parameters	First Type CHP					Second Type CHP								Units
${\eta }^{chp}$			75							75				%
${\eta }_i^{tr}$			97							97				%
		A	B	C	D			A	B	C	D	E	F
FOR	P	263	195	45	56		P	142	142	120	40	48	48	kW
	H	0	210	120	0		H	0	35	118	69	11	0	kW${}_{\mathrm{Th}}$

[13].

Table 3. Parameters of power-only units.

Parameters	First Unit	Second Unit	Third Unit	Units
Maximum output power	150	180	200	kW
Minimum output power	0	0	0	kW
${\lambda }_1^{po}$	0.030	0.037	0.044	$/kWh
${\lambda }_2^{po}$	0.036	0.040	0.049	$/kWh
${\lambda }_3^{po}$	0.039	0.045	0.054	$/kWh
$P_1^{MAX}$	60	80	100	kW
$P_2^{MAX}$	110	120	150	kW
$P_3^{MAX}$	150	180	200	kW
${MUT}_q$	2	2	2	hour
${MDT}_q$	2	2	2	hour
$R_q^{up}$	80	90	100	kW/h
$R_q^{down}$	80	90	100	kW/h

shows data related to power-only units [8], and data associated with the electrical [8] and thermal storage are given in

Table 4. Parameters of electrical energy storage.

Parameters	Value	Units
$X_b^{max}$	1000	kWh
$X_b^{min}$	50	kWh
$P_{charge}^{max}$	600	kW
$P_{disc}^{max}$	600	kW
$\chi$	90	%
$\eta$	80	%

and

Table 5. Parameters of thermal energy storage.

Parameters	Value	Units
$B_{min}$	0	kW${}_{\mathrm{Th}}$
$B_{max}$	1000	kW${}_{\mathrm{Th}}$
$H_t^{b,c,max}$	20	kW${}_{\mathrm{Th}}$
$H_t^{b,d,max}$	20	kW${}_{\mathrm{Th}}$
${\eta }^{b,c}$	95	%
${\eta }^{b,d}$	93	%

, respectively. Figure 5 displays the thermal and electrical power demand in the prosumer problem [13] and also the retailer’s fixed electrical load [10].

3.2. First Case Study: 30% Demand Response

The first case assumes that the prosumer can vary its declared consumption as much as 30% per hour; however, the sum of consumed powers after demand response in 24 h equals the sum of initial power due to the regulatory contract. In other words, only load shifting occurs during certain hours. Figure 6 illustrates the retailer pricing, the predicted initial load as well as the type of demand response to the retail prices. As is evident, from 1 to 9 a.m. and from 10 to 12 p.m., the prosumer decides to increase its demand due to low retail prices. On the contrary, during midday from 3 to 9 p.m., when the retailer raises its price, the prosumer responds to this behavior by decreasing demand through demand response. From 10 a.m. to 2 p.m. because of approximately equal retailer prices, the prosumer displays a volatile behavior around its predicted initial load.

Figure 7 shows the production share of electrical power from its production sources, electrical storage charging, and discharging and power supplied per hour by the retailer. This figure displays the prosumer’s power supply behavior from its production units considering the retailer’s prices. At 4, 5, and 24 o’clock when the retailer’s price is low, the prosumer has stored the electrical energy and consumed it at 4, 5, 6, and 8 p.m. to purchase less power from the retailer due to its high prices. It is noteworthy that as the prosumer has to deliver the initial amount of charge received from storage at the end of the day, we have storage recharging at midnight. In addition, because of the high retailer’s prices at 1 and 3 to 6 p.m., the prosumer has preferred not to purchase from the retailer. At all hours, the prosumer has supplied a portion of its power from power-only units owing to their low cost. Moreover, at hours during which the retailer’s price has increased, the prosumer has preferred to supply a portion of its power from CHP units.

The key point of this study is that the retailer’s profit and cost of the prosumer problem reach equilibrium and converge by using the proposed algorithm, as depicted in Figure 8. This means that the proposed method is effective, and demonstrates that both prosumer and retailer have reached a balance in their interactions, and do not intend to change one another’s strategies. The retailer’s profit obtains convergence to a profit equal to $50.2 after an appropriate number of iterations. As is clear, the prosumer cost converges similarly to the value of $3723 after a proper number of iterations. There are also some fluctuations in profit and cost plots before 82nd iteration, associated with the prediction error of the multivariate linear regression method.

3.3. Second Case Study: 10% Demand Response

This case assumes that a prosumer can vary its declared consumption by as much as 10% per hour. In the following, we present and discuss the obtained results for this case. Figure 9 illustrates retailer pricing and type of demand response to these prices. As is evident, during hours of 1 to 12 a.m. as well as 11 and 12 p.m., due to low retail prices, prosumer decides to increase its demand; however, in remaining hours that the retailer’s price rises, the prosumer responds to this behavior by decreasing its demand. As is evident, because of the lower demand response, the distance of demand response type from its predicted power is lower than the Figure 6.

