
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13697-3
10.1016/j.heliyon.2024.e37666
e37666
Research Article
Optimal two-stage arbitraging and scheduling strategy for distributed energy resource aggregators
Xie Peng pengxie216@163.com
a⁎
Liu Mingjun a
Chen Chun a
Tang Wenming a
Wang Han b
a School of Intelligent Manufacturing and Equipment, Shenzhen Institute of Information Technology, Shenzhen, China
b Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen, China
⁎ Corresponding author. pengxie216@163.com
12 9 2024
30 9 2024
12 9 2024
10 18 e3766617 4 2024
23 6 2024
8 9 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Given the rapid development of the distributed energy resources (DER), involving DERs into the wholesale market under the market and renewable uncertainties to achieve economic benefits is necessary but challenging. In this work, an arbitraging strategy is proposed for DER aggregators that bridge DERs with the wholesale market through energy trading. Besides, a novel self-adaptive minimax regret (MMR)-based optimal offering model is proposed for the DER aggregator to handle the uncertainties in both renewable generations and market prices. Additionally, an exact and communication-free solution methodology is proposed to resolve the formulated optimization problem with high computational efficiency. In the numerical results, the proposed methods can achieve near-to-optimal profits. Moreover, the performance of the proposed approach remains satisfactory even if the environment becomes very volatile.

Index Terms

DER aggregator
Minimax regret
Arbitrage
Uncertainty
Indirect coordination
==== Body
pmc1 Introduction

Transition from the traditional centralized generation and transmission to distributed generation is a common trend in most power systems. A common sign of this transition is the increased installation of distributed energy resources (DERs) [1]. In recent decades, the electricity industry has seen a big increase in both putting a lot of energy into the system and using smaller DERs [2]. But for DERs, their sizes and the uncertainties linked to on-and-off renewable energy sources make it hard for them to be part of the bigger wholesale market. Hence, aggregating DERs becomes an attractive solution to integrate DERS into the wholesale market [3].

Most of the current DER aggregating works either apply the centralized structure where an aggregating agent can directly control all the DERs to maximize the overall benefit or employ the decentralized structure [4,5]. However, the centralized structure requires direct control of the DERs, and wholesale market participation is not enabled in the decentralized structure. Another coordinating structure, which can overcome these drawbacks, is the distributed structure. Under the distributed structure, an aggregating agent indirectly manages the DERs through incentive signals and the DER owners make their own self-scheduling decisions according to the incentive signals. In Ref. [6], a monopoly DER aggregator is considered to participate in the electricity market representing the DERs to maximize its profit when there is no uncertainty. In Ref. [7], a microgrid (MG) operator is introduced to manage the energy sharing between the photovoltaic (PV) prosumers within the MG under renewable uncertainty. In Ref. [8], a retailer is considered to coordinate the energy sharing of PV prosumers in an MG to reduce the operational cost of the prosumers. In Ref. [9], an MG aggregator acts as a broker between the MGs and the real-time (RT) market to maximize its profit considering the renewable generation uncertainty. In Ref. [10], a virtual power plant (VPP) is formed under the distributed structure to minimize the costs of the prosumers in the VPP. In Ref. [11], a retailer is considered to maximize its profit by integrating the demand response for wholesale market participation, while the renewable generation is not considered. There are some existing transactive energy management frameworks for operating DERs. In Ref. [12], a bi-level network-constrained peer-to-peer energy trading framework for multiple connected microgrids under uncertainties is proposed using the stochastic programming. In Ref. [13], a two-stage robust stochastic model is proposed for the transactive energy-based scheduling problem of interconnected microgrids. In Ref. [14], a tri-layer risk-averse stochastic game approach for energy trading among multi-energy microgrids is proposed using the stochastic programming approach.

In DER management problems, uncertainties introduced by DER generation and market prices is a big challenge and the main focus of the operators [15]. Two common methods used for this are stochastic and robust optimization [16]. Stochastic, seen as a probabilistic method, is extensively studied. It models uncertainties in market prices and renewable generation through a large number of scenarios with specific probability distributions. In Ref. [17], an arbitrage strategy of MGs through P2P energy trading to cope with the volatility of the generation of RESs using stochastic optimization is proposed considering risk control. In Ref. [18], a stochastic optimization approach is developed to optimize the market offering and DER dispatching decisions considering energy trading. In Ref. [19], a stochastic optimization model is proposed to optimize the charging and discharging behavior of battery energy storage to maximize the profit obtained in both the energy and ancillary service markets. In Ref. [20], a three-stage stochastic bi-level optimization model is formulated to address the uncertainties in prices, load, and renewable generation.

Despite being effective, stochastic methods face challenges like requiring accurate uncertainty probability distributions are required to obtain high-quality decisions. However, such accurate probability distributions are often hard to acquire in real operation. In addition, the computational burden is also a big challenge of stochastic programming approaches because high-quality solutions need to be obtained from a relatively large number of scenarios.

Another popular alternative to address the uncertainty problem in DER management problem is robust optimization, which deals with uncertainties using uncertainty sets that requires less information compared to the stochastic programming approach. In Ref. [21], a two-stage robust optimization model of the min-max-min structure was established to deal with multiple uncertainties in a multi-energy virtual power plant. In Ref. [22], an optimal microgrid planning and energy management framework is developed using the robust optimization approach. In Ref. [23], a two-stage energy management procedure is proposed to control the responsive loads and battery storage under the price and wind uncertainties based on robust optimization. Although in robust optimization approaches, uncertainties are modeled using uncertainty sets that requires less information compared to the stochastic programming approach. However, the decisions obtained from robust optimization approach are usually too conservative to maximize the economic potential of DERs.

The minimax-regret (MMR) optimization approach has emerged. MMR is known for being distribution-free and less conservative, making it valuable in power engineering. Previous works have shown the effectiveness of MMR. For example, in Ref. [16], it successfully addressed uncertainties in a transmission expansion planning problem. In the unit commitment problem, detailed in Ref. [24], MMR was effective in handling uncertainties related to wind generation. In Ref. [25], a thermal power generator bidding strategy using MMR navigated complexities introduced by price uncertainties. Additionally, the MMR approach as used to address the uncertainties in the optimal VPP bidding and scheduling problems [26,27]. In summary, while stochastic and robust optimization have been useful for managing uncertainties, the MMR optimization approach stands out for being distribution-free and less conservative, showing promise in addressing uncertainties in power engineering applications.

