Wholesale Market Participation of DERA:
Competitive DER Aggregation

There is growing literature on DER aggregation and wholesale market participation models that broadly fall into two categories. One is through a retail market design operated by a distribution system operator (DSO) [9, 10], an aggregation/sharing platform [11, 12], or an energy coalition such as CCA [13, 14, 15]. For the most part, these works do not consider a profit-maximizing DERA’s active participation in the wholesale market. In particular, in [9, 10, 11], the DSO or an aggregation platform participates in the wholesale markets with the aggregated net demand (or possibly net production), treating the wholesale market as a balancing resource.

Our approach belongs to the second category of DER aggregations, where profit-seeking DERAs aggregate both generation and flexible demand resources, participating directly in the wholesale market with bid/offer curves. To ensure secure distribution network operation, DERA obeys the allocated distribution network access limit (or operating envelope [16]), rather than considering the computationally expensive network power flow constraints. Within the framework of FERC order 2222, this type of DER aggregation has the potential to improve the overall system efficiency and reliability.

Although the notion of competitive DER aggregation has not been formally defined, two prior works have developed competitive aggregation solutions in [13, 17]. In [13], Chakraborty et al. consider DER aggregation by a CCA, where the authors provide an allocation rule that offers its customers competitive services with respect to the regulated utility.

Most relevant to our work is the DERA’s wholesale market participation method developed by Gao, Alsheheri, and Birge [17] where the authors consider a profit-seeking DERA aggregating BTM distributed generations (DGs) and offering its aggregated generation resources to the wholesale market. In particular, the Gao-Alshehri-Birge (GAB) approach achieves a social surplus equal to that achievable by customers’ direct participation in the competitive wholesale market. In other words, GAB approaches the most economically efficient participation model. A significant difference between [17] and this paper is that we formulate a general competitive aggregation problem that includes the regulated utility. In achieving DERA’s profit maximization, our DER aggregation and market participation models are different from [17].

The approach proposed in [17] follows the earlier work of Alshehri, Ndrio, Bose, and Başar [18] where a Stackelberg game-theoretic model is used. Both approaches assume that the DERA elicits prosumer participation with an optimized (one-part or two-part) price, and the prosumer responds with its quantity to be aggregated by the DERA. The real-time wholesale market price is reflected by the variable price in [7, 8, 17]. Such a variable price conveys low but volatile wholesale prices directly to customers. To protect customers from price spikes in the real-time wholesale price, some methods like price caps [19] have been proposed. As for the wholesale market participation model of DERA, quantity bid is adopted in [17], price-quantity bid in [18, 16], and virtual power plant in [20], with which we have the same assumption that the DERA centrally schedules customers’ consumption.

In our previous work [1], we develop the first profit-maximizing competitive DER aggregation for a DERA to participate in the wholesale market. In this paper, we further explore the price stability, profitability, market efficiency, and long-run equilibrium for the competitive DER aggregation, especially under the distribution network access limits.

First, we propose a DER aggregation approach based on a constrained optimization that maximizes DERA surplus while providing higher surpluses than that offered by a competing aggregation model. In particular, we are interested in aggregation schemes that are competitive with the regulated utility rates such as NEM X,¹¹1NEM X, proposed in [21] is an inclusive parametric model that captures key features of the existing and proposed NEM tariff models. with which a customer can make cost-benefit comparisons in her decision to become a customer of the DERA. We show that such a competitive DER aggregation, despite that the aggregation involving real-time wholesale locational marginal price (LMP), has an energy cost no greater than NEM X. This implies the proposed DER aggregation mechanism ensures price stability regardless of the volatility of the wholesale market LMP, a property missing in Griddy’s pricing model [7]. Meanwhile, we establish the profitability of DERA when competing with NEM X.

Second, we propose a virtual storage model for DERA’s wholesale market participation compatible with the practical continuous storage facility participation considered by ISOs [22, 23] under FERC order 841. The DERA bidding curve is derived from the closed-form solution of the proposed DERA model. While the aggregation optimization explicitly involves wholesale market LMP, the virtual storage bidding curves do not require forecasting of LMP. We show that the proposed DERA wholesale market participation results in market efficiency equal to what is achievable when DERA’s customers participate directly in the wholesale market.

