# Imbalance Market Real Options and the Valuation of Storage in Future Energy Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**S1.**- The battery operator has a positive expected profit from the arrangement.
**S2.**- The reserve contract cannot lead to a certain financial loss for the system operator.

**S1**is also known as the individual rationality or participation condition (Fudenberg et al. 1991). While the battery operator is assumed to be a profit maximiser, the system operator may engage in the arrangement for wider reasons than profit maximisation. To acknowledge the potential additional benefits provided by batteries, for example in providing response quickly and without direct emissions, condition

**S2**is less strict than individual rationality.

#### 1.1. Objectives

- A1
- The battery operator selects a time to purchase a unit of energy on the EIM and stores it.
- A2
- With this physical cover in place, the battery operator then chooses a later time to sell the incremental reserve contract to the system operator in exchange for the initial premium ${p}_{c}$.
- A3
- The system operator requests delivery of power when the EIM price X first lies above the level ${x}^{*}$ and immediately receives the contracted unit of energy in return for the utilisation payment ${K}_{c}$.

- M1
- For the single and lifetime problems, find the highest EIM price $\stackrel{\u02c7}{x}$ at which the battery operator may buy energy when acting optimally.
- M2
- For the single and lifetime problems, find the expected value of the total discounted cash flows (value function) for the battery operator corresponding to each initial EIM price $x\ge \stackrel{\u02c7}{x}$.

#### 1.2. Approach and Related Work

## 2. Methodology

#### 2.1. Formulation and Preliminary Results

#### 2.1.1. Optimal Stopping Problems and Solution Technique

#### 2.1.2. Single Problem

#### 2.1.3. Lifetime Problem Formulation and Notation

**Lemma**

**1.**

**Proof.**

#### 2.1.4. Sustainability Conditions Revisited

**S1**and

**S2**introduced in Section 1 are our standing economic assumptions. The next lemma, proved in the appendices, expresses them quantitatively. This makes way for their use in the mathematical considerations below.

**Lemma**

**2.**

**S1**and

**S2**are equivalent to the following quantitative conditions:

**S1***:- ${sup}_{x\in (a,b)}h\left(x\right)>0$, and
**S2***:- ${p}_{c}+{K}_{c}<{x}^{*}$.

**S1***is always satisfied when $a\le 0$.

#### 2.2. Three Exhaustive Regimes in the Single Problem

**S1***and

**S2***, are in force. For completeness, the notation and general optimal stopping theory used below is presented in Appendix A.

**S2***, h is negative on $[{x}^{*},\infty )$. The following theorem completes our aim

**M2**.

**Theorem**

**1.**

**S1***and

**S2***hold. With the definition (18), there are three exclusive cases:

- (A)
- ${L}_{c}\le \frac{h\left(x\right)}{{\varphi}_{r}\left(x\right)}$ for some x$\u27f9$there is $\widehat{x}<{x}^{*}$ that maximises $\frac{h\left(x\right)}{{\varphi}_{r}\left(x\right)}$, and then, for $x\ge \widehat{x}$, ${\tau}_{\widehat{x}}$ is optimal, and$${V}_{c}\left(x\right)={\varphi}_{r}\left(x\right)\frac{h\left(\widehat{x}\right)}{{\varphi}_{r}\left(\widehat{x}\right)},\phantom{\rule{2.em}{0ex}}x\ge \widehat{x}.$$
- (B)
- $\infty >{L}_{c}>\frac{h\left(x\right)}{{\varphi}_{r}\left(x\right)}$ for all x $\u27f9{V}_{c}\left(x\right)={L}_{c}\phantom{\rule{0.166667em}{0ex}}{\varphi}_{r}\left(x\right)$ and there is no optimal stopping time.
- (C)
- ${L}_{c}=\infty \u27f9{V}_{c}\left(x\right)=\infty $ and there is no optimal stopping time.