Previous works have made significant achievements in maximizing the DER aggregator's profit. However, existing arbitraging strategies and models are still conservative and cannot fully exploit the economic potential of the wholesale market and DERs under uncertainties. Motivated by pioneering works, this work aims to enhance the DER aggregator's profit under uncertainties by developing a self-adaptive two-stage MMR-based DER aggregator arbitraging strategy. The proposed arbitraging strategy synchronizes the decisions of DERs and the aggregator. As compared to existing works that manages DERs using the transactive energy framework [[12], [13], [14]], the proposed method simplifies the negotiation between the DER aggregator and DER owners. In addition, the proposed self-adaptive MMR optimization model can maintain satisfactory economic performance without requiring accurate uncertainty probability distributions.

In this work, the arbitraging practice of a DER aggregator is investigated under the distributed structure considering price and renewable generation uncertainties. The aggregator builds up an internal market to buy energy from the DER owners and re-sell to the wholesale market. In our model, the aggregator is responsible for forecasting the uncertainties, participating in the wholesale RT market, and clearing the internal market. The DER owners are responsible for offering energy and scheduling their generations according to the internal price signals and renewable forecasts. In the literature, dual pricing schemes have been applied to the energy deviations to reduce system imbalances [19,28,29]. In this work, we propose a simple dual balancing price mechanism to ensure the participants provide the electricity allocated from the bidding process. In the internal market, since the price is given by the aggregator, the DER owners only face the risk of uncertain renewable generations when making self-scheduling decisions. In the wholesale market, the aggregator acts as a price-taker and faces risks from both price and renewable generation uncertainties when making offering decisions. The arbitraging practice of the aggregator is modeled using bi-level optimization problem, where the upper level is the arbitraging problem of the DER aggregator, and the lower level is self-scheduling problem of the DER owners. For the DER owners' self-scheduling problem, we propose an MMR-based strategy to decide their energy offers and generation scheduling. For the aggregator's arbitraging problem, we propose a novel self-adaptive two-stage MMR-based arbitraging strategies, to decide the wholesale market offer and internal market price. The proposed self-adaptive MMR strategy can adjust its conservativeness by looking back previous periods.

The contributions of this work are as follows.● A novel self-adaptive two-stage MMR arbitraging strategies is proposed for DER aggregators that bridge DERs with wholesale markets through energy trading.

● Three common types of DER owners are considered, and their operation problems are formulated based on the MMR criterion. Further, for each type of DER owners, the analytical solutions are derived to accelerate the solution efficiency.

● An exact and communication-free solution methodology for the DER aggregator arbitraging problem is developed. First, the information exchange process is avoided by reformulating the bi-level optimization problem into single-level two-stage mixed-integer minimax-regret (TSMIMMR) problem. Further, the TSMIMMR problem is transformed into an equivalent two-stage mixed-integer robust optimization (TSMIRO) problem, then resolved under the column-and-constraint-generation (C&CG) framework to obtain the exact solution.

This paper is organized as follows: Section II describes the model and provide the problem formulations for both the DER owners and the aggregator. Section III provides the analytical solutions for the DER owners and develop an exact and communication-free solution methodology for the aggregator's arbitraging problem. Section IV presents the numerical tests and our discussions. Section V concludes this paper.

2 Problem formulations

In this section, we first describe the whole operational model, then give the optimization problem formulations for the DER owners and the aggregator, respectively.

2.1 Model Descriptions

The operational model is presented in Fig. 1. The DER aggregator acts as a middleman between the wholesale RT market and the DER owners to arbitrage from their price differences. We consider three types of DER owners according to their assets: I) owners with one dispatchable generator, II) owners with one renewable generator and III) owners with one dispatchable and one renewable generator. Notably, the DER aggregator and DER owners only considers energy trading in both the wholesale and internal markets. In addition, the DER aggregator is considered as a deviator in the wholesale market and will be penalized for energy deviations caused by uncertainties.Fig. 1 The operational model and decision framework of the DER aggregator and owners.

Fig. 1

There are two markets considered in the business model including the wholesale market and the internal market. In the wholesale market, the DER aggregator represents the DER owners and participate as a price taker. The DER aggregator only needs to submit an energy offer with the generation information in the wholesale market and will take the market-clearing price for the submitted energy offer. In the internal market, the aggregator interacts with the DER owners by releasing price signals to buy energy and the DER owners responds to the price signals to sell energy. Both the wholesale market and the internal market have 24 clearing periods each day and are cleared an hour before the actual electricity energy is delivered.

The price and renewable uncertainties are modeled using intervals determined by the forecast values and an uncertainty coefficient γ, which reveals the forecast accuracy. For given price forecast λf and uncertainty coefficient γ, the actual price is assumed to reside in the interval given by [(1−γ)λf,(1+γ)λf]. Similarly, the renewable generation interval is given by [(1−γ)ωf,min{(1+γ)ωf,ωc}] for a renewable output forecast ωf, where ωc is the installed capacity of the renewable generator. In the decision process, the aggregator automatically adjusts the magnitude of the uncertainty coefficient to reflect the current forecast accuracy by looking back n previous prediction errors. The uncertainty coefficient γt at time t is chosen as the maximum percentage error of the n previous decision periods:(1a) γt=max{ωD,t−1ωf,t−1,λD,t−1λf,t−1,…ωD,t−nωf,t−n,λD,t−nλf,t−n}

Dual pricing system is currently in practice in European balancing markets. Also, some previous literatures have applied dual pricing schemes to ensure that energy deviations are undesirable for market participants [19,28,29]. In our work, a simple dual pricing scheme is proposed for both the internal and wholesale markets. Under this mechanism, the undelivered energy is settled at a price higher than the clearing price and the over-produced energy is settled at a price lower than the clearing price. The balancing prices in the internal and wholesale markets are calculated as follows.(1b) λbd,in=λin1−ρ

(1c) λsd,in=λin(1−ρ)

(1d) λbd,w=λw1−ρ

(1e) λsd,w=λw(1−ρ)

(1f) 0<ρ<1

where λin and λw are the clearing prices for the internal and wholesale markets, respectively. Terms λbd,in, λsd,in, λbd,w and λsd,w represent the buying and selling prices in the internal and wholesale markets, respectively. The penalty coefficient ρ indicates the attitude of the markets towards energy deviations. When ρ increases, the markets penalize the participants more for energy deviations.