Finally, we derive the benefit function of DERA over distribution network injection and withdrawal access limits. DERAs compete in the distribution network access auction proposed by [24] to acquire network access and we empirically evaluate the number of surviving DERAs in the long-run competitive equilibrium. Additionally, we present a set of numerical results, comparing the surplus distribution of the proposed competitive aggregation solution with those of various alternatives, including the regulated utility. Among significant insights gained are the higher social surplus, customer surplus, and DERA surplus achievable in the proposed competitive DERA model, when compared to other alternatives.

A few words are in order on the scope and limitations of this paper. First, the losses in distribution systems are not considered. Second, the contingency cases where DSO rejects cleared bids and offers from DERA for reliability concerns [2] are neglected. Under the access limit allocation framework proposed in [24], reliability concerns of DER aggregation are already satisfied under normal operating conditions. Lastly, although the proposed competitive aggregation offers higher benefits to DERA customers, it does so with a discriminative payment function, which might raise some equity concerns.

In Sec. II, we summarize the DER aggregation model and its main interactions. The problem of competitive DER aggregation is formulated in Sec. III where we derive the optimal aggregation solution. Sec. IV and Sec. V consider DERA’s wholesale market participation and its bidding strategies in the distribution network access auction, respectively. Numerical simulations are presented in Sec. VI. Sketch proofs are relegated in the appendix.

A list of major designated symbols is shown in Table I. The notations used here are standard. We use boldface letters for column vectors as in ${\bm{x}}=(x_{1},\cdots,x_{n})$. In particular, $\bm{1}$ is a column vector of all ones. The indicator function is denoted by $\mathbbm{1}\{x_{n}\leq y_{n}\}$, which equals to 1 if $x_{n}\leq y_{n}$, and 0 otherwise. $\mathbb{R}_{+}$ represents the set of all nonnegative real numbers. $[x]$ represents the set of integers from 1 to $x$, i.e., $[x]:=\{1,...,x\}$.

We assume the DERA aggregates resources in a retail market from residential, commercial, and industrial customers who have the option of being served by a regulated utility. Under a single-bill payment model, each customer settles its payment for consumption and compensation for production with the DERA. The DERA deploys an energy management system (EMS) with direct controls of the customer’s BTM generation and flexible demand resources, such as rooftop PV, HVAC, water heaters, and EV chargers.²²2The direct control in [20] is implemented through cloud-based platforms. The DERA optimizes the customer’s BTM resources under competitive aggregation optimization and provides the customer with a cost-benefit comparison with the NEM benchmark offered by the incumbent public utility. See Sec. III for details.

In this paper, we focus on DERA’s participation in the energy market based on a virtual storage model compatible with the continuous storage facility participation model [22]. See Sec. IV for the construction of bid/offer curves. To this end, the DERA submits offer/bid curves or self-scheduled quantity bids. The DERA may participate in both the day-ahead and real-time markets, although here we focus only on the real-time energy or energy-reserve market participation. The DERA may also deploy its own DG and storage capabilities to mitigate aggregation uncertainties. The strategy of incorporating DERA’s own resources in the overall DER aggregation will be discussed in future research.

We consider the DERA-DSO coordination model developed in [24] where the DERA acquires access limits at buses in the distribution network operated by a DSO. DERA’s willingness to pay for network access is explained in Sec.V. In particular, after the access limit auction or a direct bilateral contract with DSO, the DERA must aggregate DER from its customers in such a way that abides by the injection and withdrawal constraints set by the allocated access limits, i.e., (1d), in the real-time operation. That way, DERA’s aggregation has no effect on the operational reliability of the DSO under nominal operating conditions,³³3The contingency conditions can be considered by out-of-market coordination, or by extending the network access allocation in [24]. avoiding DSO intervention on ISO dispatch of DERA’s aggregation.

We consider a DERA aggregating over $N$ point of aggregations (PoAs) in the distribution network. We define PoAs⁵⁵5A diagram illustrating PoA is in Fig. 2 of [24]. as the main buses with higher voltages in the distribution network, which can be recognized with main substation information [25]. For simplicity, we illustrate the single time interval aggregation model of one prosumer at each PoA,⁶⁶6The model (1) can also be applied to the representative customer at each PoA. Such a representative customer exists when utility functions of all customers at each PoA have the Gorman form [26, P119]. and all PoAs face one zonal LMP. This model can be extended to two general cases: (i) the aggregation with multiple time intervals, whose empirical analysis is shown in Sec. VI-E, and (ii) the aggregation with multiple prosumers at one PoA and the device-level control, shown in Appendix VIII-G.

