Co-optimizing the value of storage in energy and regulation service markets

Abstract

We develop a methodology for maximizing the present value of an independently operated electric energy storage (EES) unit co-optimized to perform both energy arbitrage (EA) and regulation service (RS). While our methodology applies to all types of EES, it is particularly suitable for EES units with a finite cycle life and a high power-to-capacity ratio (e.g., grid-scale batteries). We first state the constraints of the EES that limit its ability to simultaneously provide both EA and RS and formulate a deterministic linear program to find the perfect price foresight value (PPFV) of co-optimized storage. Next, we develop the receding horizon with multi-stage forecasting (RHMF) controller, which is able to efficiently co-optimize storage across EA and RS in an online real world setting with price uncertainty. We apply this controller to two battery technologies using market data from the independent service operator of New England (ISONE). We find that RHMF significantly outperforms several simpler feasible benchmark controllers and comes close to the PPFV upper bound. We also find that co-optimizing these technologies across both EA and RS provides increased profits relative to pursuing these strategies independently. The case study demonstrates the utility of our method in understanding how storage technology parameters impact financial returns.

Appendix: Price forecasting

The RHMF controller can leverage energy and regulation price forecasts from any model, and does not depend on the internal details of the model. In this paper, we use the seasonal autoregressive integrated moving average model with exogenous variables (SARIMAX). We use two instances of this model, one for energy and one for regulation service. The model selection and training is performed independently for EA and RS. Detailed general descriptions of these models and methods are available in [29] and with specific application to energy markets in [30]. Plug and play software is available from Matlab for parameter selection and training this model [31]. Briefly, the SARIMAX model specified by $(p,D,Q)(p_s,D_s,q_s)_s$ for price $p_t$ is written as:

$$\begin{aligned} \begin{aligned} \phi (L)\Phi (L)(1-L)^D(1-L^S)^{D_s}p_t = c + x_t \beta + \theta (L)\Theta (L) \epsilon _t, \end{aligned} \end{aligned}$$

(15)

where L is the lag operator (i.e., $L^i y_t = y_{t-i}$), and the polynomials $\theta (L)$, $\Theta (L)$, $\phi (L)$, and $\Phi (L)$ are defined as:

$$\begin{aligned} \theta (L)&= (1+\theta _1 L +\theta _2 L^2+,\ldots ,\theta _q L^{q}), \\ \Theta (L)&= (1+\Theta _1 L +\Theta _2 L^2+,\ldots ,\Theta _{q_s} L^{q_s}), \\ \phi (L)&= (1+\phi _1 L +\phi _2 L^2+,\ldots ,\phi _p L^{p}), \\ \Phi (L)&= (1+\Phi _1 L +\Phi _2 L^2+,\ldots ,\Phi _{p_s} L^{p_s}). \end{aligned}$$

Fig. 9

Forecast error

For both EA and RS, the exogenous variable $x_t$ is a vector, describing the positive and negative part of the dry bulb temperature minus 68 $^{\circ }$F, the positive and negative part of the dew point minus 50 $^{\circ }$F, and an indicator variable which is equal to 1 on national holidays and 0 otherwise, i.e.,

$$\begin{aligned} x_t = \begin{pmatrix} (Temp-68^F)^+ \\ (Temp-68^F)^-\\ (Dewpoint-50^F)^+\\ (Dewpoint-50^F)^- \\ \mathbbm {1}{ (isNationalHoliday) } \end{pmatrix}. \end{aligned}$$

These exogenous variables are assumed to be three known perfectly for all hours.

For a given model SARIMA$(p,D,Q)(p_s,D_s,q_s)_s$, the coefficients $\beta$, c, $\theta _1,\ldots ,\theta _q$, $\Theta _1,\ldots ,\Theta _{q_s}$, $\phi _1,\ldots ,\phi _p$, $\Phi _1,\ldots ,\Phi _{p_s}$ are optimized in the matlab toolbox via maximum likelihood estimation. We test the goodness of fit by retraining the model after each month of data in the 2006 ISONE dataset, generating a forecast from t to $t+48$ for each t in the subsequent month, and computing the mean absolute percentage error (MAPE) relative to the actual prices.

We select the model SARIMA$(p,D,Q)(p_s,D_s,q_s)_s$ by sweeping across parameters and selecting the best goodness of fit separately for EA and RS. We found the energy price model SARIMA$(3,0,3)(3,0,3)_{168}$ performs best, while the regulation price model SARIMA$(1,0,1)(1,0,1)_{24}$ performs best. As illustrated in Fig. 9, the MAPE for the forecasted spot price for the upcoming hour is 4.6 % for both energy and regulation. The MAPE for the forecasted spot rate at $t+48$ h is 7.9 % for regulation and 6.4 % for energy.