Day-ahead to real-time price convergence. The expectation of real-time prices is approximately the day-ahead market clearing price of the same time period [1]. This convergence is facilitated by two factors. First, most suppliers and demands in electricity markets are settled in day-ahead markets, while real-time markets primarily settle the deviations to day-ahead settlements. Hence, real-time prices should be distributed around the day-ahead price. Second, virtual bids, in which market speculators can arbitrage between day-ahead and real-time markets, create incentives for participants to converge persistent price gaps between day-ahead to real-time.

Finite horizon and end value function. Given the limited time horizon market information available to storage operators, we consider the SDP bidding problem to have a finite horizon (such as end of the day) with a terminal value function \( V_T(e_T)\) representing the final value of energy stored. Note that \( V_T(e_T)\) can be simply set to zero to show no final energy value.

Storage bid curve. Given the calculated storage value functions, we generate the storage offer curve and bid curve based on the subderivatives of the cost functions, i.e., the marginal cost curve

where \( [x]^+ = \max\{0,x\}\) is used to reflect constraint (2.c) that no discharge power would be cleared during negative prices.

Storage opportunity bids. Different from generators that bid based on physical costs, the storage design bids based on opportunity costs. For example, as shown in equation (2.b), storage value \( V_{t-1}\) is dependent on its price prediction of the next period \( \hat \lambda_{t}\) as well as the value function \( V_{t}\) at period \( t\) . Similarly, \( V_{t}\) is dependent on the price prediction and its value function of period \( t+1\) . Recursively, we know storage value at period \( V_{t-1}\) is dependent over price forecasts of periods \( t, t+1, t+2,..., T\) . In practice, storage opportunity bids have been acknowledged by system operators like California ISO [2].

Concave value function. Given a concave end value function \( V_{T}(e_T)\) , then \( V_{t}(e_{t})\) is concave for all \( t\in\mathcal{T}^{'} = \{1,2,3,...,T-1\}\) and for all price distribution functions \( \hat \lambda_t \in \hat \lambda(\boldsymbol{\mu}, \boldsymbol{\sigma}^2)\) .

Convex storage bids. Given a concave end value function \( V_{T}(e_T)\) , \( {O}_t(p_t)\) monotonically increases with \( p_t\) , \( {B}_t(b_t)\) monotonically decreases with \( b_t\) , hence the market clearing model in (2) is always convex.

Deterministic storage bids. If \( \sigma_t^2=0\) in equation (2), then storage bid curves will be constant if \( V_{T}(e_{T})\) is a linear function. Hence, \( O_t(p_t)\) and \( B_t(b_t)\) as in (3) are linear.

Unbounded withholding with Gaussian distributions. Assume the storage owner anticipates future prices following Gaussian distributions with fixed expectations but unbounded variance. Given arbitrary time interval \( t \in \mathcal{T}^{'}\) , for a given price expectation over the next time period, \( \mu_{t+1}\) , for an arbitrary bid value \( \theta \geq 0\) , there exists a standard deviation \( \sigma_{t+1} \geq \underline{\sigma}\) such that

where constant \( \underline{\sigma}\) represents the lower bound essential for always ensuring the validity of inequality (6) at all times.

Given \( v_T(e_T)\geq 0, \forall e_T \in [0, E]\) and the storage’s duration \( E \geq 2P/\eta\) , then \( v_t(P\eta)\geq 0, \forall t \in \mathcal{T}^{'}\) .

Unbounded withholding for distributions of generalized types with at zero SoC. Given arbitrary time interval \( t\in\mathcal{T}^{'}\) and a concave value function \( V_{t+1}\) , for an arbitrary value \( \theta \geq 0\) and an arbitrary price expectation \( u_{t+1}\) , there exists a distributions \( \hat \lambda_{t+1}\) such that

which indicates storage economic withholding \( O_t\) is unbounded.

Unbounded withholding with extended bidding intervals. Given arbitrary time interval \( \tau\in\mathcal{T}\) and starting SoC \( e_{\tau-1}\in[0,E]\) , for an arbitrary bid value \( \theta \geq 0\) and an arbitrary trajectory of future price expectations \( u_t\) , there exists a set of future price distributions \( \hat \lambda_t\) such that for any \( p\in [0, \min\{P, e_{\tau-1}\eta\}]\)

if \( T-t \geq (e_{\tau-1}\eta-p_{\tau})/P\) .

Bounded withholding with bounded prices. Assume the storage has a linear final value function \( V_T(e) = \sigma_t \cdot e\) with \( \sigma_t \geq \Big[(\underline{ \lambda} - c )\eta\Big]^+\) , and market prices have an upper bound \( \overline{ \lambda}\) and a lower bound \( \underline{ \lambda}\) . Then, the storage bidding price has the following upper bound given \( t \in \mathcal{T}^{'}\) :

Bounded withholding with bounded expectations. Following same assumptions in Theorem 2, now adding that all price expectations have an upper bound \( \mu_t \leq \overline{\mu}\) . All assume there is a interval-based discount ratio \( \rho\in(0,1]\) , then the upper storage bid bound becomes

Bounded withholding with bounded price spikes. Assuming the price distributions are relatively centered around the expectation \( \mu_{t+1}\) , but each time period as a probability \( \alpha_{t+1}\) to observe a price spike which is significantly higher than the expectation \( \overline{\lambda} \gg \mu_{t+1}\) , then we can approximate the upper bid bound using (9.a) given \( t \in \mathcal{T}^{'}\) .

Assuming the storage capacity is negligibly small, the variance of the market clearing price \( \lambda_t\) scales linearly with the standard variance of the net demand \( D_t - w_t\) if the cost function is quadratic.

Following same assumptions in Proposition 4, if the uncertainty models of storage price forecast and systemic net demand follow the same distribution type, the storage utilizing uncertainty models for market bid formulation can potentially lower the overall system cost when the system encounters a suitable level of uncertainty.