The DERA solves in real-time for the consumption bundle of all customers ${\bm{d}}:=(d_{n})_{n\in[N]}$ and their payment functions $\hbox{\boldmath$\omega$\unboldmath}:=(\omega_{n})_{n\in[N]}$ from the following optimization.

where the objective represents the DERA profit given real-time BTM DG ${\bm{g}}:=(g_{n})_{n\in[N]}$ and LMP $\pi_{\mbox{\tiny LMP}}\in\mathbb{R}$.

To attain customers in the energy aggregation, we design the ${\cal K}$-competitive constraint (1b) to ensure that the surplus of prosumer $n$ under DERA is higher than the benchmark prosumer surplus ${\cal K}_{n}$, whose detailed setting is explained in Sec. III-B. This is the criteria for a rational customer, seeking surplus maximization, to join a DERA. Otherwise, a rational prosumer has the incentive to leave DERA and switch to the benchmark service provider for a higher customer surplus. The prosumer utility for consuming energy $d_{n}$ is represented by the function $U_{n}$, which is assumed to be concave, nonnegative, nondecreasing, continuously differentiable, and $U_{n}(0)=0$. We assume the consumer utility function $U_{n}$ is given. In practice, utility functions can be computed with parametric[21] or nonparametric[27] methods.

The rest of the constraints in (1) impose operation limits. $\underline{d}_{n}$ and $\bar{d}_{n}$ are prosumer consumption limits. The DERA has corresponding injection and withdrawal access limits at the PoAs of distribution networks, represented respectively by exogenous parameters $\overline{C}_{n},\underline{C}_{n}\in\mathbb{R}_{+}$ with details in Sec. V.

The sketch of proof is provided in Appendix VIII-D, following the convexity and Karush-Kuhn-Tucker (KKT) conditions of (1). This optimal solution has two noteworthy characteristics. First, the optimal consumption in (2a) is only a function of the wholesale LMP $\pi_{\mbox{\tiny LMP}}$ when DERA purchases enough injection and withdrawal access at the PoA $(\underline{C}_{n},\overline{C}_{n}>>1)$. Note also the difference between the optimal consumption schedule in (2a) and those in [18] where the optimally scheduled consumption always depends on the anticipated LMP and forecast of BTM DG. Second, (1) finds a Pareto efficient allocation that maximizes the surplus of the DERA, subject to the constraint that the aggregated customer has the given level of surplus ${\cal K}_{n}$. Similar optimization and the Pareto efficient allocation are also analyzed in [28, P602] for first-degree price discrimination. The payment function $\omega_{n}$ can be realized by a two-part tariff, which is explored by GAB, although it unidirectionally aggregates BTM DG [17]. Overall, such a closed-form solution allows DERA to apply simple dispatch and pricing policies over massive aggregated households.

We analyze the profitability of DERA and the energy consumption cost of aggregated prosumers for the optimal DERA aggregation competitive with a regulated NEM tariff parameterized by the retail (consumption) rate $\pi^{+}$, the sell (production) rate $\pi^{-}$, and the connection charge $\pi^{0}$. Assume $0\leq\pi_{\mbox{\tiny LMP}}\leq\pi^{-}\leq\pi^{+}$ without loss of generality. Denote the $n$-th prosumer surplus under NEM X to be $S_{n}^{\mbox{\tiny NEM}}(g_{n},\underline{C}_{n},\overline{C}_{n})$, whose computation depends on the DG generation and network access limits (formulation provided in Appendix VIII-A). We can set the benchmark prosumer surplus ${\cal K}_{n}=\zeta S_{n}^{\mbox{\tiny NEM}}(g_{n},\underline{C}_{n},\overline{C}_{n})$, $\zeta\geq 1$ to obtain competitive aggregation over the DSO’s NEM-based aggregation with the same distribution network access.⁷⁷7Customers owning DERs switch from NEM X to DERA for higher consumer surplus, granting DERs control to DERA upon joining.

The ${\cal K}$-competitive constraint in (1b) has significant implications on pricing stability, despite that the aggregation is based on real-time LMP. Price stability means the price and payment faced by customers cannot go randomly high, for which a counterexample is the real-time LMP. Because the NEM tariff has price stability, achieving a finite customer payment regardless of the wholesale LMP fluctuation, an aggregation mechanism competitive with the NEM tariff must also be stable. The proposition below formalizes this intuition.