2.2 DER owner self-scheduling problem

The DER owners only participate in the internal market by responding to the internal price signals. In the first stage of the self-scheduling problem, the DER owners need to decide the energy offers and schedule the dispatchable generators. In the second stage, after the renewable outputs are revealed, the DER owners need to balance their energy deviations through the internal market at penalty prices. Therefore, the two-stage MMR self-scheduling problem of a DER owner is:(2a) minPoin,PG{R(Poin)−C(PG)+maxω{maxPoin,ω,PGω,Pdin,ω,Psin,ω{R(Poin,ω)−C(PGω)+R(Psin,ω)−C(Pdin,ω)}−maxPdin,Psin{R(Psin)−C(Pdin)}}}

s.t.(2b) ω+PG+Pdin−Psin=Poin

(2c) PGmin≤PG≤PGmax

(2d) ω+PGω+Pdin,ω−Psin,ω=Poin,ω

(2f) PGmin≤PGω≤PGmax

(2g) (1−γ)ωf≤ω≤min{(1+γ)ω,ωc}

(2h) [Poin,Pdin,Psin,Poin,ω,Pdin,ω,Psin,ω]≥0

where ω is the renewable energy output; Poin is the energy offered in the internal market; PG is the scheduled power generation of the dispatchable generator; Pdin and Psin are the energy deficit and energy surplus due to the inaccurate renewable generation forecast. Poin,ω, PGω, Pdin,ω and Psin,ω are the optimal decisions under renewable generation scenario PR,i. In the objective function, R(Poin) and R(Psin) are revenues from selling energy to the DER aggregator in the first and second stages, respectively; C(PG) is the production cost of the dispatchable generator; C(Pdin) is the cost of buying energy from the DER aggregator in the second stage.

Constraints (2b), (2d) ensure that the energy deviations are balanced; constraints (2c), (2f) are the technical limits of the dispatchable generator; constraint (2g) specifies the range of the renewable power generation. This problem formulation is for DER owners equipped with both renewable and dispatchable generators, namely, type III DER owners. For type I and type II DER owners, we only need to remove the inapplicable parts.

2.3 DER aggregator arbitraging problem

The DER aggregator faces both the wholesale market price and internal renewable generation uncertainties. In the first stage of the arbitraging problem, according to the forecast information, the DER aggregator needs to decide the internal price for buying energy from the DER owners. After receiving the energy offers from the DER owners, the DER aggregator needs to decide its energy offer in the wholesale market. In the fMMR strategy, the aggregator can determine the energy offer in the wholesale market flexibly without being constrained by the internal energy offers; in the cMMR strategy, the energy offer of the aggregator in the wholesale market equals to the sum of energy offers from the internal market. In the second stage, the DER aggregator balances energy deviations in both the wholesale and the internal market according to the pre-defined penalty mechanism. The decision framework of the DER aggregator is illustrated in Fig. 1. Under this decision framework, the DER aggregator arbitraging practice for both strategies can be modeled as bi-level two-stage MMR optimization problem, where the upper-level is the MMR arbitraging problem of the DER aggregator, the lower-level are the self-scheduling problem of the DER owners.

Notably, this work is focused on the arbitraging problem of DER aggregators and the technical constraints of the distribution network are not considered. However, the scheduling results of DERs may violate the distribution network operation constraints in some scenarios. Regarding this issue, coordinating the DER operation with the distribution network operator (DNO) becomes important [30] because ignoring network constraint can lead to infeasible scheduling decisions. To address this problem, network-constrained transactive energy trading models can be adopted to reflect the impacts of incorporating network operational constraints [31,32] by including power flow constraints.

Let u represent an uncertainty scenario consists of a market price λw and renewable generations ω=[ω1,…ωi,…ωn]. For a DER aggregator coordinating n DER owners, the optimal solution and the corresponding objective value φ(u) for a given uncertainty scenario u can be found by solving the following deterministic problem(3a) φ(u)=maxλinu,Pow,u{Pow,uλw−∑i∈nλinuPo,iin,u+maxPd,iin,u,Ps,iin,u,Pdw,u,Psw,u{∑i∈nλbd,in,uPd,iin,u+λsd,wPsw,u−∑i∈nλsd,in,uPs,iin,u−λbd,wPdw,u}}

s.t.(3b) Po,iin,u=PG,iu+ωi+Pd,iin,u−Ps,iin,u

(3c) Pow,u=∑i∈n(PG,iu+ωi)+Pdw−Psw

(3d) λmin≤λinu

(3e) [Pow,u,Po,iin,u,Pdw,u,Psw,u,Pd,iin,u,Ps,iin,u]≥0

where φ(u) is the optimal objective value for the given uncertainty scenario u. Pow,u is the energy offer of the DER aggregator in the wholesale market; Po,iin,u is the energy offer of the ith DER owner in the internal market. It should be noted that the internal energy offers are not determined by the aggregator but the DER owners according to the internal price, therefore, the value of Po,iin,u is a function of the internal price λinu. Terms Pdw,u and Psw,u are the energy deficit and surplus of the aggregator in the wholesale market, respectively. Terms Pd,iin,u and Ps,iin,u are the energy deficit and surplus of the ith DER owner in the internal market, respectively. The penalty prices λbd,in,u, λsd,in,u, λbd,w,u and λsd,w,u in both markets can be calculated from equations in (1). The negotiated minimum internal market price λmin ensures that the renewable energy is properly rewarded. Without the restriction of λmin, the aggregator may use very low internal prices to arbitrage from the renewable generations.

In the objective function, the first-stage terms Pow,uλw and ∑i∈nλinuPo,iin,u are the revenue in the wholesale market and cost in the internal market, respectively. The terms ∑i∈Iλbd,in,uPd,iin,u and λsd,wPsw,u represent the revenues from balancing the energy deficits in the internal market and the energy surplus in the wholesale market, respectively. Terms ∑i∈Iλsd,in,uPs,iin,u and λbd,wPdw,u are the balancing cost for the energy surplus in the internal market and the energy deficit in the wholesale market, respectively. All the balancing terms in the objective function belong to the second stage of the DER aggregator arbitraging problem. Constraints (3b), (3c) ensure that energy deviations are all balanced.