**Proof.**

**S1***, $h\left(y\right)$ is positive for some $y\in I$ and the value function ${V}_{c}\left(x\right)>0$. For case A, note first that the function h is negative on $[{x}^{*},b)$ by

**S2***, see (9) and (11). Therefore, the supremum of $\frac{h}{{\varphi}_{r}}$ is positive and must be attained at some (not necessarily unique) $\widehat{x}\in (a,{x}^{*})$. The optimality of ${\tau}_{\widehat{x}}$ for $x\ge \widehat{x}$ then follows from Lemma A1. Case B follows from Lemma A2 and the fact that ${L}_{c}>0$. Lemma A2 proves case C. The continuity of ${V}_{c}$ follows from Lemma A3. ☐

**Lemma**

**3.**

**S1${}^{*}$**holds, then:

- 1.
- The equality ${L}_{c}=0$ implies case A of Theorem 1.
- 2.
- Any of the following conditions is sufficient for ${L}_{c}=0$:
- (a)
- $a>-\infty $,
- (b)
- $a=-\infty $ and ${lim}_{x\to -\infty}\frac{x}{{\varphi}_{r}\left(x\right)}=0$.

**Proof.**

**S1${}^{*}$**ensures that h takes positive values. Hence, the ratio $\frac{h\left(x\right)}{{\varphi}_{r}\left(x\right)}>0={L}_{c}$ for some x. For assertion 2(a), recall from Section 2 that ${\varphi}_{r}(a+)=\infty $ since the boundary a is not-exit. Then, we have ${L}_{c}={lim\; sup}_{x\to a}(-x)/{\varphi}_{r}\left(x\right)=0$ as $a>-\infty $. In 2(b), the equality ${L}_{c}=0$ is immediate from the definition of ${L}_{c}$. ☐

**M1**, we have

**Corollary**

**1.**

- (a)
- the quantity$$\stackrel{\u02c7}{x}:=max\left\{x\in I:\phantom{\rule{4pt}{0ex}}\frac{h\left(x\right)}{{\varphi}_{r}\left(x\right)}=\underset{y\in I}{sup}\frac{h\left(y\right)}{{\varphi}_{r}\left(y\right)}\right\}$$
- (b)
- there is no price at which it is optimal for the battery operator to purchase energy. In this case, the single problem’s value function may either be infinite (case C) or finite (case B).

**Proof.**

#### 2.3. Two Exhaustive Regimes in the Lifetime Problem

**M1**for the lifetime problem.

**Corollary**

**2.**

**S1***and

**S2***hold. In the lifetime problem with $\widehat{\zeta}=\widehat{V}$, either:

- (a)
- the quantity$$\stackrel{\u02c7}{x}:=max\left\{x\in I:\phantom{\rule{4pt}{0ex}}\frac{\widehat{h}(x,\widehat{\zeta})}{{\varphi}_{r}\left(x\right)}=\underset{y\in I}{sup}\frac{\widehat{h}(y,\widehat{\zeta})}{{\varphi}_{r}\left(y\right)}\right\}$$
- (b)
- there is no price at which it is optimal for the battery operator to purchase energy. In this case, the lifetime value function may either be infinite (case 3) or finite (case 2b in Lemma A4 in Appendix B).

**Proof.**

**S2${}^{*}$**, this strategy would lead to unbounded losses almost surely in the lifetime problem started at EIM price ${x}^{*}$ leading to $\widehat{V}\left({x}^{*}\right)=-\infty $. This would contradict the fact that $\widehat{V}>0$, so we conclude that $\stackrel{\u02c7}{x}<{x}^{*}$. ☐

**M2**, we will show now that there are two regimes in the lifetime problem: either the lifetime value function is strictly greater than the single problem’s value function (and the cycle A1–A3 is repeated infinitely many times), or the lifetime value equals the single problem’s value. Although the latter case appears counterintuitive, it is explained by the fact that the lifetime problem’s value is then attained only in the limit when the purchase of energy (action A1) is made at a decreasing sequence of prices converging to a, the left boundary of the process $\left({X}_{t}\right)$. In this limit, the benefit of future payoffs becomes negligible, equating the lifetime value to the single problem’s value.1

**Theorem**

**2.**

- ($\alpha $)
- $\widehat{V}\left(x\right)>{V}_{c}\left(x\right)$ for all $x\ge {x}^{*}$,
- ($\beta $)
- $\widehat{V}\left(x\right)={V}_{c}\left(x\right)$ for all $x\ge {x}^{*}$ (or both are infinite for all x).

**Proof.**

**Corollary**

**3.**

**M1**and

**M2**for the lifetime problem, in the next section, we provide results for the construction and verification of the lifetime value function and corresponding stopping time. For this purpose, we close this section by summarising results obtained above (making use of additional results from Appendix C).

