Financial storage rights in electric power networks

Abstract

The decreasing cost of energy storage technologies coupled with their potential to bring significant benefits to electric power networks have kindled research efforts to design both market and regulatory frameworks to facilitate the efficient construction and operation of such technologies. In this paper, we examine an open access approach to the integration of storage, which enables the complete decoupling of a storage facility’s ownership structure from its operation. In particular, we analyze a nodal spot pricing system built on a model of economic dispatch in which storage is centrally dispatched by the independent system operator (ISO) to maximize social welfare. Concomitant with such an approach is the ISO’s collection of a merchandising surplus reflecting congestion in storage. We introduce a class of tradable electricity derivatives—referred to as financial storage rights (FSRs)—to enable the redistribution of such rents in the form of financial property rights to storage capacity; and establish a generalized simultaneous feasibility test to ensure the ISO’s revenue adequacy when allocating such financial property rights to market participants. Several advantages of such an approach to open access storage are discussed. In particular, we illustrate with a stylized example the role of FSRs in synthesizing fully hedged, fixed-price bilateral contracts for energy, when the seller and buyer exhibit differing intertemporal supply and demand characteristics, respectively.

Appendix

1.1 Proof of Lemma 1

Let $(V,U,{\varLambda })$ denote an efficient market equilibrium throughout. And, let $(\varvec{{\lambda }}(k), \gamma (k), \varvec{{\mu }}(k), \overline{\varvec{{\nu }}}_i, \underline{\varvec{{\nu }}}_i)$ denote the corresponding Lagrange multipliers satisfying the KKT conditions (6)–(13) for all $k$ and $i$. We prove the desired result by establishing nonnegativity of both the TCS and SCS.

Proposition 1

$\mathrm{TCS}= \sum _{k=0}^{N-1} \varvec{{\mu }}(k)^{\top } {\mathbf {c}} \ge 0$.

Proof

We have that $\mathrm{TCS}= -\sum _{k=0}^{N-1} \varvec{{\lambda }}(k)^{\top } ({\mathbf {v}}(k)+{\mathbf {u}}(k))$ based on its definition in (16). Substituting for $\varvec{{\lambda }}(k)$ according to Eq. (8), and using the fact that ${\mathbf {1}}^{\top }({\mathbf {v}}(k)+{\mathbf {u}}(k)) = 0$ for all $k$, we have that

$$\begin{aligned} \mathrm{TCS}= \sum _{k=0}^{N-1} \varvec{{\mu }}(k)^{\top } H ({\mathbf {v}}(k)+{\mathbf {u}}(k)). \end{aligned}$$

The complementary slackness condition (11) yields $\mathrm{TCS}= \sum _{k=0}^{N-1} \varvec{{\mu }}(k)^{\top } {\mathbf {c}}$, which is clearly nonnegative. $\square$

Proposition 2

$\mathrm{SCS}= \sum _{i=1}^{n} \overline{\varvec{{\nu }}}_i^{\top } {\mathbf {b}}_i \ge 0$.

Proof

We have that $\mathrm{SCS}= \sum _{i=1}^{n} \varvec{{\lambda }}_i^{\top } {\mathbf {u}}_i$ based on its definition in (17). A direct substitution of the stationarity condition (10) and complementary slackness conditions (12)–(13) yields

$$\begin{aligned} \mathrm{SCS}= \sum _{i=1}^{n} \overline{\varvec{{\nu }}}_i^{\top } L {\mathbf {u}}_i - \underline{\varvec{{\nu }}}_i^{\top } L {\mathbf {u}}_i = \sum _{i=1}^{n} \overline{\varvec{{\nu }}}_i^{\top } {\mathbf {b}}_i, \end{aligned}$$

which is clearly nonnegative. $\square$

The desired result follows from Propositions 1 and 2, as $\mathrm{MS}= \mathrm{TCS}+ \mathrm{SCS}$.