In this formulation, the wholesale market energy offer is unconstrained, namely, this formulation is for the fMMR strategy. For the cMMR strategy, we need to add one more constraint to make sure that the wholesale market offer equals to the sum of internal market offers:(3f) Pow,u=∑i∈nPo,iin,u

The aim of the MMR approach is to find the first-stage decisions [λin,Pow] such that the maximum profit difference between our solution and the optimal solution is minimized. For the fMMR strategy, the problem is:(4a) minλin,Powmaxu{φ(u)−maxPd,iin,Ps,iin,Pdw,Psw{Powλw−∑i∈nλinPo,iin+∑i∈Iλbd,inPd,iin+λsd,wPsw−∑i∈nλsd,inPs,iin−λbd,wPdw}}

s.t.(4b) Po,iin=PG,i+ωi+Pd,iin−Ps,iin

(4c) Pow=∑(PG,i+ωi)+Pdw−Psw

(4d) (1−γ)λwf≤λw≤(1+γ)λwf

(4e) (1−γ)ωif≤ωi≤min{(1+γ)ωif,ωic}

(4g) λmin≤λin

(4h) [Pow,Pdw,Psw,Pd,iin,Ps,iin]≥0

For the cMMR strategy, we need to add the constraint:(4i) Pow=∑i∈nPo,iin

In this formulation, the internal energy offer Po,iin is an optimized result from the ith DER owner's self-scheduling problem. Therefore, the optimization process of the aggregator requires information exchange with the DER owners through the internal market.

3 Solution methodology

In this section, the analytical solutions for the DER owners’ self-scheduling problem are derived, then the results are leveraged to transform the energy offering decisions Po,iin of the lower-level self-scheduling problem into decision variables of the upper-level arbitraging problem. By doing so, we can reformulate the communication-required bi-level arbitraging problem into single-level communication-free TSMIMMR optimization problem. Currently, there is no algorithm that can directly solve the reformulated TSMIMMR problem. To obtain exact solutions of the reformulated problem, we firstly transform the TSMIMMR optimization problem into equivalent TSMIRO problem, then resolve it under the C&CG framework.

3.1 Analytical solutions for DER owners

In our work, we assume that the production costs of the renewable generators are negligible, the dispatchable generators have linear cost functions and their ramping limits are not considered. Under this assumption, we conclude that, the optimal solutions for the DER owners MMR-based self-scheduling problem are summarized as follows:(5a) Po,iin=PG,i+ωif−ωifγρ2−ρ,(1+γ)ωif≤ωic

(5b) Po,iin=PG,i+ωif,m−ωif,mγmρ2−ρ,(1+γ)ωif>ωic

(5c) PG,i=PG,imax,λin≥ai

(5d) PG,i=0,λin<ai

(5e) ωif,m=ωic+(1−γ)ωif2

(5f) γm=ωic−(1−γ)ωif2ωif,m

where for type I DER owners, ωif,m=0, for type II DER owners, PG,imax=0. The proofs are provided in Appendices.

3.2 Single-level Reformulation and TSMIRO transformation

Now we can make use of the analytical solutions to turn the optimization results Po,iin of the lower-level problem into decision variables of the upper-level arbitraging problem. The reformulated arbitraging problem for the fMMR strategy is:(6a) minλin,Pow,Po,iin,yimaxu∈U{φ(u)−maxPd,iin,Ps,iin,Pdw,Psw{Powλw−∑i∈nλinPo,iin+∑i∈nλbd,inPd,iin+λsd,wPsw−∑i∈nλsd,inPs,iin−λbd,wPdw}}

s.t.(6b) yi(ai−λin)≤0

(6c) Po,iin=yiPG,imax+ωif−ωifγρ2−ρ

(6d) yi∈(0,1)

And constraints (4b) - (4h). The binary variable yi indicates the on/off status of the ith dispatchable generator. Constraint (6b) controls the on/off status of the dispatchable generators. Constraint (6c) gives the energy offer of the DER owners. It should be noted that the uncertainty coefficient and renewable generation forecast in (6c) should be modified according to (5e) and (5f) when needed. For the cMMR strategy, we also need to add constraint (4i).

The reformulated DER aggregator arbitraging problem are single-level TSMIMMR problem, which do not require information exchange between the DER aggregator and owners. To obtain their exact solutions, we transform the TSMIMMR problem into equivalent TSMIRO problem, the objective function of the transformed problem is:(7) minλin,Pow,Po,iin,yi{∑i∈nλinPo,iin+max[u,λinu,Pow,u,Po,iin,u,Pd,iin,u,Ps,iin,u,Pdw,u,Psw,u]∈U′{minPd,iin,Ps,iin,Pdw,Psw{λbd,wPdw+∑i∈nλsd,inPs,iin−Powλw−∑i∈nλbd,inPd,iin−λsd,wPsw+Pow,uλw−∑i∈nλinuPo,iin,u+∑i∈Iλbd,in,uPd,iin,u+λsd,wPsw,u−∑i∈nλsd,in,uPs,iin,u−λbd,wPdw,u}

where λinu,Pow,u,Po,iin,u,Pd,iin,u,Ps,iin,u,Pdw,u and Psw,u are the corresponding optimal decision variables of the optimal solution under scenario u; U′ is the enlarged uncertainty set. The detailed transformation process is presented in Appendix D. In previous research, the C&CG algorithm has shown its effectiveness in solving a two-stage robust optimization problem [28]. Next, a comprehensive C&CG framework is developed to precisely solve the TSMIRO problem.