The sketch of proof is provided in Appendix VIII-B. Such price stability comes directly from the ${\cal K}$-competitive constraint, which enforces a lower bound for customer surplus and thus naturally limits the maximum customer payment. Note that the two-part pricing of Griddy [7] is not a stable pricing mechanism because the retail rate is tied directly to the real-time LMP.

In the ${\cal K}$-competitive constraint (1b), $\zeta$ controls the surplus distribution among DERA and its aggregated prosumers. A larger $\zeta$ rebates more benefits to prosumers and incentivizes prosumers to join DERA, although it increases the deficit risk of DERA. DERA has to set $\zeta$ properly to avoid deficits while attracting prosumers. In the proposition below, we guarantee DERA’s profitability when $\zeta$ is bounded above and all prosumers’ surpluses are nonnegative under NEM X.

The sketch of proof is provided in Appendix VIII-C. So, if NEM X is not crediting BTM DG of prosumer at a higher production rate $\pi^{-}$ than wholesale LMP, our aggregation method is profitable when competing with NEM X to attain customers. In practice, $\pi_{\mbox{\tiny LMP}}=\pi^{-}$ happens since many states including California are setting the export rate $\pi^{-}$ at the avoided cost rate, which is, in many cases equivalent to the wholesale LMP $\pi_{\mbox{\tiny LMP}}$ [29], to mitigate cross-subsidies [6, 4].

As a virtual storage participant in the real-time energy market, the DERA is either self-scheduled or scheduled by ISO/RTO according to its bids and offers. This work focuses on developing price-quantity bid/offer curves that define DERA’s willingness to produce and consume.⁸⁸8For the quantity bid, one would forecast the LMP, and compute the optimal net production $\sum_{n}(g_{n}-d_{n}^{*})$ with $d_{n}^{*}$ defined in (2a). In a competitive market, such curves are the marginal cost of production and the marginal benefit of consumption derived from the optimal DERA decision in Theorem 1.

Let $Q$ be the aggregated quantity to buy (when $Q<0$) or sell (when $Q>0$) for the DERA and $\pi_{\mbox{\tiny LMP}}$ be the wholesale market LMP. Let $G=\sum_{n=1}^{N}g_{n}$ be the BTM DG aggregated by DERA. In a competitive market, a price-taking DERA participant bids truthfully with its aggregated supply function $Q=\mathscr{F}(\pi_{\mbox{\tiny LMP}})$ given by

where $d^{*}_{n}$ is defined in (2a). Note that the inverse of the DERA supply function $\mathscr{F}^{-1}(Q)$ defines the offer/bid curves of the DERA. By submitting a bidding curve as virtual storage, DERA does not need to forecast LMP. Note also that the supply function depends on the aggregated BTM generation $G$, which is not known to the DERA at the time of the market auction. In practice, $G$ can be approximated by using historical data or $N\mathbb{E}(g_{n})$ vis the Law of Large Numbers involving $N$ independent prosumers or via the Central Limit Theorem for independent and dependent random variables [30].

We now establish that the DERA’s participation in the wholesale market achieves the same social welfare as that when all profit-maximizing prosumers participate in the wholesale market individually under certain conditions. We assume the wholesale market is competitive, where all participants are price takers with truthful bidding incentives.

Consider a transmission network with $M$ buses. Without loss of generality, we assume $N$ PoAs at the distribution network are connected to each bus of the transmission network and all prosumers are aggregated by the proposed DERA model. Denote $U_{in}$ as the aggregated prosumer utility function at the $n$-th PoA of the $i$-th transmission network bus, ${\bm{g}}_{n}:=(g_{in})_{i\in[M]}$ and ${\bm{d}}_{n}:=(d_{in})_{i\in[M]}$ are respectively the BTM DG generation and energy consumption for the prosumer, $E_{i}(D_{i})$ is the bid-in benefit function of the flexible demand at bus $i$ when consuming $D_{i}$, and $C_{i}(P_{i})$ is the bid-in cost function for the generator at bus $i$ when producing $P_{i}$. For simplicity, we assume each transmission bus has bids from one elastic demand and one generator. Denote ${\bm{D}}:=(D_{i})_{i\in[M]}$, ${\bm{P}}:=(P_{i})_{i\in[M]}$, ${\bm{S}}\in\mathbb{R}^{L\times M}$ as the parameter for the DC power flow model of the transmission network, and ${\bm{F}}\in\mathbb{R}^{L}$ the line flow limit for $L$ branches of the transmission network.

