Power Markets with Information-Aware Self-scheduling Electric Vehicles

Abstract

We consider multi-period (24-h day-ahead) multi-commodity (energy and regulation reserves) decentralized electricity transmission and distribution (T&D) market designs. Whereas conventional centralized generators with uniform price quantity offers are scheduled by a transmission system operator, low-voltage network-connected distributed energy resources (DERs) with complex preferences and requirements, such as electric vehicles (EVs), are allowed to self-schedule adapting to spatiotemporal marginal cost-based prices. We model the salient characteristics of interconnected T&D networks, and we consider self-scheduling DER responses under alternative distribution network information-aware or information-unaware market designs. Moreover, we consider a single (EV load aggregator) network information-aware scheduler market design. Our contribution is the characterization and comparative analysis—analytic as well as numerical—of equilibria, using game-theoretical approaches to prove existence and uniqueness, and the investigation of the role of information on self-scheduling and EV aggregator coordinated EV scheduling. Finally, we derive conclusions on the impact to social welfare and distributional equity of information-aware/information-unaware self-scheduling as well as single EV aggregator scheduling and implications that are relevant to market design and policy considerations.

A Appendix

1.1 A.1 Proof of Proposition 1 (Key Impact of Decentralized Designs on Scheduling Decisions)

For the purposes of the proof, we reduce the multi-commodity (energy and reserves) decisions to a single-commodity decision. To this end, we note that reserve provision $q_{j,t}^R$ either increases with energy consumption $q_{j,t}^P$ if constraint (10) is not binding, or decreases if constraint (11) is not binding. If the former is true, then (i) $\nu _{j,t}^1=0$ and, (ii) $\frac{\partial q_{j,t}^R}{\partial q_{j,t}^P}=1$. In that case, rearranging the optimality conditions w.r.t. $q_{j,t}^P$ and $q_{j,t'}^P$ yields:

• $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$: $\lambda _{n,t}^P-\lambda _{n,t}^R+2\delta q_{j,t}^P=\lambda _{n,t'}^P-\lambda _{n,t'}^R+2\delta q_{j,t'}^P,$

• $\mathbf{EV^{ind}_A}$: $(\lambda _{t}^P-\lambda _t^R)[\gamma _n q_{j,t}^P+1+m_{n,t}(q_{-j,t}^P,q_{j,t}^P)] +2\delta q_{j,t}^P = = (\lambda _{t'}^P-\lambda _{t'}^R)[\gamma _n q_{j,t'}^P+1+m_{n,t'}(q_{-j,t'}^P,q_{j,t'}^P)] +2\delta q_{j,t'}^P,$

• $\mathbf{EV^{agg}_A}$: $( \lambda _{t}^P-\lambda _t^R)\left[ \gamma _n \sum _j q_{j,t}^P+1+m_{n,t}(\varSigma q_{j,t}^P) \right] +2\delta q_{j,t}^P = = ( \lambda _{t'}^P-\lambda _{t'}^R ) \left[ \gamma _n \sum _j q_{j,t'}^P +\right. \left. 1+m_{n,t'}(\varSigma q_{j,t'}^P) \right] +2\delta q_{j,t'}^P$.

If, on the other hand, reserve provision decreases with energy consumption, then (i) $\nu _{j,t}^2=0$ and (ii) $\frac{\partial q_{j,t}^R}{\partial q_{j,t}^P}=-1$. Rearranging the optimality conditions w.r.t. $q_{j,t}^P$ and $q_{j,t'}^P$ yields:

• $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$: $\lambda _{n,t}^P+\lambda _{n,t}^R+2\delta q_{j,t}^P=\lambda _{n,t'}^P+\lambda _{n,t'}^R+2\delta q_{j,t'}^P$,

• $\mathbf{EV^{ind}_A}$: $(\lambda _{t}^P+\lambda _t^R)\left[ \gamma _n q_{j,t}^P+1+m_{n,t}(q_{-j,t}^P,q_{j,t}^P)\right] +2\delta q_{j,t}^P= = (\lambda _{t'}^P+\lambda _{t'}^R)\left[ \gamma _n q_{j,t'}^P+1+\right. \left. m_{n,t'} (q_{-j,t'}^P,q_{j,t'}^P)\right] +2\delta q_{j,t'}^P$,

• $\mathbf{EV^{agg}_A}$: $(\lambda _{t}^P+\lambda _t^R)\left[ \gamma _n \sum _j q_{j,t}^P+1+m_{n,t}(\varSigma q_{j,t}^P)\right] +2\delta q_{j,t}^P= = (\lambda _{t'}^P+\lambda _{t'}^R)\left[ \gamma _n \sum _j q_{j,t'}^P+\right. \left. 1+m_{n,t'}(\varSigma q_{j,t'}^P)\right] +2\delta q_{j,t'}^P$.

The equality of marginal cost reduction in hour t to the marginal cost increase in hour $t'$ is thus shown and quantified by expressing $q_{j,t}^R$ in terms of $q_{j,t}^P$.

