Network-Aware and Welfare-Maximizing Dynamic Pricing for Energy Sharing

While BTM DER are primarily adopted to provide prosumer services such as bill savings and backup power, they can also be leveraged, under proper consumer-centric mechanism design, to provide various grid services such as voltage control, system support during contingencies, and new capacity deferrals [1]. Harnessing the flexibility of BTM DER participation in grid services is usually challenged by the DSO’s lack of visibility and controllability on BTM DER alongside the absence of network-aware pricing mechanisms that can induce favorable prosumer behaviors.

The rising notion of energy sharing of a group of prosumers under the DSO’s tariff presents a compelling solution to optimize DER utilization, comply with dynamic grid constraints, and promote renewable energy integration. A major barrier facing the practical implementation of energy-sharing markets is the incorporation of network constraints into their pricing, and aligning the objectives of the self-interested energy-sharing prosumers with the global objective of maximizing the coalition’s welfare.

Despite the voluminous literature on energy-sharing systems’ DER control and energy pricing, network constraints are rarely considered due to the theoretical complexity they introduce. A short list of recent works on energy communities and energy sharing that neglected network constraints can be found here [2, 3, 4, 5]. Some works considered a coarse notion of network constraints by incorporating operating envelopes (OEs) at the point of common coupling between the energy-sharing system and the DSO [6, 7] that limit the export and imports between the two entities. Others considered OEs at the prosumer’s level [8, 9]. Few papers considered network-aware pricing mechanisms in distribution networks, such as [10, 11, 12] and the line of literature on distribution locational marginal prices (dLMP), e.g. [13, 14].

Our work differs from the existing literature in two important directions. Firstly, we consider network-aware pricing under a generic DSO NEM tariff constraint that charges the energy-sharing platform different prices based on its aggregate net consumption. Secondly, the dynamic network-aware pricing of a platform that is subject to the DSO’s fixed and exogenous NEM price gives rise to a market manager’s profit/deficit that needs to be re-allocated to the coalition members. We shed light on a unique re-allocation rule that makes the prosumers’ payment functions uniform, even if they are located on different buses and the network constraints are binding. Such a re-allocation rule is highly relevant when charging end-users, as it avoids ‘undue discrimination’, which is one of the key principles of rate design outlined by Bonbright [15].

In this paper, we present a network-aware and welfare-maximizing pricing policy for energy-sharing coalitions that aggregate DER downstream of a DSO’s revenue meter that charges the energy-sharing platform based on a generic NEM tariff. The pricing policy announces an ex-ante locational, threshold-based, and dynamic price to induce a collective prosumer response that decentrally maximizes the social welfare, while abiding by the network voltage constraints. An ex-post charge/reward is then used to ensure the market operator’s profit neutrality. We show that the market mechanism achieves an equilibrium to the Stackelberg game between the energy-sharing market operator and its prosumers. Although network constraints couple the decisions of the energy-sharing prosumers, which give rise to locational marginal prices (LMP), we show that by adopting a unique proportional re-allocation rule, the payment function becomes uniform for all prosumers, even if they are located at different buses in the energy-sharing network. Numerical simulations using the IEEE 13-bus test feeder and real BTM DER data shed more light on how the pricing policy influences prosumers’ response to ensure safe network operation.

This paper extends our work on Dynamic NEM (D-NEM) without OEs [3] and with OEs [9] by incorporating network constraints, which add substantial complexity, primarily due to coupling of the DER decisions across network buses.

For the column vector $\bm{x}$, $[\bm{x}]^{+}$, and $[\bm{x}]^{-}$ represent its positive and negative elements. The notation $[x]_{\underline{x}}^{\overline{x}}$ represents the projection of $x$ into the closed and convex set $[\underline{x},\overline{x}]$ as per the rule $[x]_{\underline{x}}^{\overline{x}}:=\max\{\underline{x},\min\{x,\overline{x}\}\}$. The notation is also used for vectors, i.e., $[\bm{x}]_{\underline{\bm{x}}}^{\overline{\bm{x}}}$.