3.3 Solution methodology

In the C&CG framework, a TSMIRO problem is broken down into main and sub problems. The master problem is created by easing certain constraints of the TSMIRO problem, and these constraints are incrementally reintroduced by solving the slave problem with various first-stage decisions. By progressively adding constraints to the master problem, its optimal objective value converges toward the optimal objective value of the original problem. The master problem, in the context of the decomposed TSMIRO problem for the fMMR strategy is:(8a) minλin,Pow,Po,iin,yi,θ∑i∈IλinPo,iin+θ

s.t.(8e) Po,iin=ωi,lv+Pd,i,lin−Ps,i,lin+yiPG,imax

(8f) Pow=∑i∈nωi,lv+Pd,lw−Ps,lw+∑i∈nyiPG,imax

(8g) θ≥(λb,ld,w,vPd,lw−Powλw,lv−λs,ld,w,vPs,lw+Po,lw,u,vλw,lv+∑i∈nλsd,inPs,i,lin−∑i∈nλbd,inPd,i,lin−∑i∈nλin,lu,vPo,i,lin,u,v+∑i∈nλb,ld,in,u,vPd,i,lin,u,v+λsd,w,vPs,lw,u,v−∑i∈nλs,ld,in,u,vPs,i,lin,u,v−λb,ld,w,vPd,lw,u,v)

And constraints (4g), (4h). The auxiliary variable θ represents the optimal objective value of the slave problem. Terms Pd,i,lin, Ps,i,lin, Pd,lw and Ps,lw are the new variables created in the (l−1)th iteration. The superscript v and subscript l for terms λw,lv, ωi,lv, Po,lw,u,v, Po,i,lin,u,v, Pd,i,lin,u,v, Ps,i,lin,u,v, Ps,lw,u,v and Pd,lw,u,v mean that they are values where the slave problem is optimized in the (l−1)th iteration. For the cMMR strategy, we should add constraint (4i).

The slave problem of the decomposed TSMIRO problem for the fMMR strategy is defined as:(9a) max[u,λinu,Pow,u,Po,iin,u,Pd,iin,u,Ps,iin,u,Pdw,u,Psw,u]∈U′{minPd,iin,Ps,iin,Pdw,Psw{λbd,wPdw+∑i∈nλsd,inPs,iin−Pow,vλw−∑i∈nλbd,inPd,iin−λsd,wPsw+Pow,uλw−∑i∈nλinuPo,iin,u+∑i∈nλbd,in,uPd,iin,u+λsd,wPsw,u−∑i∈Iλsd,in,uPs,iin,u−λbd,wPdw,u}

s.t.(9b) Po,iin,v=yivPG,imax+ωi+Pd,iin−Ps,iin

(9c) Pow,v=∑i∈n(yivPG,imax+ωi)+Pdw−Psw

(9d) Pow,u=∑i∈n(yiuPG,imax+PR,i)+Pdw,u−Psw,u

(9e) Po,iin,u=yiuPG,imax+PR,i+Pd,iin,u−Ps,iin,u

(9h) yiu(ai−λinu)≤0

(9i) Po,iin,u=yiuPGmax+PR,if−PRfγρ2−ρ

(9j) λmin≤λinu

(9k) yiu∈(0,1)

(9l) [Pd,iin,Ps,iin,Pdw,Psw,Pdw,u,Psw,u,Pd,iin,u,Ps,iin,u]≥0

And constraints (4d), (4e). The terms Po,iin,v, yiv, λinv and Pow,v are input values from the master problem in the same iteration. For the cMMR strategy, we should add the following constraint.(9m) Pow,u=∑i∈nPo,iin,u

In the C&CG framework, we use the results from solving the slave problem to set up constraints for the master problem. There are usually two types of constraints added to the master problem – one for feasibility when the slave problem is not feasible, and another for optimality when the slave problem is feasible. In our scenario, deviations can always be balanced in both the internal and wholesale markets. Hence, the slave problem is always feasible, and the concern is mainly about the optimality constraints.

The adjustment of lower and upper bounds in the TSMIRO problem involves solving both the master and slave problems. Solving the master problem yields lower bounds for the TSMIRO problem as the master problem relaxes the constraints of the original minimization problem, resulting in overly optimal outcomes. Similarly, solving the slave problem provides upper bounds since the feasible space in the slave problem is restricted (by fixing the values of first-stage decision variables), leading to sub-optimal results compared to the original problem.

The complete C&CG algorithm for solving this problem is described as follows.1) Set the lower bound LB to −∞, the upper bound UB to ∞, the iteration number l=0, the tolerance ε=10−3.

2) Solve the master problem (8) and derive an optimal solution of (λin,l+1v,Po,l+1w,v,Po,i,l+1in,v,yiv,θl+1v). Update the lower bound LB to max{LB,∑i∈Iλin,l+1vPo,i,l+1in,v+θl+1v}.

3) Solve the slave problem by substituting the obtained optimal first-stage decisions. Derive the values of λw,l+1v, ωi,l+1v, Po,l+1w,u,v, Po,i,l+1in,u,v, Pd,i,l+1in,u,v, Ps,i,l+1in,u,v, Ps,l+1w,u,v and Pd,l+1w,u,v. Derive the optimal objective value of the slave problem θ(λin,l+1v,Po,l+1w,v,Po,i,l+1in,v,yi). Update the upper bound UB to min{UB,∑i∈Iλin,l+1vPo,i,l+1in,v+θ(λin,l+1v,Po,l+1w,v,Po,i,l+1in,v,yi)}

4) If UB−LB≤ε, terminate the iterations and return λin,l+1v, Po,l+1w,v, Po,i,l+1in,v and yiv; otherwise, create new variables Pd,i,l+1in, Ps,i,l+1in, Pd,l+1w and Ps,l+1w, add constraints (8e)-(8g) to the master problem, go to step 2.

The flowchart of the C&CG algorithm is provided in Fig. 2.Fig. 2 The C&CG solution process flowchart.

Fig. 2

4 Case studies

In this section, we first describe the details of the datasets we used, then numerically evaluate the performances of the proposed MMR arbitraging strategies.

4.1 Basic data

The DER owners and their generator characteristics are summarized in Table I. The considered renewable generators in our work are wind generators, which are close in geography locations such that they can share the same generation forecast profile. The negotiated minimum internal price is 10 £/MWh. The penalty coefficient ρ is set to be 0.5.Table 1 DER owners and their generator characteristics.

Table 1DER owner	DER type	Thermal unit production cost (£/MWh)	Thermal unit capacity (MW)	Wind generator capacity (MW)	
1	I	34	6	/	
2	I	31	11	/	
3	I	32	9	/	
4	II	/	/	4	
5	II	/	/	6	
6	II	/	/	5	
7	III	26	15	7	
8	III	29	13	5	
9	III	27	14	6	

Without loss of generality, the day-ahead UK energy price is used, and it is assumed that the prices are unknown until each market-clearing time slot. The wind generation forecast data we use is the actual aggregated onshore UK wind generation data [33] in the same day divided by 100. The reason to divide the wind generation data by 100 is to make the data suitable for the generator capacities in our case study problem. We set the uncertainty coefficient γ to be 0.5. The forecast data and intervals for market price and aggregated wind power generation are presented in Fig. 3.Fig. 3 The forecast values and intervals for (a) the aggregated wind generation and (b) the market price in Jan 3rd, 2021.