1.2 A.2 Proof of Lemma 1

• (i) The result follows directly from the optimality conditions (23). Since the equality is not satisfied when $\nu _{j,t}^1=\nu _{j,t}^2=0$, at least one of these dual variables must be nonzero, assuming that $\lambda _{t}^R>0, \forall t$. Note that we do not consider the unlikely event that $\lambda _{t}^R=0$.

• (ii) The result follows from (24) for $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$, from (25) for $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ and from (26) for $\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}$. Note that $\underline{\zeta }_j$ is positive when $\varDelta \lambda _{t}=\lambda _{t}^P-\lambda _{t}^R>0$. Therefore, the minimum charging demand constraint in (8) is binding. That is, $\sum _j q_{j,t}^P=\underline{s}_j$.

• (iii) For $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$, the result follows from (23) and (25). Note that when $q_{j,t}^R<q_{j,t}^P$, constraint (11) is not binding and $\nu _{j,t}^2=0$. Then, combining (23) and (25), we obtain: $\underline{\zeta }_{j}= \lambda _{t}^P(1+m_{n,t})+ \lambda _{t}^R(1+m_{n,t}) +\gamma _n(\lambda _{}^Pq_{j,t}^P-\lambda _{t}^Rq_{j,t}^R)+2\delta q_{j,t}^P+\overline{\zeta }_j$. Therefore, $\underline{\zeta }_{j}>0$ and constraint (8) is binding. The positivity of $\underline{\zeta }_{j}$ in the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ and $\mathbf{EV^{agg}_A}$ market designs is shown similarly.

• (iv) For $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$, the result follows from (23) and (25). When $\underline{\zeta }_{j}=\nu _{j,t}^2=0$, we obtain $\overline{\zeta }_j=-\varDelta \lambda _{t}+\gamma _n(\lambda _{t}^Rq_{j,t}^R-\lambda _{t}^Pq_{j,t}^P)-2\delta q_{j,t}^P$. Since a non-binding constraint (8) implies $\lambda _{t}^R-\lambda _{t}^P>0$, we conclude that $\overline{\zeta }_j>0$ and constraint (9) is binding. The positivity of $\overline{\zeta }_j$ in the scheduling problems solved in Step 2 of Algorithms 2 ($\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$) and 4 ($\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}$) can be shown similarly.

1.3 A.3 Proof of Theorem 2 (Nash Equilibrium Uniqueness)

Theorem 2 employs the potential function approach to show uniqueness of an equilibrium. A game in strategic form is called a potential game if the change in the payoff of a player due to change in strategy is equal to the change in the global potential function due to unilateral change of strategy of the player. That is, a game where each player i has payoff function $f_i(q_i,q_{-i})$, is a potential game if for every $q_i \in S$, $f_i(q_i,q_{-i})-f_i(q_i',q_{-i})=\overline{P}(q_i)-\overline{P}(q_i'), \ \forall i, q_i$. The function $\overline{P}$ that couples all players is called the potential function. Constructing it is key to invoke Theorem 2. The relation between the Nash equilibrium condition and the optimization problem with objective function replaced by $\overline{P}$ is based on [38, Definition 1]. The conditions under which the Nash equilibrium is unique are given by [26, Theorem 2] and restated in [8]. In fact (see [8]), the diagonal strict convexity condition for equilibrium uniqueness given by [32, Theorem 2] is satisfied when the potential function is strictly convex. Lastly, it is stated in [21] that there is no clear economic interpretation of the potential function, in other words, the function that players jointly maximize.

1.4 A.4 Proof of Lemma 2 (Single-Commodity, High Charging Flexibility)

We prove the result by contradiction for the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$, $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ and, $\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}$ market designs and illustrate it for $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$. Assume the opposite is true, i.e., $\exists t$ s.t. $q_{j,t}^P>\frac{\overline{q}_j}{2}$. By Assumption 2 and Lemma 1(ii), $\sum _t q_{j,t}^P=\underline{s}_j$. Then, given (27), there should exist some hour $t^{\prime }{}$ s.t. $q_{j,t^{\prime }{}}^P < \frac{\overline{q}_j}{2}$. Comparing the expressions of $\underline{\zeta }_j$ obtained from the first-order optimality conditions associated with $q_{j,t}^P$ and $q_{j,t^{\prime }{}}^P$, we get: $\underline{\zeta }_j= (\lambda _{t}^P+\lambda _{t}^R)(1+m_{n,t}) +\gamma _n(\lambda _{t}^Pq_{j,t}^P-\lambda _{t}^Rq_{j,t}^R)+2\delta q_{j,t}^P+ \overline{\zeta }_j$, and $\underline{\zeta }_j=\varDelta \lambda _{t'}(1+m_{n,t'}) +\gamma _n(\lambda _{t^{\prime }{}}^Pq_{j,t^{\prime }{}}^P-\lambda _{t^{\prime }{}}^Rq_{j,t^{\prime }{}}^R)+2\delta q_{j,t'}^P+\overline{\zeta }_j$. Note that the terms $\gamma _n(\lambda _{t}^Pq_{j,t}^P-\lambda _{t}^Rq_{j,t}^R)$ include energy and reserve decisions of EV j only. Hence, these quantities, including the degradation term, are small enough, implying that the two expressions for $\underline{\zeta }_j$ cannot be identical since $\lambda _{n,t}^P+\lambda _{n,t}^R>\varDelta \lambda _{n,t'}$ from Assumption 1. Therefore, the optimal solution of the EV scheduling problem satisfies $q_{j,t}^P\le \frac{\overline{q}_j}{2}$, $\forall t$.