Proof of this Lemma in Appendix VIII-E relies on showing that pricing and dispatch results from (4) are at the bidding curve of DERA, i.e., (3). With the optimal dual variable $\lambda^{\star}\in\mathbb{R}$ for the power balance constraint (4b) and $\hbox{\boldmath$\mu$\unboldmath}^{\star}\in\mathbb{R}^{L}$ for the line flow limit (4c), the market clearing LMP over $M$ buses is defined by $\hbox{\boldmath$\pi$\unboldmath}_{\mbox{\tiny LMP}}:=\mathbf{1}\lambda^{\star}-{\bm{S}}^{\intercal}\hbox{\boldmath$\mu$\unboldmath}^{\star}$, where $\hbox{\boldmath$\pi$\unboldmath}_{\mbox{\tiny LMP}}:=(\pi_{i}^{\mbox{\tiny LMP}})_{i\in[M]}$. The aggregated prosumer utility in (5), and constraints for energy consumption and distribution network access in (4d) (4e) come from DERA’s offer/bid (3).

As for the prosumer’s direct participation in the wholesale market, we know that a price-taking prosumer at PoA $n$ and bus $i$ constructs her offer/bid curves by solving the following surplus maximization problem with the given LMP $\pi^{\mbox{\tiny LMP}}_{i}$:

where ${\cal D}_{in}:=[\max\{\underline{d}_{in},g_{in}-\overline{C}_{in}\},\min\{\bar{d}_{in},g+\underline{C}_{in}\}]$ because we assume prosumers participating in the wholesale market directly are facing the same network injection and withdrawal access limits at each PoA for the fairness of comparison. So, the bid/offer curve of the prosumer at PoA $n$ and bus $i$ is

where $d^{*}_{in}(\pi^{\mbox{\tiny LMP}}_{i})$ takes the same definition as that in (2a).

Let $\mbox{\sf SW}_{\mbox{\rm\tiny Direct}}$ and $S_{\mbox{\rm\tiny PRO}}$ be, respectively, the optimal social welfare and prosumers’ surplus when all prosumers directly participate in the wholesale market. The following theorem is parallel to [17], although we have different aggregation methods, under the model that the DERA (and prosumers) submits its bids and offers to the wholesale electricity market.

The sketch of proof is provided in Appendix VIII-F, which relies on the fact that the proposed DERA has its bidding curve (3) equal to the sum of the prosumer’s bidding curve in (7). From this, we can establish that the wholesale market clearing problem with the direct participation of all prosumers has the same market-clearing results as (4).

Although the proposed DER aggregation model only focuses on DERA’s profit maximization in the objective (50), the competitive constraint (1b) aligns the aggregated prosumer’s surplus maximization with DERA’s profit maximization. So, the proposed competitive DER aggregation has the incentive to maximize prosumers’ surpluses and get the maximum total surplus that can be split among DERA and its aggregated prosumers. Essentially, DERA acts as a middleman and brings all prosumers to indirectly participate in the wholesale market. Since DERA is profiting from this middleman business, the individual prosumer receives less payment in the indirect wholesale market participation through DERA than from direct wholesale market participation (shown by Fig. 2). This is not unreasonable. After all, individual prosumers cannot participate directly in the wholesale market.

The following Proposition provides an expression for the benefit function of DERA, $\varphi(\overline{{\bm{C}}},\underline{{\bm{C}}})$, which can be used as the bid curve of access limits submitted to the auction in [24].

The sketch of proof is provided in Appendix VIII-D. The optimal DERA surplus is decomposed into three terms: the surplus dependent on the withdraw access $\underline{\phi}_{n}({\underline{C}}_{n})$, the surplus dependent on the injection access $\overline{\phi}_{n}({\overline{C}}_{n})$, and the surplus independent of the network access $h_{n}$. When there are no binding constraints for network access limits, $\hat{d}_{n}$ is the optimal consumption of prosumer $n$, defined below (2a); when there are binding access limits, the optimal consumption $d^{*}_{n}$ equals $\hat{d}_{n}$ truncated by the distribution network access limits, and DERA’s benefit is modified by $\overline{\phi}_{n}({\overline{C}}_{n})$ or $\underline{\phi}_{n}({\underline{C}}_{n})$.