Fig. 3

4.2 Results and discussions

The benchmark employed to assess the proposed arbitraging strategies is the perfect information approach (PIA) [34], renowned for its capacity to achieve the maximum possible profit. In addition, the maximin profit (MMP) strategy, characterized as a distribution-free approach, is applied to the same arbitraging problem for comparative purposes. The MMP approach seeks to maximize the worst-case profit for the aggregator. Following the derivation of solutions from the MMR and MMP strategies, the assessment of their performances involves the generation of 10,000 random scenarios based on forecast data and the uncertainty coefficient γ, ensuring statistically significant results. The evaluation metric utilized is profit percentages (PP), where the hourly PP for each arbitraging strategy is defined as follows:(9a) PPcMMR,th=ProfitcMMR,tProfitPIA,t100%

(9b) PPfMMR,th=ProfitfMMR,tProfitPIA,t100%

(9c) PPMMP,th=ProfitMMP,tProfitPIA,t100%

Fig. 3 depicts the hourly PPs of the arbitraging strategies over 24 clearing periods. From Fig. 3, Fig. 4 we can see that the hourly PP patterns of the fMMR and cMMR strategies comply with the wholesale market price pattern, which indicates that the proposed strategies are able to seize the arbitraging opportunities in the wholesale market. As a comparison, the hourly PP pattern of the MMP strategy sometimes violates the market price pattern, such as the hours between 6 and 9 o'clock, where the hourly PPs of the MMP strategy shows an opposite trend from the market price forecast. This violation is because in these hours the market prices are very low and the extremely bad situations for which the MMP strategy prepares are more likely to happen.Fig. 4 The hourly profit percentages of different arbitraging strategies.

Fig. 4

Also, both the proposed fMMR and cMMR strategies can achieve near-to-optimal profits most of the time. In 19 clearing periods of the day, both MMR arbitraging strategies achieved hourly PPs higher than 80 %. In comparison, the MMP arbitraging strategy only achieved hourly PPs higher than 80 % in 2 clearing periods. Between the two MMR strategies, we observe that the cMMR strategy always slightly outperforms the fMMR strategy. This shows the influence of different wholesale market offering decisions. Fig. 5 shows that the offering behavior of the cMMR strategy is more conservative than the fMMR strategy. Since our penalty mechanism punishes the energy deficit more than energy surplus, as a result, the performance of the cMMR strategy is slightly better than the fMMR strategy.Fig. 5 The aggregator's energy offer decisions in the wholesale market under different arbitraging strategies.

Fig. 5

The resource utilization rate is a crucial factor that affects the profit-making ability of the arbitraging strategy, larger resource utilization rate means more opportunity to earn profits. In our problem, the utilization rate of the dispatchable resource is the ratio between the utilized capacity and the total available capacity. The dispatchable resource utilization rate of each arbitraging strategy is displayed in Fig. 6. The utilization rates of both MMR strategies are 100 % except for hours 7 and 9. In hours 7 and 9, the wholesale market price forecasts are very low and the utilization rate for both MMR strategies is 91 %. As a comparison, the utilization rates of the MMP strategy are 0 % in more than half of the time, and its maximum utilization rate is 91 %, which only happened high price hours 18 and 19.Fig. 6 Utilization rate of the dispatchable generators for different arbitraging strategies.

Fig. 6

4.3 Uncertainty level Sensitivity Analysis

(2) Uncertainty Level

The uncertainty coefficient γ reveals the variability of the environment, larger γ represents a more volatile environment and worse predictability. In this subsection, we evaluate the ability of the proposed MMR strategies to maintain economic performances when the environment becomes more uncertain. To evaluate this ability of the proposed strategies, we use daily PPs to show the averaged economic performances over a day, the daily PPs are defined as:(10a) PPfMMRd=∑t=1:24ProfitfMMR,t∑t=1:24ProfitPIA,t

(10b) PPcMMRd=∑t=1:24ProfitcMMR,t∑t=1:24ProfitPIA,t

(10c) PPMMPd=∑t=1:24ProfitMMP,t∑t=1:24ProfitPIA,t

In Fig. 7, as the penalty coefficient is fixed at 0.5 and increase the uncertainty coefficient from 0.3 to 0.7, the daily PP reductions for the fMMR, cMMR, and MPP strategies are 11.35 %, 11.26 % and 58.06 %, respectively. Comparing with the MMP strategy, when the uncertainty level increases significantly, the small daily PP reductions indicate that both the cMMR and the fMMR arbitraging strategies can maintain satisfactory performances when the environment becomes more uncertain.Fig. 7 The daily PPs of the arbitraging strategies for different uncertainty levels.

Fig. 7

5 Conclusion

This work studied the arbitraging problem of DER aggregators under renewable and market price uncertainties considering the wholesale energy market and DER internal market. This work first proposed a two-stage self-adaptive MMR model to optimally make DER market-bidding and internal clearing decisions. Then, different types of DER owners are modeled and integrated into the operation of the DER aggregator. In addition, an exact and communication-free solution methodology for the DER aggregator arbitraging problem is developed to simplify the solution process. As compared to the existing approaches, the proposed methods can achieve higher profit without required accurate uncertainty probability distributions. Moreover, the proposed methods can maintain its financial performance even if the environment becomes more volatile as compared to conventional robust optimization. Some possible future extension of this work can include the coordination with the distribution network operator and include more emerging energy sources such as batteries and electric vehicles.