1.2 Proof of Lemma 2

Let $(V,U,{\varLambda })$ denote an efficient market equilibrium throughout the proof. Also, let $\gamma (k)$ and $\varvec{{\mu }}(k)$ denote the corresponding Lagrange multipliers associated with the power balance and line flow capacity constraints for each time period $k=0, \dots , N-1$. The maximum rent achievable by any simultaneously feasible collection of transmission rights is given by the optimal value of

$$\begin{aligned}&\text {maximize}&\mathrm{{\Phi }}({{\mathcal {T}}}, {{\mathcal {F}}})&\end{aligned}$$

(22)

$$\begin{aligned}&\text {subject to}&{\mathbf {t}}(k) + {\mathbf {q}}(k) \in \ {\mathcal {P}}({\mathbf {c}}-{\mathbf {f}}(k)),&k=0,\dots ,N-1 \end{aligned}$$

(23)

$$\begin{aligned}&{\mathbf {q}}_{i} \in \ {\mathcal {U}}({\mathbf {b}}_i),&i =1,\dots ,n, \end{aligned}$$

(24)

where $\mathrm{{\Phi }}({{\mathcal {T}}}, {{\mathcal {F}}})= \sum _{k=0}^{N-1} -\varvec{{\lambda }}(k)^{\top } {\mathbf {t}}(k) \ + \ \varvec{{\mu }}(k)^{\top } {\mathbf {f}}(k)$. It is not difficult to show (by induction) that any feasible solution of problem (22)–(24) must satisfy ${\mathbf {q}}_i={\mathbf {0}}$ for $i = 1, \dots , n$. This stems from the injection/extraction symmetry required by our definition of FTRs (which implies that ${\mathbf {1}}^{\top } {\mathbf {t}}(k) = 0$ for $k= 0, \dots , N-1$), and our assumption of zero initial stored energy. One can, therefore, equivalently reformulate problem (22)–(24) as

$$\begin{aligned}&\text {maximize}&\mathrm{{\Phi }}({{\mathcal {T}}}, {{\mathcal {F}}})&\end{aligned}$$

(25)

$$\begin{aligned}&\text {subject to}&{\mathbf {t}}(k) \in \ {\mathcal {P}}({\mathbf {c}}-{\mathbf {f}}(k)), \qquad k=0,\dots ,N-1.&\end{aligned}$$

(26)

This is a convex optimization problem with linear constraints in the decision variables ${\mathbf {t}}(k)\in \mathbb {R}^n$ and ${\mathbf {f}}(k) \in \mathbb {R}^{2m}_+$ ($k=0, \dots , N-1$). It follows that a primal optimal solution is characterized by the existence of Lagrange multipliers ${\tilde{\gamma }}(k)\in \mathbb {R}$ and $\tilde{\varvec{{\mu }}}(k) \in \mathbb {R}^{2m}_+$ ($k=0, \dots , N-1)$ such that the KKT conditions (26)–(29) hold. The stationarity condition is given by:

$$\begin{aligned} {\tilde{\gamma }}(k) {\mathbf {1}} - H^{\top } \tilde{\varvec{{\mu }}}(k) - \varvec{{\lambda }}(k)=&\ 0, \qquad k=0,\dots , N-1 \end{aligned}$$

(27)

$$\begin{aligned} \tilde{\varvec{{\mu }}}(k) - \varvec{{\mu }}(k) =&\ 0, \qquad k=0,\dots , N-1. \end{aligned}$$

(28)

The complementary slackness condition is given by:

$$\begin{aligned} \tilde{\varvec{{\mu }}}(k) \circ \left( H {\mathbf {t}}(k) - \mathbf {c}+ {\mathbf {f}}(k) \right) = 0,&\qquad k=0, \dots , N-1. \end{aligned}$$

(29)

Recall that $(V,U,{\varLambda })$ and $\gamma (k), \varvec{{\mu }}(k)$ satisfy the KKT conditions (6), (8) and (11) associated with multi-period economic dispatch problem. It follows that a primal optimal solution to problem (25)–(26) is given by

$$\begin{aligned} {\mathbf {t}}(k) = {\mathbf {v}}(k) + {\mathbf {u}}(k) \quad \text {and} \quad {\mathbf {f}}(k)={\mathbf {0}}, \qquad k=0, \dots , N-1. \end{aligned}$$

This optimal solution yields a collection of transmission rights with an associated rent of

$$\begin{aligned} \mathrm{{\Phi }}({{\mathcal {T}}}, {{\mathcal {F}}})= \sum _{k=0}^{N-1} -\varvec{{\lambda }}(k)^{\top } {\mathbf {t}}(k) = \sum _{k=0}^{N-1} -\varvec{{\lambda }}(k)^{\top } ({\mathbf {v}}(k)+{\mathbf {u}}(k)). \end{aligned}$$

Upon examination of Eqs. (14)–(17), it is straightforward to verify that the right-hand side equals the $\mathrm{TCS}$ associated with $(V,U,{\varLambda })$, thus completing the proof.

