On Efficient Aggregation of Distributed Energy Resources

Consider a power network with $n$ nodes (locations). At each location $i$, there are $N_{i}$ number of prosumers. We assume that each prosumer $j$ at location $i$ can purchase energy at the wholesale price $\lambda^{i}$. Furthermore, prosumer $j$ at location $i$ is endowed with a capacity ${C}^{i}_{j}\geq 0$ of power production²²2$C^{i}_{j}$ can be zero for consumers with no DER supply, so the model includes those regular consumers. from a collection of resources, such as solar panels, wall-mounted batteries, and plug-in electric vehicles. The power produced can be consumed locally by prosumer $j$ or sold (partially) back to the wholesale market, again at the wholesale price $\lambda^{i}$. The prosumer observes $\lambda^{i}$ and decides the amount of energy to buy/sell. Let $u^{i}_{j}$ be prosumer $j$’s utility of power consumption. We make the following assumption on prosumers’ utility of consumption.

We can then write prosumer $j$’s optimization problem as

where $d^{i}_{j}$ is the amount of energy prosumer $j$ purchases and $x^{i}_{j}$ is the amount of energy she sells. For selling $x^{i}_{j}$ at the wholesale price, prosumer $j$ receives $\lambda^{i}x^{i}_{j}$; for buying $d^{i}_{j}$ at the wholesale price, she is charged $\lambda^{i}d^{i}_{j}$. The prosumer leaves with $C^{i}_{j}+d^{i}_{j}-x^{i}_{j}$ to consume and her utility from consumption would be $u^{i}_{j}(C^{i}_{j}+d^{i}_{j}-x^{i}_{j})$. While Assumption 1 further imposes strict concavity of prosumers’ utilities, our analysis throughout this paper remains largely applicable to generic concave utilities, but strict concavity allows us to derive unique analytical solutions and gain deep insights.

We note that since prosumers at location $i$ can buy and sell energy at the same price $\lambda^{i}$, prosumer $j$ essentially has $x^{i}_{j}-d^{i}_{j}$ as the single decision to make in solving (1), while we keep both $x^{i}_{j}$ and $d^{i}_{j}$ in the notation to clearly represent the amount of selling and buying, and to be consistent with the notation in the following sections.

The above Lemma directly follows from the properties of problem (1). Without loss of generality, and for ease of exposition, we restrict our attention to the case in which at most one of $x^{i}_{j}$ and $d^{i}_{j}$ can be nonzero. We first solve (1) for $(x^{i}_{j}-d^{i}_{j})^{*}$. Then, we let ${x^{i}_{j}}^{*}=\left[(x^{i}_{j}-d^{i}_{j})^{*}\right]^{+}$ and ${d^{i}_{j}}^{*}=\left[-(x^{i}_{j}-d^{i}_{j})^{*}\right]^{+}$. We also use the notation ${x^{i}_{j}}^{*}(\lambda^{i})$ and ${d^{i}_{j}}^{*}(\lambda^{i})$ to denote the optimal response of prosumer $j$ at location $i$ for a given wholesale market price $\lambda^{i}$.

Let $G_{i}$ denote the number of dispatchable conventional generators at location $i$. Generator $j$ at location $i$ chooses to supply $y^{i}_{j}\in\left[\underline{y}^{i}_{j},\overline{y}^{i}_{j}\right]$. Let $c^{i}_{j}(y^{i}_{j})$ be denote the production cost. We have the following assumption.

By selling $y^{i}_{j}$, generator $j$ earns a compensation $\lambda^{i}y^{i}$. Given a wholesale price $\lambda^{i}$, generator $j$ maximizes its payoff by solving

The above Lemma directly follows from the properties of problem (3). We also use the notation ${y^{i}_{j}}^{*}(\lambda^{i})$ to denote the optimal response of generator $j$ at location $i$ at a given wholesale price $\lambda^{i}$.