CRediT authorship contribution statement

Peng Xie: Writing – review & editing, Writing – original draft, Funding acquisition. Mingjun Liu: Conceptualization. Chun Chen: Writing – review & editing, Data curation. Wenming Tang: Writing – review & editing. Han Wang: Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A Deduction for solutions of type 1 DER owners:

Type 1 DER owners only have dispatchable generators with linear production cost, therefore, as long as the internal price is higher than their production cost, they will offer and schedule the generators at the maximum power to earn profit. That is:(11a) Po,iin,type1=PG,imax,λin≥ai

(11b) PG,itype1=PG,imax,λin≥ai

When the internal price is lower than their production cost, they will not turn on the generators and the energy offers are also zero:(11c) Po,iin,type1=0,λin≤ai

(11d) PG,itype1=0,λin≤ai

B Deduction for solutions of type 2 DER owners:

For type 2 DER owners, they do not have dispatch generators, thus, their dispatchable generation is 0. That is:(12a) PG,itype2=0

Case 1 The upper bound of the renewable generation is smaller than the installed capacity

Now assume the energy offer of a type 2 owner is Po,iin,type2∈[(1−γ)ωif,(1+γ)ωif].

When the actual renewable generation ωi is larger than Po,iin,type2, the regret of this owner is:(12b) Reg[ωi>Po,iin,type2]=λinωi−[λinPo,iin,type2+λin(1−ρ)×(ωi−Po,iin,type2)]=λinρ(ωi−Po,iin,type2)

When the actual renewable generation ωi is smaller than Po,iin,type2, the regret of this owner is:(12c) Reg[ωi>Po,iin,type2]=λinωi−[λinPo,iin,type2−λin1−ρ(Po,iin,type2−ωi)]=λinρ1−ρ(Po,iin,type2−ωi)

Since both regrets are linear functions of the actual renewable generation ωi. The maximum regrets can be found at the ending points of the renewable generation interval, that is:(12d) maxReg[ωi>Po,iin,type2]≡Reg[ωi=(1+γ)ωif>Po,iin,type2]≡λinρ[(1+γ)ωif−Po,iin,type2]

Which leads to:(12e) maxReg[ωi<Po,iin,type2]≡Reg[ωi=(1−γ)ωif<Po,iin,type2]≡λinρ1−ρ(Po,iin,type2−(1−γ)ωif)

The solution that can minimize the maximum regret can be found when both regrets are equal, that is:(12f) λinρ[(1+γ)ωif−Po,iin,type2]=λinρ1−ρ(Po,iin,type2−(1−γ)ωif)

Which gives:(12g) Po,iin,type2=ωif(1−ργ2−ρ)

Case 2 The upper bound of the renewable generation is larger than the capacity

When the upper bound of the renewable generation is larger the installed capacity, we simply need to change the upper bound to ωic, and recalculate the regrets at both ending points, by equating the new ending point regrets:(12h) λinρ[ωic−Po,iin,type2]=λinρ1−ρ(Po,iin,type2−(1−γ)ωif)

We have:(12i) Po,itype2=ωif,m(1−ργm2−ρ)

where the modified renewable generation forecast ωif,m and the uncertainty coefficient γm are given as:(12j) ωif,m=ωic+(1−γ)ωif2

(12k) γm=ωic−(1−γ)ωif2ωif,m

C Proof for optimal solutions of type 3 DER owners:

We conclude that the optimal self-scheduling solution for a type 3 owners is.Case 1 λin≥aiand(1+γ)ωif≤ωic :(13a) Po,itype3=PG,imax+ωif(1−ργ2−ρ)

(13b) PG,itype3=PG,imax

Case 2 λin<aiand(1+γ)ωif≤ωic :(13c) Po,itype3=ωif(1−ργ2−ρ)

(13d) PG,itype3=0

Case 3 λin≥aiand(1+γ)ωif>ωic :(13e) Po,itype3=PG,imax+ωif,m(1−ργm2−ρ)

(13f) PG,itype3=PG,imax

Case 4 λin<aiand(1+γ)ωif>ωic :(13d) Po,itype3=ωif,m(1−ργm2−ρ)

(13f) PG,itype3=0

We present the proof for case 1, for other cases, the proofs can be obtained in a similar way. To prove our conclusion, we first find the optimal generator scheduling decision for a given energy offer decision, then find the optimal offering decision.

I The optimal generator scheduling decision

For an energy offer Po,i, a renewable generation forecast ωif, assume the scheduled power generation is PG,i and the actual renewable generation is ωi. Since λin≥aiand(1+γ)ωif≤ωic, the optimal generation scheduling PG,iu is:(14a) PG,iu=Po,i−ωi

(1) When PG,i+ωi<Po,i :

(14b) Reg[PG,i]=Profit[PG,iu]−Profit[PG,i]≡(Po,iλin−PG,iuai)−[Po,iλin−PG,iai−(Po,i−PG,i−ωi)λin1−ρ]=ρλin1−ρ(Po,i−PG,i−ωi)

(2) When PG,i+PR,i>Po,i :

(14c) Reg[PG,i]=Profit[PG,iu]−Profit[PG,i]≡(Po,iλin−PG,iuai)−[Po,iλin−PG,iai+(PG,i+ωi−Po,i)(1−ρ)λin=(PG,i+ωi−Po,i)ρλin

Since both regrets are linear functions of ωi, we can find the maximum regrets at the ending points, and the generation scheduling solution that minimizes the maximum regret can be found when both regrets are equal, namely:(14d) ρλin1−ρ(Po,i−PG,i−(1−γ)ωif)=(PG,i+(1+γ)ωif−Po,i)ρλin

Which gives the optimal MMR scheduling solution:(14e) PG,i=Po,i−ωif+γρωif2−ρ

II The optimal energy offer decision

For two energy offer decisions Po,i,1 and Po,i,2 where Po,i,1>Po,i,2, from (14e) we know that their MMR-based generator scheduling decisions are given as Po,i,1−ωif+γρωif2−ρ and Po,i,2−ωif+γρωif2−ρ , respectively. For a given actual renewable generation ωi , the energy deviations for both Po,i,1 and Po,i,2 are:(14f) Pdeviation,1=Po,i,1−(Po,i,1−ωif+γρωif2−ρ)−ωi=ωif+γρωif2−ρ−ωi

(14g) Pdeviation,2=Po,i,2−(Po,i,2−PR,if+γρωif2−ρ)−ωi=ωif+γρωif2−ρ−ωi

The profits for Po,i,1 and Po,i,2 are:(14h) Profit1=Po,i,1λin−(Po,i,1−ωif+γρωif2−ρ)ai−Costdeviation,1