We consider the problem of designing a welfare-maximizing and network-aware pricing policy for an energy sharing system that bidirectionally transacts energy and money with the DSO under a general NEM tariff. Under NEM, the energy sharing platform imports from the DSO at an import rate if it is net-consuming, and collectively exports from the DSO at the export rate if it is net-producing. A budget-balanced market operator is responsible for announcing the market’s pricing policy. The market operator uses spatially varying pricing signals to adhere to its network’s operational constraints communicated by the DSO.¹¹1We posit that such DER aggregation schemes are informed by the DSO about their networks’ information, including OEs, line thermal limits, voltage limits, among others. We assume that the timescale of community members’ decision is equivalent to that of the NEM netting period [16], which allows us to adopt a single time step formulation.

A radial low voltage distribution network flow model is used to model the network power flow [17, 18]. Consider a radial distribution network described by $\mathcal{G}=({\cal B},\mathcal{L})$, with ${\cal B}=\{1,\ldots,B\}$ as the set of energy sharing buses, excluding bus 0, and $\mathcal{L}=\{(i,j)\}\subset{\cal B}\times{\cal B}$ as the set of distribution lines between the buses, with $i,j$ as bus indices. The root bus $0$ represents the secondary of the transformer and is referred to as the slack bus (substation bus). The natural radial network orientation is considered, with each distribution line pointing away from bus $0$.

For each bus $i\in{\cal B}$, denote by $\mathcal{L}_{i}\subseteq\mathcal{L}$ the set of lines on the unique path from buses 0 to $i$, and by $Z_{i},q_{i}$ the active and reactive power consumptions of bus $i$, respectively. The magnitude of the complex voltage at bus $i$ is denoted by $v_{i}$, and we denote the fixed and known voltage at the slack bus by $v_{0}$. For each line $(i,j)\in\mathcal{L}$, denote by $r_{ij}$ and $x_{ij}$ its resistance and reactance. For each line, $(i,j)\in\mathcal{L}$, denote by $P_{ij}$ and $Q_{ij}$ the real and reactive power from buses $i$ to $j$, respectively. Let $\ell_{ij}$ denote the squared magnitude of the complex branch current from $i$ to $j$.

We adopt the distribution flow (DistFlow) model, introduced in [17], to model steady state power flow in a radial distribution network, as

	$\displaystyle P_{ij}$	$\displaystyle=-Z_{j}+\sum_{k:(j,k)\in\mathcal{L}}P_{jk}+r_{ij}\ell_{ij}$		(1a)
	$\displaystyle Q_{ij}$	$\displaystyle=-q_{j}+\sum_{k:(j,k)\in\mathcal{L}}Q_{jk}+x_{ij}\ell_{ij}$		(1b)
	$\displaystyle v_{j}^{2}$	$\displaystyle=v_{i}^{2}-2\left(r_{ij}P_{ij}+x_{ij}Q_{ij}\right)+\left(r_{ij}^{2}+x_{ij}^{2}\right)\ell_{ij},$		(1c)

where $\ell_{ij}=(P^{2}_{ij}+Q_{ij}^{2})/v_{i}^{2}$ is the line losses, (1a)-(1b) are the active and reactive power balance equations, and (1c) is the voltage drop. We exploit a linear approximation of the DistFlow model above that ignores line losses, given that in practice $\ell_{ij}\approx 0$ for all $(i,j)\in\mathcal{L}$. Therefore, the linearized Distflow (LinDistFlow) equations are given by re-writing (1a)-(1c) to

	$\displaystyle P_{ij}$	$\displaystyle=-\sum_{k\in{\cal O}(j)}Z_{k},\quad\quad\quad Q_{ij}=-\sum_{k\in{\cal O}(j)}q_{k}$
	$\displaystyle v_{j}^{2}$	$\displaystyle=v_{i}^{2}-2\left(r_{ij}P_{ij}+x_{ij}Q_{ij}\right),$

where ${\cal O}(j)$ represent the set node $j$’s descendants, including node $j$, i.e., ${\cal O}(j):=\left\{i:\mathcal{L}_{j}\subseteq\mathcal{L}_{i}\right\}$. This gives a solution for $v_{i}^{2}$ in terms of $v_{0}^{2}$, as

$$v_{0}^{2}-v_{i}^{2}=-2\sum_{j\in{\cal B}}\tilde{R}_{ij}Z_{j}-2\sum_{j\in{\cal B}}\tilde{X}_{ij}q_{j},$$

where

$$\tilde{R}_{ij}:=\sum_{(h,k)\in\mathcal{L}_{i}\cap\mathcal{L}_{j}}r_{hk},\quad\tilde{X}_{ij}:=\sum_{(h,k)\in\mathcal{L}_{i}\cap\mathcal{L}_{j}}x_{hk}$$