Wholesale electricity markets in the United States and other countries are managed by independent system operators (ISOs). An ISO clears the market by matching supply and demand via social welfare maximization (in practice, this is often done by production cost minimization to meet fixed system demands [4, 5, 6]), while ensuring that the power flows satisfy the network and line capacity constraints. Specifically, let $X^{i}=\sum_{j\in[N_{i}]}x^{i}_{j}$ be the total power supply at node $i$ from prosumers; let $Y^{i}=\sum_{j\in[G_{i}]}y^{i}_{j}$ be the total power supply at node $i$ from conventional generators; and let $D^{i}=\sum_{j\in[N_{i}]}d^{i}_{j}$ be the total demand at node $i$. Furthermore, we let ${\bf f}$ be the vector of capacities of transmission lines in the power network, and let ${\bf B}$ be the line-to-node incidence matrix. The system operator chooses a vector ${\bf h}$, where each element $h^{i}$ is the net injection to node $i$. We have the following network constraints:

We note that (5a) ensures the total supply matches the total demand at each node; (5b) ensures that the total net injection by the system operator is zero over the power network (here, ${\bf 1}$ is a vector of ones), and the total power transmission at each line does not exceed its capacity. In addition to the network constraints, the ISO needs to also consider all participant-specific constraints described earlier in problems (1) and (3):

The objective of the system operator is to maximize the social welfare, which includes the prosumer surplus (PS), generator surplus (GS), and merchandizing surplus (MS):

where we have imposed the relationship (5a) in deriving (7a). The social welfare that the system operator optimizes is the sum of PS, GS, and MS. After canceling terms, the social welfare can be written as

The system operator’s economic dispatch problem is then:

The above proposition states that solving the system operator’s problem (9) leads to a competitive equilibrium. From the prospective of prosumer $j$ at node $i$, this means that given the wholesale market price $\lambda^{i}$ (the optimal Lagrange multiplier of  (5a) for the same node), the corresponding solution to her problem, which satisfies (2), is the same as the optimal decision made by the ISO via solving  (9). This is also true for all other prosumers and generators. Having all market participants being satisfied with the the competitive equilibrium, it serves as a good benchamrk for market efficiency.

Consider prosumer $j$ at location $i$. Upon seeing the prices $(P^{i}_{j},p^{i}_{j})$ announced by $\mathcal{A}$, prosumer $j$ decides if she would sell part of her capacity to $\mathcal{A}$. If she chooses so, she would pay a fee $P^{i}_{j}$ to $\mathcal{A}$, and receives $p^{i}_{j}$ for each unit of energy sold. We may write prosumer $j$’s payoff as

Given $(P^{i}_{j},p^{i}_{j},\lambda^{i})$, prosumer $j$ solves:

where $d^{i}_{j}$ is the amount of energy prosumer $j$ purchases at wholesale market price $\lambda^{i}$, ⁴⁴4In real markets, end-use customers are represented in the wholesale market via utility companies. Here, for ease of exposition, we omit the analysis of the relationship between utility companies and end-use customers and assume that they represent the true preferences of their customers in the wholesale market without modifications. This allows us to focus on the strategic aggregation component of the problem. Nevertheless, our insights remain largely applicable if this assumption is relaxed. and $x^{i}_{j}$ is the amount of energy she sells to the aggregator. For buying $d^{i}_{j}$ at the wholesale price, prosumer $j$ is charged $\lambda^{i}d^{i}_{j}$. If the prosumer does not sell ($x^{i}_{j}=0$), she has a total of $d^{i}_{j}+C^{i}_{j}$ to consume, and her utility from consumption would be $u^{i}_{j}(d^{i}_{j}+C^{i}_{j})$. If the prosumer chooses to sell $x^{i}_{j}>0$ to $\mathcal{A}$, she is charged a participation fee $P^{i}_{j}$, and receives a compensation $p^{i}_{j}x^{i}_{j}$. The prosumer would have $d^{i}_{j}+C^{i}_{j}-x^{i}_{j}$ to consume in this case and her utility from consumption would be $u^{i}_{j}(d^{i}_{j}+C^{i}_{j}-x^{i}_{j})$.