(14i) Profit2=Po,i,2λin−(Po,i,2−ωif+γρωif2−ρ)ai−Costdeviation,2

As can be seen from (14f), (14g), the energy deviations for Po,i,1 and Po,i,2 are the same, thus, their balancing costs (or revenue) in the second stage will be the same. The profit difference (PD) between Po,i,1 and Po,i,2 is:(14j) PD=Profit1−Profit2=Po,i,1λin−(Po,i,1−ωif+γρωif2−ρ)ai−[Po,i,2λin−(Po,i,2−ωif+γρωif2−ρ)ai]=(Po,i,1−Po,i,2)(λin−ai)>0

Therefore, larger energy offer always makes more profit than smaller energy offers in case 1. Since we used the result in (14e), this conclusion is only valid for energy offers satisfying the following condition:(14k) Po,i≤PG,imax+ωif−γρωif2−ρ

Therefore, in the range between [0, PG,imax+ωif−γρωif2−ρ], the energy offering decision Po,i=PG,imax+ωif−γρωif2−ρ is the optimal choice and we denote it as Po,istar. However, there are cases when PG,imax+ωif−γρωif2−ρ≤Po,i≤PG,imax+(1+γ)ωif. We denote an energy offering decision in the range [PG,imax+ωif−γρωif2−ρ,PG,imax+(1+γ)ωif] as Po,ilarge and prove that the maximum regret of any Po,ilarge is larger than the maximum regret of Po,istar.

The regrets of Po,istar at both ending points are:(14l) Reg[Po,istar,ωi=(1+γ)ωif]=ρλin[PG,imax+(1+γ)ωif−Po,istar]

(14m) Reg[Po,istar,ωi=(1−γ)ωif]=ρλin1−ρ[Po,istar−PG,imax−(1−γ)ωif]

Subtracting (14l) from (14m) gives:(14n) ρλin1−ρ[Po,istar−PG,imax−(1−γ)ωif]−ρλin[PG,imax+(1+γ)ωif−Po,istar]=2−ρ1−ρ(Po,istar−PG,imax−ωif+γρωif2−ρ)=0

Therefore, both regrets are the same, the maximum regret for Po,istar can be represented by (14l) or (14m).

For Po,ilarge, it is reasonable to set its corresponding energy generation as PG,imax since smaller PG,i will result in larger regret. Still, we find the maximum regrets at both ending points of the renewable generation:(14n) Reg[Po,ilarge,ωi=(1+γ)ωif]=ρλin[PG,imax+(1+γ)ωif−Po,ilarge]

(14o) Reg[Po,ilarge,ωi=(1−γ)ωif]=ρλin1−ρ[Po,ilarge−PG,imax−(1−γ)ωif]

To compare the two regrets, we subtract (14m) from (14n):(14p) ρλin1−ρ[Po,ilarge−PG,imax−(1−γ)ωif]−ρλin[PG,imax+(1(1+γ)ωif−Po,ilarge]=2−ρ1−ρ(Po,ilarge−PG,imax−ωif+γρωif2−ρ)>0

Therefore, the largest regret for Po,ilarge is given by (14o).

Now we compare the maximum regret of Po,ilarge with Po,istar, by subtracting (14m) from (14o) we have:(14p) Reg(Po,ilarge)−Reg(Po,istar)=ρλin1−ρ[Po,ilarge−PG,imax−(1−γ)ωif]−ρλin1−ρ[Po,istar−PG,imax−(1−γ)ωif]=ρλin1−ρ(Po,ilarge−Po,istar)>0

This implies that the maximum regret of Po,istar is smaller than the maximum regret of Po,ilarge.

Therefore, the optimal energy offering solution for a type 3 DER owner is Po,i=Po,istar and PG,i=PG,imax. For other cases, proofs can be obtained through a similar way.

D TSMIRO Transformation of TSMIMMR Problem

The complete objective function of the TSMIMMR problem is:(15a) minλin,Pow,Po,iinmaxu{maxλinu,Pow,u,Po,iin,u,Pd,iin,u,Ps,iin,u,Pdw,u,Psw,u{Pow,uλw−∑i∈nλinuPo,iin,u+∑i∈nλbd,in,uPd,iin,u+λsd,wPsw,u−∑i∈nλsd,in,uPs,iin,u−λbd,wPdw,u}−maxPd,iin,Ps,iin,Pdw,Psw{Powλw−∑i∈nλinPo,iin+∑i∈nλbd,inPd,iin+λsd,wPsw−∑i∈nλsd,inPs,iin−λbd,wPdw}}

To integrate the inner optimization problem, we enlarge the uncertainty set U to U′ by adding the optimal solution decision variables into it. The problem becomes:(15b) minλin,Pow,Po,iin,yi{max[u,λinu,Pow,u,Po,iin,u,Pd,iin,u,Ps,iin,u,Pdw,u,Psw,u]∈U′{Pow,uλw−∑i∈nλinuPo,iin,u+∑i∈nλbd,in,uPd,iin,u+λsd,wPsw,u−∑i∈nλsd,in,uPs,iin,u−λbd,wPdw,u}−maxPd,iin,Ps,iin,Pdw,Psw{Powλw−∑i∈nλinPo,iin+∑i∈nλbd,inPd,iin+λsd,wPsw−∑i∈nλsd,inPs,iin−λbd,wPdw}}}

(15c) ≡minλin,Pow,Po,iin,yi{∑i∈nλinPo,iin+max[u,λinu,Pow,u,Po,iin,u,Pd,iin,u,Ps,iin,u,Pdw,u,Psw,u]∈U′{minPd,iin,Ps,iin,Pdw,Psw{λbd,wPdw+∑i∈nλsd,inPs,iin−Powλw−∑i∈nλbd,inPd,iin−λsd,wPsw+Pow,uλw−∑i∈nλinuPo,iin,u+∑i∈Iλbd,in,uPd,iin,u+λsd,wPsw,u−∑i∈nλsd,in,uPs,iin,u−λbd,wPdw,u}

From (15b) to (15c), we first take the internal buying cost ∑i∈IλinPo,iin out, then rewrote the profit-maximization problem into a cost-minimization problem. The objective function (15c) is now a typical two-stage robust optimization problem objective function.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant No. 72401205 ), the Special Innovation Projects of Ordinary Colleges and Universities in Guangdong Province (Grant No. 2024KTSCX258 ), and the Shenzhen Fundamental Research Program Stability Support Program for Higher Education Institutions (Grant No. 20231127142912001 ).
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