DER Pricing Power in the Presence of Multi-Location Consumers with Load Migration Capabilities

The MLEC aims to minimize the total cost of supplying electricity to its different locations. The MLEC has a base demand $D_{n}^{base}$ at each location $n\in\mathcal{N}$ of which a fraction $\alpha$ can be shifted between locations.

We assume that the base demand $D_{n}^{base}$ and DERs’ production capacities $E_{p}^{gen}$, $p\in\mathcal{P}$, are fixed over the study period. We do not take load migration costs into account. The optimization problem of the MLEC in iteration $i$ of the dynamic game then reads as

subject to

The decision variables $\Theta_{i}$ of this problem are the purchased energy from the grid $E^{g}_{n,i}$ at location $n$, the purchased energy $E_{p,i}$ from each DER $p\in\mathcal{P}$, and the demand at location $n$ $D_{n,i}$. The MLEC’s objective (1) is to minimize the total cost of purchasing electricity from the grid and DERs, where $\pi_{p,i}$ denotes the price of electricity sold by the owner of DER $p\in\mathcal{P}$ to the MLEC. The constraints ensure the following conditions: demand-supply balance is maintained at each location (2); the total demand $D^{tot}$ of the MLEC is met (3); the purchased amount from each DER is less than its capacity $E_{p}^{gen}$ (4); the MLEC’s demand at location $n$ can only deviate by a fraction $\alpha\in[0,1]$ from the base demand $D_{n}^{base}$ at that location (5); electricity purchased from the grid $E^{g}_{n,i}$ at each location is non-negative (6).

Each DER owner aims to strategically maximize its profit from electricity sales, i.e., it estimates the reaction of the MLEC to different price offers and chooses its price bids accordingly.

We assume that the DERs are renewable power plants with zero marginal production cost. Therefore, the problem formulation for the DER owner $p$ in iteration $i$ of the dynamic game is

subject to

The decision variable is the offered energy price to the MLEC $\pi_{p,i}$. The objective (7) is to maximize the profit earned by selling electricity to the MLEC and the rest to the grid. The constraints imply that the DER owner estimates its market share depending on the bid price $\pi_{p,i}$ using a linear function (8) with parameters $a_{p,i}$ and $b_{p,i}$ and that the plausible DER owner offers are limited by the grid’s buying and selling prices (9).

To find the equilibrium between the MLEC and the DER owners, we simulate the negotiation process with Algorithm 1.

During the iterative negotiation process, each DER owner $p\in\mathcal{P}$ stores its current knowledge about the MLEC’s reaction to its possible price offers, i.e., an estimate of the price-demand curve, as a set $\mathcal{X}_{p}$ of pairs of offer prices and the corresponding energy amounts bought by the MLEC. The set $\mathcal{X}_{p}$ is initialized using the following rationale (Line 1): each DER owner assumes that by offering a price slightly lower (by $\epsilon>0$) than the grid’s selling price, it can sell all its generation to the MLEC, and by offering a price slightly higher than the grid’s price, it cannot sell anything. The DER owner also stores the optimal price and the optimal own profit seen so far (Line 2).

In each iteration $i$, all DER owners first (Line 4-11) estimate the parameters $a_{p,i},b_{p,i}$ of the linear price-demand curve used in (8) using linear regression. If the linear regression worked well, i.e., the coefficient of determination $R^{2}_{p,i}$ of the linear regression is high, the DER owners decide to choose the next offer price $\pi_{p,i}$ by optimally exploiting their current knowledge $\mathcal{X}_{p}$. In case their existing knowledge in inconclusive, i.e., $R^{2}_{p,i}$ is low, they try to better explore the MLEC’s reaction to different offer prices. To this end, they choose a new offer price at random from a normal distribution centered around the best price choice so far. The variance of this distribution decreases as the iteration number increases, following a decreasing function $\sigma_{i}^{2}=\sigma_{0}^{2}/i$.

Next (Line 12), the MLEC determines the energy amounts to buy from each DER. Lastly (Line 13-18), the DER owners update their knowledge about the MLEC’s behavior as well as their storage of the optimal price and profit seen so far. The actual profit is computed as $\text{Profit}(\pi_{p,i},E_{p,i})=E_{p,i}\pi_{p,i}+(E_{p}^{gen}-E_{p,i})\pi^{g,b}$ in this step.

The algorithm terminates (Line 19-23) when all DER owners’ price decisions change by less than $3\text{ ct/kWh}$ in consecutive iterations. This entire process is repeated ten times, and the results are averaged to mitigate the influence of random factors.

To conduct our experiments, we set the MLEC’s base demand to $D_{n}^{base}=5\text{ kW}$ and the total demand to $D^{tot}=N\times D_{n}^{base}$. The market prices are chosen as $\pi^{g,b}=15\text{ ct/kWh}$ and $\pi^{g,s}_{n}=50\text{ ct/kWh}-10\text{ ct/kWh}\times\dfrac{n-1}{N-1}$. All DERs have $E_{p}^{gen}=10\text{ kW}$, as is typical for decentral PV plants.

The game is iterated until the prices converge, which is defined as a change of no more than $0.05\text{ ct/kWh}$ between 30 consecutive iterations. We implement the dynamic game in Python and solve the linear program of MLEC and the quadratic program of each DER using Gurobi [gurobi].

The offered prices and the MLEC’s expenses for $N=1$ and $P\in\{1,2,100\}$ are shown in Fig. 1. For $P=1$ the observed offer prices equal the grid’s selling price. Increasing the number of production competitors lowers the offered prices towards the plausible minimum, i.e., the grid’s buying price. This behavior is as expected since for $P=1$ we have a local production monopoly. The producers’ pricing power decreases when there is more local competition. Here, as the capacity of each DER is higher than the demand, even for the case of p=2, the offered prices at equilibrium are equal to the minimum price.

With these simulations, we investigate the effect of load migration capabilities of the MLEC. We assume $P=N$, i.e., one DER per location which is realistic for many behind-the-meter microgrids. We change the MLEC’s demand shift capability parameter $\alpha$ in steps of $0.1$.

The resulting offer prices and the MLEC’s expenses are shown in Fig. 2 for $N=3$ locations and in Fig. 3 for $N=10$ locations.

When the value of $\alpha$ is zero, it means that load cannot be shifted between locations. In this case, the prices are at their maximum level, i.e., the grid’s selling prices. In this situation the locations are independent of each other and we observe local production monopolies as in the previous experiment.

However, as the value of $\alpha$ increases and the MLEC gains the ability to shift fractions of the load to cheaper locations, the offer prices decrease. This indicates that load migration can break the pricing power of local producers. The MLEC purchases most of its energy from these cheaper locations, which significantly reduces its electricity bill. Specifically, we observe a reduction in the MLEC’s bill by $65\%$ for $N=3$ and $67\%$ for $N=10$. This implies that MLECs with many locations can particularly benefit from load migration capabilities.