In our aggregation model, the optimal DERA benefit $\varphi({\underline{{\bm{C}}}},{\overline{{\bm{C}}}})$ is separable across injection and withdrawal access, and across prosumers at different distribution buses. Furthermore, at most one of $\overline{\phi}_{n}({\overline{C}}_{n})$ and $\underline{\phi}_{n}({\underline{C}}_{n})$ can be nonzero, depending upon the renewable generation $g_{n}$ for prosumer $n$. When $g_{n}\leq q^{+}_{n}$, $\underline{\phi}_{n}({\underline{C}}_{n})$ is nonzero and the prosumer at bus $n$ is a consumer with binding network withdrawal access constraints; when $q^{-}_{n}\leq g_{n}$, $\overline{\phi}_{n}({\overline{C}}_{n})$ is nonzero and the prosumer at bus $i$ is a producer with binding network injection access constraints. Related simulation is shown in Sec. VI-D.

In a long-run competitive industry, we explore how many DERA can survive. DERAs compete to attract customers, attain distribution network access, and participate in the wholesale market. We assume all DERAs adopt the proposed competitive DER aggregation method. The condition for a competitive long-run equilibrium [31, P193]⁹⁹9In [31], question 4.26 focuses on long-run competitive equilibrium; equations (4.21) and (4.22) focus on long-run monopolistic equilibrium. has two components: (i) the marginal benefit of DERA equals the marginal cost of DSO for providing the distribution network access, and (ii) all DERAs have profits equal to zero, i.e., DERA’s profit in conducting aggregation equals DERA’s payment to acquire distribution network access. Related derivations and simulations are in Sec. VI-E and Appendix VIII-H.

Assume homogeneous utility function [21] for the aggregated customers, i.e., $\forall n\in[N]$,

where $\alpha=\$0.4/\mbox{kWh},\beta=\$0.1/(\mbox{kWh})^{2}$. The marginal utility had $V_{n}^{-1}\in[\underline{d}_{n},\bar{d}_{n}]$, $\forall n$ for the consumption boundaries.

We used NEMa and NEMp to represent the DER aggregation under NEM X when prosumers were active and passive, respectively. Passive customers are not responsive to the retail prices, but active customers will optimize their energy consumption given the retail price and the BTM DG generations. Based on PG&E residential rate, we set $\pi^{+}=\$0.3/\mbox{kWh}$ for the NEM X. We assumed $\pi^{-}=\pi_{\mbox{\tiny LMP}}$ and the fixed cost of NEM X was covered by extracting fixed payment from DERA, so we simulated with $\pi^{0}=\$0$. GAB represented the two-part pricing in [17] which allowed customers to sell BTM DG to the DERA while purchasing energy from its incumbent utility company. Detailed models for NEMa, NEMp, and GAB are provided in Appendix VIII-A. Our DER aggregation method was simulated in Co.NEMa and Co.GAB, competitive to NEMa and GAB respectively. We set $\zeta$ at the upper bound in Proposition 2 for Co.NEMa. For Co.GAB, we set $\zeta=1.05$ to provide 5% more customer surplus than the GAB competitor.

LMP $\pi_{\mbox{\tiny LMP}}$ was assumed to be a Gaussian random variable with mean $0.05/kWh and standard deviation (STD) $0.01/kWh, truncated by $(0,\pi^{+})$. The BTM DG generation $g$ was assumed to be a Gaussian random variable with STD 0.2 kWh, truncated by $(0,+\infty)$. Data sources for the wholesale LMP and BTM DG came from CAISO and Pecan Street Dataport [32], respectively. We sampled 10,000 scenarios for the LMP and BTM DG. At a certain PoA, we analyzed the per-customer level surplus distributions on expectation based on the sample mean of these scenarios.

Four observations below were drawn when all aggregators received plenty of distribution network access.