Figure 10 shows the production share of electrical power from its production sources, electrical storage charging, and discharging and power supplied per hour by the retailer. Concerning the reduced amount of demand response in this case, as compared with Figure 7, the prosumer has purchased an amount of its electrical power from the retailer in all hours. Moreover, the storage is charged at 5, 6, and 24 o’clock, when the retailer’s price is low and is discharged at 17 and 18 o’clock when the retailer’s price is high. As shown in Figure 11, in this case, similar to the first case study, both the retailer’s profit and the cost of the prosumer problem converge. Due to the decreased demand response, the retailer’s profit rises by a value of $55.66; however, the prosumer costs also grow by an amount of $3908. This means that the more activity and price sensitivity of the prosumer decreases, the higher the retailer’s profit and the higher the prosumer’s costs.

Figure 12 shows the prediction error of the multivariate linear regression method. Here, prediction error is evaluated by the mean absolute error (MAE) method. As is evident, the prediction error is low at initial iterations owing to the insufficient data, which reflects the accuracy of the prediction made by this method. As the iterations proceed, which also increases data, the error slightly rises; however, it eventually achieves convergence due to sufficient data. The prediction error converges to appropriate values of 5% and 2% for 30% and 10% demand responses, respectively. Needless to say, as prosumer behavior is more limited with regard to demand response, the retailer’s prediction error also declines in the 10% demand response case study. Notably, the maximum number of iterations is considered as n = 1000 to ensure convergence of the prediction error. Figure 13 displays the types of retailer’s pricing with respect to the purchase prices from wholesale. The correlation coefficient for wholesale and retail price plots is calculated to be 0.8836 and 0.8657 for 30% and 10% demand response, respectively. The results show a decrease in the calculated coefficient with demand response. The reason is that the increases in demand response force the retailer to declare prices closer to wholesale prices to the prosumer for increasing its maximum profit. Prosumer cost, retailer’s profit, and correlation coefficient of retail and wholesale prices are compared in

Table 6. Comparison of demand response with prosumer cost, retailer profit, and correlation coefficient.

Demand Response	Prosumer Cost ($)	Retail Profit ($)	Wholesale & Retail Price Covariance
30%	3723	50.20	0.8836
10%	3908	55.66	0.8657
No demand response	4053	58.18	0.8590

. As is evident from the table, the retailer’s profit has decreased, along with the prosumer cost, with an increase in demand response. The retailer has been forced to declare prices closer to wholesale prices to achieve its expected maximum profit. As an example, in the 10% demand response case study, the prosumer cost decreased by 3.57%, compared to the case of no-demand response, whereas in the 30% demand response case study, this reduction reached 8.14%. Furthermore, as a result of the proposed distributed approach in this paper, the problems of the retailer’s profit maximization and prosumer cost minimization have been solved separately. With more demand response to its suggested prices, the retailer’s profit also dropped by 4.33% and 13.71% for 10% and 30% demand response case studies, respectively. In both case studies, the retailer’s profit and prosumer costs decreased with more demand response. Another issue is the preservation of prosumer privacy by the retailer. Although the retailer had no information about prosumer’s behavior and types of their demand response, still, with data increasing over iterations, it provided a more accurate estimation of the prosumer behavior model, and the proposed approach led to problem convergence for both case studies.

4. Conclusions

There has been an increase in data privacy concerns as new capabilities are enabled in the retail sector. Therefore, the purpose of this research is to respond to the important issue of data privacy and prosumer behavioral model privacy by using a heuristic method based on machine learning. The results and convergence of the presented algorithm allow us to provide a framework in which retailer and prosumer can communicate with minimal information exchange, in a way that the goals of both are achieved. In this paper, we developed a heuristic distributed modeling approach for studying the retailer’s behavior in day-ahead pricing, along with price-based demand response in prosumer problems. To evaluate the efficacy of the proposed model, two case studies were investigated, namely 30% and 10% demand response. By utilizing a multivariate linear regression method from the field of machine learning, the proposed approach was employed for distributed modeling at various levels. In spite of the fact that the retailer had no knowledge of the prosumer’s behavior and its decision-making model in the proposed model, with the increase of data in online learning, the retailer reached to a more accurate estimate of the prosumer’s behavior over time. The obtained results confirmed the efficacy of the proposed method. In general, as the demand response to retailer pricing increased, there was a subsequent decrease in the retailer’s profit and prosumer costs. The main results are summarized as follows.

• The proposed distributed model, in both case studies, led to the convergence of prosumer cost and retailer’s profit.
• Such an approach adheres to the problem of preserving privacy and private information of the prosumer and the retailer.
• Increasing demand response to the retailer’s suggested prices prevented the retailer from taking advantage of the passive behavior of its prosumers and also reduced the prosumer’s cost by up to almost 10%.
• By using online learning, without prior information and during the replication of the algorithm, the retailer learned the model behavior of the prosumer with appropriate accuracy.
• It was shown that with increasing demand response, the correlation coefficient of retailer’s prices with prices taken from the wholesale market increased due to the greater activation of the prosumer to the prices.