1.3 Proof of Theorem 1

Let $(V,U,{\varLambda })$ denote an efficient market equilibrium, and let $\gamma (k), \varvec{{\mu }}(k)$ ($k=0, \dots , N-1$) and ${\mathbf {{\overline{\nu }}}}_i, {\mathbf {{\underline{\nu }}}}_i$ ($i =1, \dots , n$) denote the corresponding Lagrange multipliers. The maximum rent achievable by any simultaneously feasible collection of transmission and storage rights is given by the optimal value of

$$\begin{aligned}&\text {maximize}&\mathrm{{\Phi }}({{\mathcal {T}}}, {{\mathcal {F}}})+ \mathrm{{\Sigma }}({{\mathcal {S}}}, {{\mathcal {E}}})&\end{aligned}$$

(30)

$$\begin{aligned}&\text {subject to}&{\mathbf {t}}(k) - {\mathbf {s}}(k) + {\mathbf {q}}(k) \in \ {\mathcal {P}}({\mathbf {c}}-{\mathbf {f}}(k)),&k=0,\dots ,N-1 \end{aligned}$$

(31)

$$\begin{aligned}&{\mathbf {q}}_{i} \in \ {\mathcal {U}}({\mathbf {b}}_i-{\mathbf {e}}_i),&i =1,\dots ,n, \end{aligned}$$

(32)

where the objective function is given by

$$\begin{aligned} \mathrm{{\Phi }}({{\mathcal {T}}}, {{\mathcal {F}}})+ \mathrm{{\Sigma }}({{\mathcal {S}}}, {{\mathcal {E}}})&= \sum _{k=0}^{N-1} \left( -\varvec{{\lambda }}(k)^{\top } {\mathbf {t}}(k) + \varvec{{\mu }}(k)^{\top } {\mathbf {f}}(k) \right) \ + \ \sum _{i=1}^n \left( \varvec{{\lambda }}_i^{\top } {\mathbf {s}}_i + \overline{\varvec{{\nu }}}_i^{\top } {\mathbf {e}}_i \right) . \end{aligned}$$

This is a convex optimization problem with linear constraints in the decision variables ${\mathbf {t}}(k), {\mathbf {f}}(k), {\mathbf {q}}(k), {\mathbf {s}}(k)$ ($k=0, \dots , N-1$) and ${\mathbf {e}}_i$ ($i =1, \dots , n$). Using arguments identical to those employed in the proof of Lemma 2, it is straightforward to verify that an optimal solution to problem (30)–(32) is given by

$$\begin{aligned} {\mathbf {t}}(k)={\mathbf {v}}(k)+{\mathbf {u}}(k), \quad {\mathbf {f}}(k)={\mathbf {0}}, \quad {\mathbf {q}}(k)={\mathbf {u}}(k), \quad {\mathbf {s}}(k)={\mathbf {u}}(k), \quad \text {and} \quad {\mathbf {e}}_{i}={\mathbf {0}} \end{aligned}$$

for all $k= 0, \dots , N-1$ and $i = 1, \dots , n$. This optimal solution yields a collection of transmission and storage rights with an associated rent of

$$\begin{aligned} \mathrm{{\Phi }}({{\mathcal {T}}}, {{\mathcal {F}}})+ \mathrm{{\Sigma }}({{\mathcal {S}}}, {{\mathcal {E}}})= -\sum _{k=0}^{N-1} \varvec{{\lambda }}(k)^{\top } {\mathbf {v}}(k). \end{aligned}$$

According to (14), this equals the $\mathrm{MS}$ associated with $(V,U,{\varLambda })$, thus completing the proof.