Let ${x^{i}_{j}}^{*}(P^{i}_{j},p^{i}_{j},\lambda^{i})$ and ${d^{i}_{j}}^{*}(P^{i}_{j},p^{i}_{j},\lambda^{i})$ denote the optimal response of prosumer $j$ given aggregator’s announced prices $(P^{i}_{j},p^{i}_{j})$ and the wholesale market price $\lambda^{i}$.⁵⁵5Sometimes, we drop arguments for simplicity. Note that if $p^{i}_{j}>\lambda^{i}$, the prosumer can arbitrage by buying at the price $\lambda^{i}$ and selling at a higher price $p^{i}_{j}$. This will result in the prosumer earning infinite payoff and aggregator losing infinite profit, which would be avoided by the aggregator. Therefore, we may without loss of generality restrict our discussions to the case when $p^{i}_{j}\leq\lambda^{i},\forall i\in[n],j\in[N_{i}]$. In the case $p^{i}_{j}=\lambda^{i}$, similar to the direct participation model, we may enforce that $x^{i}_{j}$ and $d^{i}_{j}$ cannot both be nonzero. We then have the following lemma on the optimal response of prosumers.

The above Lemma states that if the capacity exceeds a certain value $z_{2}$ (note that $z_{2}$ is the value at which the marginal utility of consumption is equal to the aggregator’s marginal price offer $p^{i}_{j}$) and the upfront fee $P^{i}_{j}$ is not too high, then prosumers have an incentive to sell DER supply to the aggregator. If the capacity is too small or the upfront fee is too high, then prosumers would not sell DER supply and prefer to consume it locally.

The DER aggregator $\mathcal{A}$ collects power from prosumers and sells it in the wholesale market. By offering the prices $(P^{i}_{j},p^{i}_{j})$ to each prosumer $j$ at location $i$, $\mathcal{A}$ procures a total capacity of $\sum_{j\in[N_{i}]}{x^{i}_{j}}^{*}(P^{i}_{j},p^{i}_{j})$ from location $i$. $\mathcal{A}$ then sells it at the wholesale market price $\lambda^{i}$. Given the wholesale market price $\lambda^{i}$, the aggregator’s profit from prosumer $j$ at location $i$ is

Anticipating the response functions ${x^{i}_{j}}^{*}(P^{i}_{j},p^{i}_{j})$’s, she seeks to maximize her overall profit:

Aggregator $\mathcal{A}$’s profit is composed of two parts; the total participation fees charged to prosumers who sell positive amount of energy and the profits earned by reselling the procured energy in the wholesale market. We note that ${\cal A}$’s profit from prosumer $j$ depends only on the response and the prices of prosumer $j$, and there is no coupling among prosumers’ problems. It should be clear that the Aggregator’s problem (14) can be decomposed to optimizing $(P^{i}_{j},p^{i}_{j})$ for each prosumer at each location:

Solving (15) for each $i\in[n]$ and $j\in[N_{i}]$ would lead to the vectors $\bf(P^{*},p^{*})$, which constitute an optimal solution for (14). We have the following result on the optimal decisions of the aggregator.

Lemma 4 provides an optimal solution $({P^{i}_{j}}^{*},{p^{i}_{j}}^{*})$ to (15), and the collection  $\bf(P^{*},p^{*})$ form an optimal solution to (14). We also note that the optimal solution may not be unique in general: ${\cal A}$ may deviate from (16) by further decreasing ${p^{i}_{j}}^{*}$ and increasing ${P^{i}_{j}}^{*}$ to earn the same profit, while keeping the response of the prosumer $j$ being unchanged. This lemma states that there is an optimal pricing scheme which sets the marginal price $p^{i}_{j}$ to be the wholesale market price $\lambda^{i}$, and all prosumers at the same location will be offered the same $p^{i}_{j}=\lambda^{i}$. The participation fees $P^{i}_{j}$, however, will be charged differently for prosumers with different utility functions. A keen reader would observe that in view of Lemmas 1 and 3, the optimal DER supply to the wholesale market is the same with and without the aggregator ${\cal A}$, thus making this aggregation model economically efficient with socially optimal market outcomes.