**Theorem**

**3.**

#### 2.4. Construction of the Lifetime Value Function

**Lemma**

**4.**

**Proof.**

## 3. Results

**S1***and

**S2***hold. In the examples of this section, the stopping sets $\mathsf{\Gamma}$ for the single and lifetime problems take the form $(a,\stackrel{\u02c7}{x}]$ although, in general, stopping sets may have much more complex structure. Interestingly, the stopping sets for the single and lifetime problem are either both half-lines or both compact intervals.

**S2***is ensured by the explicit choice of parameters. Verification of condition

**S1${}^{*}$**is straightforward by checking, for example, if the left boundary a of the interval I satisfies $a<{p}_{c}+{lim}_{x\to a}\frac{{\psi}_{r}\left(x\right)}{{\psi}_{r}\left({x}^{*}\right)}{K}_{c}$, i.e., that ${lim\; sup}_{x\to a}h\left(x\right)>0$. In particular,

**S1${}^{*}$**always holds if $a=-\infty $.

#### 3.1. OU Price Process

#### 3.2. General Mean-Reverting Processes

**Lemma**

**5.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Remark**

**1.**

#### 3.3. Shifted Exponential Price Processes

**Theorem**

**4.**

**S1***and

**S2***hold. Then,

- (i)
- (Single problem) There exists $\widehat{z}<{z}^{*}$ that maximises $\frac{{h}_{f}\left(z\right)}{{\varphi}_{r}^{Z}\left(z\right)}$, the stopping time ${\tau}_{\widehat{z}}$ is optimal for $z\ge \widehat{z}$, and$${V}_{c}\left(z\right)={\varphi}_{r}^{Z}\left(z\right)\frac{{h}_{f}\left(\widehat{z}\right)}{{\varphi}_{r}^{Z}\left(\widehat{z}\right)},\phantom{\rule{2.em}{0ex}}z\ge \widehat{z}.$$
- (ii)
- (Lifetime problem) The lifetime value function $\widehat{V}$ is continuous and a fixed point of $\widehat{\mathcal{T}}$. There exists $\tilde{z}\in (\widehat{z},{z}^{*})$ which maximises $\frac{\widehat{h}(z,\widehat{V})}{{\varphi}_{r}^{Z}\left(z\right)}$ and ${\tau}_{\tilde{z}}$ is an optimal stopping time for $z\ge \tilde{z}$ with$$\widehat{V}\left(z\right)=\widehat{\mathcal{T}}\widehat{V}\left(z\right)={\varphi}_{r}^{Z}\left(z\right)\frac{\widehat{h}(\tilde{z},\widehat{V})}{{\varphi}_{r}^{Z}\left(\tilde{z}\right)},\phantom{\rule{2.em}{0ex}}z\ge \tilde{z}.$$

**Proof.**

**S1${}^{*}$**may be given in terms of the problem parameters. Assume that ${a}^{Z}=-\infty $ as in the examples studied below. If ${p}_{c}>D$ and ${K}_{c}\ge 0$, this is sufficient for the condition

**S1${}^{*}$**to be satisfied as then ${h}_{f}\left(z\right)\ge -f\left(z\right)+{p}_{c}>0$ for sufficiently small z. When ${p}_{c}=D$ and ${K}_{c}>0$, it is sufficient to verify that ${e}^{bz}=o\left({\psi}_{r}^{Z}\left(z\right)\right)$ as $z\to -\infty $ since then ${h}_{f}\left(z\right)=-d{e}^{bz}+{\psi}_{r}^{Z}\left(z\right)\phantom{\rule{0.166667em}{0ex}}{K}_{c}/{\psi}_{r}^{Z}\left({z}^{*}\right)$ for $z<{z}^{*}$. On the other hand, our assumption that

**S1${}^{*}$**holds necessarily excludes parameter combinations with ${p}_{c}-D={K}_{c}=0$, since the reserve contract writer then cannot make any profit because ${h}_{f}\left(z\right)\le 0$ for all z.