First, Co.NEMa and Co.GAB were at the Pareto front in Fig. 2 achieving the maximum social surplus as if all prosumers directly participated in the wholesale market without middlemen like DERA. This verified Theorem 2. Note that we computed the Pareto front by adding up surpluses of DERA and customers, omitting surpluses of other units. This was because we adopted the price taker assumption in the wholesale market, thus surpluses of other units stayed the same in different DERA models. The blue dot, named Direct, represented the ideal case that prosumers directly participated in the wholesale market with bidding curve (7). The green rectangular contained aggregation methods achieving less DERA surplus and customer surplus than Co.NEMa, thus dominated by our proposed competitive DER aggregation method. Similarly, the orange rectangular was dominated by its top right corner, Co.GAB. This was because our aggregation methods efficiently participated in the wholesale market with aggregated resources and scheduled the aggregated customers at a consumption level with a higher customer surplus. When the expected BTM DG increased from 1.1kW to 5.1kW, comparing the left and right panels in Fig. 2, we observed the expected social surplus, which was the sum of DERA and customer surpluses, increased, because more BTM DG was sold to the wholesale market.

Second, customers had the highest expected surplus in Co.NEMa and Co.GAB shown by the top of Fig. 3. Passive customers in NEMp had the least surplus because its scheduling had no awareness of DG generation. Customer surpluses almost overlapped in all cases at a low DG adopter ratio with fewer producers, since most aggregation benefits came from BTM DG of producers. When the DG adopter ratio increased, the expected customer surplus increased in all cases.

Third, when DG adopter ratio or the DG generation was low, Co.NEMa and Co.GAB achieved the highest expected DERA surplus, as is shown in the bottom of Fig. 3. When the DG adopter ratio and DG generation were high, GAB achieved the highest DERA surplus because GAB only aggregated producers.¹⁰¹⁰10GAB achieved the Pareto front when all prosumers were producers, e.g. DG adopter ratio equal 100% and $\mathbb{E}[g]=5.1$kW. Co.NEMa always had DERA profit no less than zero since we chose $\zeta$ based on Proposition 2. NEMp and NEMa were completely overlapped when comparing DERA surplus because we set $\pi^{-}=\pi^{\mbox{\tiny LMP}}$.

Fourth, since NEM X provided a higher surplus to customers with BTM DG, DERAs must commensurately reduce their own profits and share them with the customers to remain competitive with NEM X. Therefore, in most cases of Fig. 3, the expected DERA surplus decreased when the DG adopter ratio increased. However, GAB witnessed an increasing DERA surplus when the DG adopter ratio increased because GAB only aggregated producers.

Set distribution network access limits for each prosumer by $\overline{C}=\underline{C}=8\delta$ kW and vary the network access ratio $\delta$ from 0 to 1 to analyze the influence of limited distribution network access. First, as is shown in Fig. 4, either Co.NEMa or Co.GAB achieved the highest customer surplus and DERA surplus under a limited network access ratio. Second, when the network access ratio increased, customer surplus increased in most cases except NEMp which passively controlled DG. Third, the DERA surplus in all cases increased when the network access ratio increased. This was intuitive because aggregators needed distribution network access to deliver the aggregated resources and participate in the wholesale market.

We computed the bid-in benefit function of the proposed DERA model, i.e., (8), with $\zeta=1.01$ and 50 prosumer aggregated at a certain PoA. DERA was competing with NEM X and prosumers were passive. We plot the expected benefit $\varphi$ of DERA with respect to the injection and withdrawal access in Fig. 5 with varying expected BTM DG generations.

In the left panel of Fig. 5, DERAs with lower expected DG generations had higher benefits and chose higher bid prices to purchase withdrawal access, as shown by the slope of the benefit function. Because, with less BTM DG, DERA relied more on the withdrawn electricity from the network. The slope of the right panel in Fig. 5 showed that DERAs with higher DG chose higher prices for injection access. The benefit function was lower when the DG generation was higher. This counterintuitive phenomenon happened because NEM X provided higher surpluses to customers with higher DG, DERAs must reduce their profits and share them with the customers to remain competitive with NEM X.