Let $\mathcal{G}(\boldsymbol{\lambda})$ denote the Stackelberg game among the aggregator and the prosumers for a given vector of wholesale market prices $\boldsymbol{\lambda}$. Aggregator $\mathcal{A}$ acts as a Stackelberg leader and sets the prices $\bf(P,p)$. The prosumer follows by responding with ${x^{i}_{j}}^{*}(P^{i}_{j},p^{i}_{j},\lambda^{i})$. We now define the equilibrium of the game.

We then have the following Stackelberg equilibrium for the game ${\cal G}(\boldsymbol{\lambda})$, which follows directly from the prosumer’s optimal response (Lemma 3) and the aggregator’s optimal pricing (Lemma 4).

We note that the game ${\cal G}(\boldsymbol{\lambda})$ may admit other Stackelberg equilibria, but Proposition 2 provides the most economically efficient one. In case of non-uniqueness, this economically efficient equilibrium corresponds to the case where prosumers slightly prefer participation, i.e., if selling $x^{i}_{j}$ amount of energy earns the prosumer $j$ the same $\pi^{i}_{j}$ as not selling, then she chooses to sell this $x^{i}_{j}$. ⁶⁶6Alternatively, one can impose a slight perturbation of ${P^{i}_{j}}^{*}$ to ${P^{i}_{j}}^{*}-\epsilon$, where $\epsilon>0$, to ensure maximum DER supply. The above equilibrium is quite intuitive, ${\cal A}$ passes the location marginal price $\lambda^{i}$ obtained from wholesale market outcomes as is to prosumers, so ${\cal A}$ has no marginal profits from ${\bf p}^{*}$. Instead, ${\cal A}$ makes all of the profits from the upfront participation fees ${\bf P^{*}}$.

For a given wholesale market price $\lambda^{i}$, the conventional generators solve the same problem as described in Section II-B, and the result of Lemma 2 still applies under the current aggregation model.

The system operator solves an optimization problem similar to that in the direct participation model as described in Section II-C. The network constraints (5) remain valid. Besides, the ISO also considers participant-specific constraints:

The objective of the system operator is to maximize the social welfare, which now includes the prosumer surplus (PS), aggregator surplus (AS), generator surplus (GS), and merchandizing surplus (MS):

The social welfare is the sum of the above four terms. By the supply-demand balance ${\bf h=D-Y-X}$, and after canceling terms, we write the social welfare as

which is the same as ${\cal W}_{B}$ in (8). The system operator’s economic dispatch problem is then:

As the system operator solves (21), the wholesale market prices $\boldsymbol{\lambda}$ are given by the optimal Lagrange multiplier of the constraint (5a). We then have the following proposition.

We now present the following theorem, which states that our proposed aggregation model achieves the same market efficiency as the benchmark direct participation model.

In summary, under the proposed aggregation model, the aggregator procures energy from prosumers using two-part pricing, the aggregator would optimally pay the wholesale market price to the prosumers for each unit of energy procured, while the participation fee is differently charged to each prosumer as the additional consumer surplus when she sells those energy compared with not selling (which is dependent on her utility of consumption). Theorem 1, together with Proposition 3, implies that under the aggregator’s two-part differential pricing scheme, the prosumers’ optimal buying and selling behavior is exactly the same as those in the direct participation model. As a result, the social welfare achieved under the aggregation model matches that of the direct participation model, i.e., there is no loss of market efficiency from the aggregation.