#### 3.3.1. Brownian Motion Imbalance Process

- Assume first that $r>\frac{1}{2}{b}^{2}$:
- (i)
- We may exclude the subcase ${p}_{c}\le D$, since then $H\left(y\right)=\frac{\widehat{h}(z,\widehat{\zeta})}{{\varphi}_{r}^{Z}\left(z\right)}{|}_{z={\left({F}^{Z}\right)}^{-1}\left(y\right)}$ is strictly convex on $(0,{F}^{Z}\left({z}^{*}\right))$ for any $\widehat{\zeta}$ and $\mathsf{\Gamma}$ cannot intersect this interval, contradicting Theorem 4 and, consequently, violating
**S1${}^{*}$**or**S2${}^{*}$**. - (ii)
- If ${p}_{c}>D$, H is concave on $(0,{F}^{Z}\left(B\right))$ and convex on $({F}^{Z}\left(B\right),\infty )$, where$$B=\frac{1}{b}log\left(\frac{r({p}_{c}-D)}{d(r-\frac{1}{2}{b}^{2})}\right).$$By Theorem 4 and the positivity of H on $(0,{F}^{Z}\left(\widehat{z}\right)),$ we have $\mathsf{\Gamma}=(-\infty ,\widehat{z}]$ and $\mathsf{\Gamma}=(-\infty ,\tilde{z}]$ for the single and lifetime problems, respectively, with $\tilde{z}<\widehat{z}<B$.

- Suppose that $r<\frac{1}{2}{b}^{2}$.
- (i)
- When ${p}_{c}\ge D$, the function H is concave on $(0,\infty )$. Hence, the stopping sets $\mathsf{\Gamma}$ for single and lifetime problems have the same form as in case 1(ii) above.
- (ii)
- If ${p}_{c}<D$, the function H is convex on $(0,{F}^{Z}\left(B\right))$ and concave on $({F}^{Z}\left(B\right),\infty )$. The set $\mathsf{\Gamma}$ must then be an interval, respectively $[{\widehat{z}}_{0},\widehat{z}]$ and $[{\tilde{z}}_{0},\tilde{z}]$. For explicit expressions for the left and right endpoints for the single problem, as well as sufficient conditions for
**S1${}^{*}$**, the reader is refered to Moriarty and Palczewski (2017).

- In the boundary case $r=\frac{1}{2}{b}^{2}$, the convexity of H is determined by the sign of the difference $D-{p}_{c}$. As above, the possibility $D>{p}_{c}$ is excluded since then H is strictly convex. Otherwise, H is concave and the stopping sets $\mathsf{\Gamma}$ have the same form as in case 1(ii) above.

#### 3.3.2. OU Imbalance Process

- If ${p}_{c}\ge D$, then the function $\eta $ is negative on $(-\infty ,u)$, where u is the unique root of $\eta $. Hence, H is concave on $(0,{F}^{Z}\left(u\right))$ and convex on $({F}^{Z}\left(u\right),\infty )$. The stopping sets $\mathsf{\Gamma}$ for the single and lifetime problems must then be of the form $(-\infty ,\widehat{z}]$ and $(-\infty ,\tilde{z}]$, respectively, cf. case 1(ii) in Section 3.3.1.
- The case ${p}_{c}<D$ is more complex:
- (i)
- Let ${z}^{\diamond}\ge {z}^{*}$. We exclude the possibility $\eta \left({z}^{*}\right)\ge 0$, since then the function H is convex on $(0,{F}^{Z}\left({z}^{*}\right))$ and the set $\mathsf{\Gamma}$ has empty intersection with this interval, contradicting Theorem 4 and, consequently, violating
**S1${}^{*}$**or**S2${}^{*}$**. When $\eta \left({z}^{*}\right)<0$, H is convex on $(0,{F}^{Z}\left(u\right))$ and concave on $({F}^{Z}\left(u\right),{F}^{Z}\left({z}^{*}\right))$, where u is the unique root of $\eta $ on $(0,{z}^{*})$. Therefore, the stopping sets $\mathsf{\Gamma}$ for the single and lifetime problems are of the form $[{\widehat{z}}_{0},\widehat{z}]$ and $[{\tilde{z}}_{0},\tilde{z}]$, respectively, with $min({\widehat{z}}_{0},{\tilde{z}}_{0})>u$, cf. case 2(ii) in Section 3.3.1. - (ii)
- Consider now ${z}^{\diamond}<{z}^{*}$. As above, we exclude the case $\eta \left({z}^{\diamond}\right)\ge 0$, since then H is convex on $(0,{F}^{Z}\left({z}^{*}\right))$. The remaining case $\eta \left({z}^{\diamond}\right)<0$ implies that the stopping sets $\mathsf{\Gamma}$ have the same form as in case 2(i) above, as H is convex and then concave if $\eta \left({z}^{*}\right)\le 0$, and convex–concave–convex if $\eta \left({z}^{*}\right)>0$.