In the long-run competitive equilibrium analysis with multi-interval aggregation of DERs, we assumed 200 DERAs initially existed and computed the expected number of surviving DERAs in the long run. For simplicity, we assumed DERAs were homogeneous and had the same setting as Sec. VI-D. Prosumers had the same expected DG generation created from the 24-hour roof-top solar data in Pecan Street [32].¹¹¹¹11Detail DG trajectories and the long-run equilibrium results for single-interval aggregation are shown in Appendix VIII-H, providing intuitions about long-run equilibrium for multi-interval aggregation here. We multiplied the mean of 24-hour DG by $\epsilon_{1}\in\mathbb{R}_{+}$ to simulate different DG installation capacities and sampled 10,000 random DG scenarios. DERA submitted the benefit function, as in Fig. 5, to acquire hourly distribution network access. Same as [24], the DSO cost function for providing distribution network access was assumed to be the sum of quadratics, $J(x)=\frac{1}{2}bx^{2}+ax$ with $a=\$0.009/\mbox{kWh},b=\$0.0005/(\mbox{kWh})^{2}$ for both the injection and withdrawal access. We multipied DSO’s cost $J$ by $\epsilon_{2}\in\mathbb{R}_{+}$ to simulate different levels of DSO’s costs.

Two observations were drawn from results in Fig. 6. First, when the DG capacity ratio $\epsilon_{1}$ was about 0.4-1.4, all initial 200 DERAs survived because DERAs can internally balance customer demands with its aggregated DG, thus relying and paying less to the network access. This was validated by the yellow dot curve from Fig. 6 (right), which required almost zero network access over 24 hours. Second, when the DG capacity ratio decreased from 0.4 to 0 in Fig. 6 (left), the number of surviving DERA decreased. In this case, DG was lower than the aggregated customers’ consumption, and not all DERAs can survive when competing and paying for the network withdrawal access over 24 hours, shown by the blue solid curve of Fig. 6 (right). In the green dash curve of Fig. 6 (left), DSO’s cost for providing network access was lower, so more DERAs survived than other curves. Similar reasons applied when DG capacity ratio increased beyond 1.4.

A prosumer in a distribution system can choose to enroll in a NEM X retail program offered by her utility or a DERA providing energy services. In this context, a summary of short-run analysis over several existing models for the participation of prosumers in the regulated utility and different DERA schemes is presented.

If $g_{n}\leq d_{n}^{+}$, we have

Here, (a) comes from the optimal solution in Theorem 1. (b) relies on the setting that ${\cal K}_{n}=\zeta S_{n}^{\mbox{\tiny NEM}}(g_{n},\underline{C}_{n},\overline{C}_{n})$. (c) replies on $\zeta\geq 1$ and the assumption that $S_{n}^{\mbox{\tiny NEM}}\geq 0$. (d) comes from the definition of $S_{n}^{\mbox{\tiny NEM}}$ given in (17).¹³¹³13We ignore the fix charge $\pi^{0}$ under NEM X here for simplicity.(e) comes from

which can be derived from the optimality of

If $g_{n}>d_{n}^{+}$, for passive prosumer, we have

Here,(a) comes from the optimal solution in Theorem 1. (b) relies on the setting that ${\cal K}_{n}=\zeta S_{n}^{\mbox{\tiny NEM}}(g_{n},\underline{C}_{n},\overline{C}_{n})$. (c) replies on $\zeta\geq 1$ and the assumption that $S_{n}^{\mbox{\tiny NEM}}\geq 0$. (d) comes from the definition of $S_{n}^{\mbox{\tiny NEM}}$ given in (17). (e) replies on $g_{n}>d_{n}^{+}$ and $\pi^{-}\geq 0$. (f) comes from (20). (g) holds because $d_{n}^{+}\geq 0$ and $\pi^{+}\geq 0$.

If $d_{n}^{+}<g_{n}\leq d_{n}^{-}$, for active prosumer, we have

Here,(a) comes from Theorem 1 and the assumption that $S_{n}^{\mbox{\tiny NEM}}\geq 0$. (b) comes from the definition of $S_{n}^{\mbox{\tiny NEM}}$ given in (17) when $d_{n}^{+}<g_{n}\leq d_{n}^{-}$ for active prosumer. (c) comes from the optimality of $d^{0}={\rm arg~{}max}_{d\in{\cal D}}U(d)$, $d_{n}^{*}\geq 0$, and $\pi^{+}\geq 0$.

If $g_{n}>d_{n}^{-}$, for active prosumer, we have

Here,(a) comes from Theorem 1 and the assumption that $S_{n}^{\mbox{\tiny NEM}}\geq 0$. (b) comes from the definition of $S_{n}^{\mbox{\tiny NEM}}$ given in (17) when $g_{n}\geq d^{-}_{n}$. (c) holds because $g_{n}\geq d^{-}_{n}$. (d) comes from