(3)

The LinDistFlow can be compactly written as,

$$\bm{v}=-\bm{R}\bm{Z}-\bm{X}\bm{q}+v_{0}^{2}\bm{1},$$

(4)

where $\bm{v}:=(v_{1}^{2},\ldots,v_{B}^{2}),\bm{Z}:=(Z_{1},\ldots,Z_{B}),\bm{q}:=(q_{1},\ldots,q_{B})$, and $\bm{R}:=[2\tilde{R}_{ij}]_{B\times B}$ and $\bm{X}:=[2\tilde{X}_{ij}]_{B\times B}$ are the resistance and reactance matrices, respectively. We treat the reactive power $\bm{q}$ as given constants rather than decision variables, which allows us to write (3) as

$$\bm{v}=-\bm{R}\bm{Z}+\bm{\hat{v}},$$

(5)

where $\bm{\hat{v}}:=-\bm{X}\bm{q}+v_{0}^{2}\bm{1}$. The voltage magnitude vector above is constrained as

$$\bm{v}_{\text{min}}\preceq\bm{v}\preceq\bm{v}_{\text{max}},$$

(6)

where $\bm{v}_{\text{min}}:=v_{\text{min}}^{2}\bm{1}$ and $\bm{v}_{\text{max}}:=v_{\text{max}}^{2}\bm{1}$. Given that the second term in (5) is fixed, we re-write (6) to

$$\bm{\underline{v}}\preceq\bm{v}\preceq\bm{\overline{v}},$$

(7)

where $\overline{\bm{v}}:=\bm{v}_{\text{max}}-\bm{\hat{v}}$ and $\underline{\bm{v}}:=\bm{\hat{v}}-\bm{v}_{\text{min}}$. We will impose (7) on the operation of the energy-sharing market.

Let ${\cal N}:=\{1,\ldots,N\}$ denote the set of energy-sharing system’s prosumers. Every prosumer $n$ is connected to one of the $B$ buses in the considered radial network through its revenue meter that measures the prosumer’s net consumption and BTM generation. Figure 1 shows an example 4-bus energy-sharing platform. We denote the set of prosumers connected to bus $i\in{\cal B}$ by ${\cal N}_{i}$, hence, ${\cal N}=\bigcup_{i\in{\cal B}}{\cal N}_{i}$. In this section, we model prosumers’ DER in $\S$III-A, and payment and surplus functions in $\S$III-B, followed by a formulation of the proposed bi-level program in $\S$III-C.

At the beginning of each pricing period, the operator communicates the price to each prosumer. Given the price, prosumers simultaneously move to solve their own surplus maximization problem. At the end of the netting period, and given the resulting $Z_{0}$, the DSO charges the energy sharing operator based on the NEM X tariff in (10). We propose the network-aware pricing policy and delineate its structure in $\S$IV-A, followed by solving the optimal response of prosumers in $\S$IV-B. We discuss the operator’s profit/deficit redistribution in $\S$IV-C and $\S$IV-E. In $\S$IV-D, we establish the market equilibrium result.

Our network-aware market mechanism was validated on the IEEE 13-bus feeder converted to a single-phase equivalent [21], see Figure 2. Bus 1 is the substation and represents the network slack bus. Buses 2 to 13 instead host 23 prosumers. For every $n\in{\cal N}$, the following utility function was chosen

$$U_{n}(d_{n})=\left\{\begin{array}[]{ll}\alpha_{n}d_{n}-\frac{1}{2}\beta_{n}d_{n}^{2},&0\leq d_{n}\leq\frac{\alpha_{n}}{\beta_{n}}\\ \frac{\alpha_{n}^{2}}{2\beta_{n}},&d_{n}>\frac{\alpha_{n}}{\beta_{n}},\end{array}\right.$$

(21)

where the parameters $\alpha_{n},\beta_{n}$ were learned and calibrated using historical retail prices³³3The retail prices were taken from Data.AustinTexas.gov historical residential rates in Austin, TX. and consumptions⁴⁴4For the historical consumption data, we used pre-2018 PecanStreet data for households in Austin, TX. and by assuming an elasticity of 0.21 taken from [22] (see appendix D in [16]). The minimum demand was set to ${\underline{\bm{d}}_{n}}=\bm{0}$ for every $n\in{\cal N}$, whereas the maximum demands ${\overline{\bm{d}}_{n}}$ and DER generations were obtained using data from the PecanStreet dataset. We set $\bm{v}_{\text{min}}=0.95$ p.u. and $\bm{v}_{\text{max}}=1.05$ p.u.