## 4. Benchmark Case Study and Economic Implications

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Lemmas and Proofs from Section 2

**Lemma**

**A1.**

- 1.
- the stopping time ${\tau}_{\widehat{x}}$ is optimal,
- 2.
- $v\left(x\right)=\frac{\vartheta \left(\widehat{x}\right)}{{\varphi}_{r}\left(\widehat{x}\right)}{\varphi}_{r}\left(x\right),$
- 3.
- any stopping time τ with ${\mathbb{P}}^{x}\left\{\vartheta \left({X}_{\tau}\right)/{\varphi}_{r}\left({X}_{\tau}\right)<\vartheta \left(\widehat{x}\right)/{\varphi}_{r}\left(\widehat{x}\right)\right\}>0$ is strictly suboptimal for the problem $v\left(x\right)$.

**Proof.**

**Lemma**

**A2.**

- 1.
- If $L=\infty $, then the value function is infinite and there is no optimal stopping time.
- 2.
- If $L<\infty $ and $L>\vartheta \left(x\right)/{\varphi}_{r}\left(x\right)$ for all $x\in I$, then there is no optimal stopping time and the value function equals $v\left(x\right)=L{\varphi}_{r}\left(x\right)$.

**Proof.**

**Assertion 1.**

**Assertion 2.**Recall that, due to the supremum of $\frac{\vartheta}{{\varphi}_{r}}$ being strictly positive, we have $L>0$. From the proof of Lemma A1, for an arbitrary stopping time $\tau $, we have

**Corollary**

**A1.**

**Lemma**

**A3.**

**Proof**

**of Lemma 2.**

**S1***does not hold, then the payoff from cycle A1–A3 is not profitable (on average) for any value of the EIM price x, so

**S1**does not hold. Conversely, if

**S1***holds, then there exists x such that $\widehat{\mathcal{T}}\mathbf{0}\left(x\right)\ge h\left(x\right)>0$. For any other ${x}^{\prime}$, consider the following strategy: wait until the process X hits x and proceed optimally thereafter. This results in a strictly positive expected value: $\widehat{\mathcal{T}}\mathbf{0}\left({x}^{\prime}\right)>0$ and, by the arbitrariness of ${x}^{\prime}$, we have $\widehat{\mathcal{T}}\mathbf{0}>0$.

**S2***holds. Then, the system operator makes a profit on the reserve contract (relative to simply purchasing a unit of energy at the power delivery time ${\widehat{\tau}}_{e}$, at the price $X\left({\widehat{\tau}}_{e}\right)\ge {x}^{*}$) in undiscounted cash terms. Considering discounting, the system operator similarly makes a profit provided the EIM price reaches the level ${x}^{*}$ (or above) sufficiently quickly. Since this happens with positive probability for a regular diffusion, a certain financial loss for the system operator is excluded. When

**S2***does not hold, suppose first that ${p}_{c}+{K}_{c}>{x}^{*}$: then, the system operator makes a loss in undiscounted cash terms, and if the reserve contract is sold when $x\ge {x}^{*}$, then this loss is certain. In the boundary case ${p}_{c}+{K}_{c}={x}^{*}$, the battery operator can only make a profit by purchasing energy and selling the reserve contract when ${X}_{t}<{x}^{*}$, in which case the system operator makes a certain loss. This follows since instead of buying the reserve contract, the system operator could invest ${p}_{c}>0$ temporarily in a riskless bond, withdrawing it with interest when the EIM price rises to ${x}^{*}={p}_{c}+{K}_{c}$. The loss in this case is equal in value to the interest payment. ☐

## Appendix B. Lemmas for the Lifetime Problem

**Definition**

**A1.**

**Lemma**

**A4.**

**S1***and

**S2***hold. If $\widehat{\zeta}$ is an admissible continuation value function, then

- 1.
- In case A, there exists $\widehat{x}\le {x}^{*}$ which maximises $\frac{\widehat{h}(x,\widehat{\zeta})}{{\varphi}_{r}\left(x\right)}$ and ${\tau}_{\widehat{x}}$ is an optimal stopping time for $x\ge \widehat{x}$ with value function$$v\left(x\right)=\widehat{\mathcal{T}}\widehat{\zeta}\left(x\right)={\varphi}_{r}\left(x\right)\frac{\widehat{h}(\widehat{x},\widehat{\zeta})}{{\varphi}_{r}\left(\widehat{x}\right)},\phantom{\rule{2.em}{0ex}}x\ge \widehat{x}.$$Denoting by ${\widehat{x}}_{0}$ the corresponding $\widehat{x}$ in case A of Theorem 1, we have ${\widehat{x}}_{0}\le \widehat{x}$.
- 2.
- In case B, either
- (a)
- there exists ${x}_{L}\in (a,b)$ with $\frac{\widehat{h}({x}_{L},\widehat{\zeta})}{{\varphi}_{r}\left({x}_{L}\right)}\ge {L}_{c}$: then, there exists $\widehat{x}\in (a,{x}^{*}]$ which maximises $\frac{\widehat{h}(x,\widehat{\zeta})}{{\varphi}_{r}\left(x\right)}$, and ${\tau}_{\widehat{x}}$ is an optimal stopping time for $x\ge \widehat{x}$ with value function $v\left(x\right)={\varphi}_{r}\left(x\right)\frac{\widehat{h}(\widehat{x},\widehat{\zeta})}{{\varphi}_{r}\left(\widehat{x}\right)}$ for $x\ge \widehat{x}$; or
- (b)
- there does not exist ${x}_{L}\in (a,b)$ with $\frac{\widehat{h}({x}_{L},\widehat{\zeta})}{{\varphi}_{r}\left({x}_{L}\right)}\ge {L}_{c}$: then, the value function is $v\left(x\right)={L}_{c}\phantom{\rule{0.166667em}{0ex}}{\varphi}_{r}\left(x\right)$ and there is no optimal stopping time.

- 3.
- In case C, the value function is infinite and there is no optimal stopping time.

**Proof.**

**S1${}^{*}$**, is attained on $(a,{x}^{*}]$ or asymptotically when $x\to a$. In cases 1 and 2a, the optimality of ${\tau}_{\widehat{x}}$ for $x\ge \widehat{x}$ then follows from Lemma A1. To see that ${\widehat{x}}_{0}\le \widehat{x}$ in case 1, take $x<{\widehat{x}}_{0}$. Then, from (A6), we have

**Remark**

**A1.**

**Remark**

**A2.**

**Lemma**

**A5.**

**Proof.**

- their expected net present value is given by an optimal stopping problem, namely, the timing of the next action A1:$$\underset{\tau \ge {\sigma}^{*}}{sup}{\mathbb{E}}^{x}\left\{{e}^{-r\tau}{h}_{\left(iii\right)}\left({X}_{\tau}\right){\mathbf{1}}_{\tau <\infty}\right\},$$
- the choice ${\tau}_{A2}=0$ minimises the exercise time ${\sigma}^{*}$ and thus maximises the value of component (iii), since the supremum in (A7) is then taken over the largest possible set of stopping times.

**Lemma**

**A6.**

- 1.
- For each $n\ge 1$, the function ${\widehat{\zeta}}_{n}:={\widehat{\mathcal{T}}}^{n}\mathbf{0}$ is an admissible continuation value function and is decreasing on $[{x}^{*},b)$.
- 2.
- The functions ${\widehat{\mathcal{T}}}^{n}\mathbf{0}$ are strictly positive and uniformly bounded in n.
- 3.
- The limit $\widehat{\zeta}={lim}_{n\to \infty}{\widehat{\mathcal{T}}}^{n}\mathbf{0}$ exists and is a strictly positive bounded function. Moreover, the lifetime value function $\widehat{V}$ coincides with $\widehat{\zeta}$.
- 4.
- The lifetime value function $\widehat{V}$ is a fixed point of $\widehat{\mathcal{T}}$.