In our simulations, we considered four scenarios, described in the following, that differ in the DER generation levels. For each scenario, we used the exact AC power flow solver to obtain bus voltages, whereas we solved the optimization problems relying on the power flow equation linearization (4). The results are shown in Figure 3.

Scenario 1: the DER generation here is zero for each prosumer. Hence, $G_{0}=0$ and $G_{0}<\sigma_{1}(\bm{g})$. The energy-sharing system is importing energy. In this case, the energy sharing optimization problem solutions are such that $\underline{\eta}^{\ast}_{i}\neq 0$, i.e., some voltages are on the lower bound $\bm{v}_{\text{min}}$. The resulting prices are in general higher than $\pi^{+}$.

Scenario 2: the DER generation $G_{0}$ is non-zero but still not enough to cover the demand, i.e., $G_{0}<\sigma_{1}(\bm{g})$. Hence, the energy-sharing system is importing energy. However, the optimum demands are such that all the voltages are within the desired bounds and the energy prices are equal $\pi^{+}$.

Scenario 3: the DER generation was further increased in this scenario and $\sigma_{1}(\bm{g})\leq G_{0}\leq\sigma_{2}(\bm{g})$. That is, the energy-sharing platform did not exchange active power with the external network. The energy sharing platform optimization problem provides an energy price within $\pi^{+}$ and $\pi^{-}$; voltage limits are satisfied at the optimal consumption.

Scenario 4: here, we increased the generation until $G_{0}\geq\sigma_{2}(\bm{g})$. The platform exports power to the grid. The energy sharing optimization problem solution is such that the voltages in some locations are exactly $\bm{v}_{\text{max}}$ and the Lagrange multipliers vector $\overline{\bm{\eta}}^{\ast}$ is different from zero. The energy prices are smaller than $\pi^{-}$ and close to zero, i.e., consumption is incentivized to take full advantage of generation.

Some observations are in order. In general, we observe that increasing the DER generation $G_{0}$ results in the decrease of energy prices. The energy prices can in principle be greater than $\pi^{+}$, see Scenario 1. This is to ensure that the voltage constraints are satisfied by decreasing the power demand. Finally, we note a slight difference between the true and the expected (i.e., the ones computed by the energy-sharing platform optimization problem) voltage magnitudes. Indeed, we see that the voltages in Scenario 4 are all strictly lower than $\bm{v}_{\text{max}}$ even though we obtained $\overline{\eta}^{\ast}_{i}\neq 0$, see the middle panel of Figure 3. This can be explained by the fact that (11) was solved relying on the linearized equations (4) rather than on the true power flow equations. Note, however, that using the true equation would result in a nonconvex energy sharing optimization problem possibly displaying multiple local minima.

In this work, we propose a network-aware and welfare-maximizing market mechanism for energy-sharing coalitions that aggregate small but ubiquitous BTM DER downstream of a DSO’s revenue meter, charging the energy-sharing systems using a generic NEM tariff. The proposed pricing policy has ex-ante and ex-post pricing components. The ex-ante locational and threshold-based price decreases as the energy-sharing generation-to-demand ratio increases. The price is used to induce a collective prosumer reaction that decentrally maximizes social welfare while being network-cognizant. On the other hand, the ex-post charge/reward is used to enforce the market operator’s profit-neutrality condition. We show that the market mechanism achieves an equilibrium to the Stackelberg game between the operator and its prosumers. We also show that a unique proportional rule to re-allocate the operator’s profit/deficit can make the payment function of all energy-sharing prosumers uniform, even when the network constraints are binding. Our simulation results leverage real DER data on an IEEE 13-bus test feeder system to show how the dynamic pricing drives the energy sharing’s flexible consumption to abide by the network voltage limits.

The work of Ahmed S. Alahmed and Lang Tong was supported in part by the National Science Foundation under Award 2218110 and the Power Systems Engineering Research Center (PSERC) under Research Project M-46. The work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding is provided by the U.S. DOE Office of Energy Efficiency and Renewable Energy Building Technologies Office, United States. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.