**Proof.**

## Appendix C. Uniqueness of Fixed Points

**Lemma**

**A7.**

**Proof**

**of Lemma A7.**

**Lemma**

**A8.**

**Proof.**

**Corollary**

**A2.**

**Proof.**

## Appendix D. Note on Lemma 3

## Appendix E. Verification Theorem for the Lifetime Value Function

- (I)
- $\widehat{\mathcal{T}}{V}_{1}={V}_{2}$,
- (II)
- ${V}_{1}\left({x}^{*}\right)={V}_{2}\left({x}^{*}\right)$,
- (III)
- for $i=1,2$, the highest price at which the battery operator buys energy in the problem $\widehat{\mathcal{T}}{V}_{i}$ is not greater than ${x}^{*}$.

**Lemma**

**A9.**

**Proof.**

**S2***implies that the payoff $\mathfrak{h}(x,y)$ of (A11) is negative for $x>{x}^{*}$, which establishes property (iii) for problem (A12).

**Corollary**

**A3.**

**Proof.**

**S2***leads to $\widehat{\mathcal{T}}{\widehat{\xi}}^{y}\left({x}^{*}\right)<{\widehat{\xi}}^{y}\left({x}^{*}\right),$ which is a contradiction. Thus, from Lemma A8, ${\widehat{\mathcal{T}}}^{n}\mathbf{0}$ converges to ${\widehat{\xi}}^{y}$ as $n\to \infty $. As the limit of ${\widehat{\mathcal{T}}}^{n}\mathbf{0}$ is the lifetime value function, we obtain $\widehat{V}={\widehat{\xi}}^{y}$.

## Appendix F. Facts about the OU Process

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1 | If the lifetime value is infinite then so is the single problem’s value and they are equal in this sense. When the lifetime value is zero then it is optimal not to enter the contract, and so the single problem’s value is also zero. |

**Figure 2.**Sensitivity of the expected value in the single problem with respect to the stopping boundary. The EIM price is modelled as an Ornstein-Uhlenbeck process $d{X}_{t}=3.42(47.66-{X}_{t})dt+30.65d{W}_{t}$ (time measured in days, fitted to Elexon Balancing Mechanism price half-hourly data from July 2011 to March 2014). The interest rate $r=0.03$, power delivery level ${x}^{*}=60$, the initial premium ${p}_{c}=10$, and the utilisation payment ${K}_{c}=40$. The initial price is ${X}_{0}$ is set equal to ${x}^{*}$.

**Figure 3.**Results obtained with the Ornstein-Uhlenbeck model fitted in Section 4, as functions of the total premium, with interest rate $3\%$ per annum. Solid lines: ${x}^{*}=100$, dotted: ${x}^{*}=75$, dashed: ${x}^{*}=50$. Left: lifetime value $\widehat{V}\left({x}^{*}\right)$. Right: the stopping boundary $\stackrel{\u02c7}{x}$, the maximum price for which the battery operator can buy energy optimally.

**Figure 4.**Lifetime value $\widehat{V}\left({x}^{*}\right)$ as a function of ${x}^{*}$ with the Ornstein-Uhlenbeck model fitted in Section 4, with interest rate $3\%$ per annum. Dashed line: ${p}_{c}+{K}_{c}=20$, solid: ${p}_{c}+{K}_{c}=30$, dotted: ${p}_{c}+{K}_{c}=40$, mixed: ${p}_{c}+{K}_{c}=50$. The horizontal grey line indicates the current price of lithium-ion battery storage per MWh (IRENA 2017, Figure 33).

**Table 1.**Summary statistics for the 15 min balancing group price per MWh in the German Amprion area, 1 June 2012 to 31 May 2016.

Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. |
---|---|---|---|---|---|

−6002.00 | 0.27 | 33.05 | 31.14 | 66.97 | 6344.00 |

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**MDPI and ACS Style**

Moriarty, J.; Palczewski, J.
Imbalance Market Real Options and the Valuation of Storage in Future Energy Systems. *Risks* **2019**, *7*, 39.
https://doi.org/10.3390/risks7020039

**AMA Style**

Moriarty J, Palczewski J.
Imbalance Market Real Options and the Valuation of Storage in Future Energy Systems. *Risks*. 2019; 7(2):39.
https://doi.org/10.3390/risks7020039

**Chicago/Turabian Style**

Moriarty, John, and Jan Palczewski.
2019. "Imbalance Market Real Options and the Valuation of Storage in Future Energy Systems" *Risks* 7, no. 2: 39.
https://doi.org/10.3390/risks7020039