1.5 A.5 Proof of Proposition 2 (Existence and Uniqueness, High Charging Flexibility)

Existence of equilibrium for the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ and $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ market designs follows from Lemma A1 and Theorem 1.

Lemma A1

Consider the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ and $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ market designs. The feasible set $S_j^{HF}$ of EV j in the EV self-scheduling problems, under High Charging Flexibility, is closed, bounded and convex. In addition, given Assumption 2, the cost functions ${f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ and ${{f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}}$ of EV j are both strictly convex and continuous in $q_{j}^P$.

Proof

Due to the positivity of $\varDelta \lambda _{t}$ from Assumption 2, $f_j^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}(q_{j}^{P},q_{-j}^{P})$ and $f_j^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}(q_{j}^{P})$ given in (31) and (32) are strictly convex in $q_{j}^P$ and continuous in both $q_{j}^P$ and $q_{-j}^P$. In addition, the feasible set $S_j^{HF}$ of EV j is closed and bounded. Without the inter-temporal minimum charging demand constraint (8), the feasible set of EV j is the box constraint $q_{j,t}^P \in [0,\frac{\overline{q}_j}{2}]$. Therefore, the feasible set of EV j without constraint (8), denoted by $\overline{S}_j$, is given by $\overline{S}_j=\overline{S}_j^1 \times \cdots \times \overline{S}_j^t \times \cdots \times \overline{S}_j^T$ where $\overline{S}_j^t=[0,\frac{\overline{q}_j}{2}]$ and T is the number of periods EVs are plugged in. Therefore, $\overline{S}_j$ is closed and bounded. The feasible set in the presence of constraint (8), $S_j^{HF}$, is then a closed subset of $\overline{S}_j$ and hence also bounded. $S_j^{HF}$ is convex as well since the constraints are linear. $\square$

Consider the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ market design. The potential function $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ for the game with information-aware EVs, where EV j has the cost function ${f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ in (31) and strategy set $S_j^{HF}$, is given by:

$$\begin{aligned} \overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}= \sum \limits _{j,t}{\varDelta \lambda _{t}\left[ 1+m_{n,t}(\varSigma q_{j,t}^P)\right] q_{j,t}^{P}}+\delta (q_{j,t}^P)^2-\sum \limits _{j,j'|j'>j,t}{\varDelta {{\lambda }_{t}}\gamma _n q_{j,t}^{P}q_{j',t}^{P}}. \end{aligned}$$

(A.1)

In addition, $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ is strictly convex in $q_{j,t}^P, \forall j$. The $J\times J$ Hessian matrix of the potential function $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}(q_t^P)$ in (A.1) for a given t has the following form:

$$\begin{aligned} H_t=\begin{bmatrix} 2\varDelta \lambda _t \gamma _n +2\delta &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \dots &{}\quad \varDelta \lambda _t \gamma _n \\ \varDelta \lambda _t \gamma _n &{}\quad 2\varDelta \lambda _t \gamma _n +2\delta \, &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \dots &{}\quad \varDelta \lambda _t \gamma _n \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \varDelta \lambda _t \gamma _n &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \dots &{}\quad 2\varDelta \lambda _t \gamma _n +2\delta \end{bmatrix}, \end{aligned}$$

which can be written as the sum of a positive semi-definite matrix and the identity matrix multiplied by a positive scalar. Since the identity matrix is positive definite, and the sum of a positive semi-definite and positive definite matrix is also positive definite, $H_t$ is positive definite, and hence, the potential function (A.1) is strictly convex, which implies uniqueness. We then proceed to show that the equilibrium conditions are identical to the optimality conditions of the equilibrium recovery problem (Nash equilibrium is recovered by minimizing the potential function $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ subject to $S_j^{HF}, \forall j$). The first-order optimality condition of the EV self-scheduling problem (Step 2 of Algorithm 3) w.r.t. $q_{j,t}^P$ is written as:

$$\begin{aligned} \varDelta \lambda _{t}(1+m_{n,t})+\varDelta {{\lambda }_{t}}\gamma _n q_{j,t}^{P}+2\delta q_{j,t}^P-{{\underline{\zeta } }_{j}}+{{\nu }_{j,t}^1}=0. \end{aligned}$$

(A.2)

The optimality condition w.r.t. $q_{j,t}^P$ of the problem with the objective function replaced by the potential function $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ given in (A.1) is:

$$\begin{aligned} \varDelta {{\lambda }_{t}}(1+m_{n,t})+\varDelta {{\lambda }_{t}}\gamma q_{j,t}^{P}+\varDelta {{\lambda }_{t}}\gamma q_{-j,t}^{P} +2\delta q_{j,t}^P-\varDelta {{\lambda }_{t}}\gamma q_{-j,t}^{P}-{{\underline{\zeta }}_{j}}+{{\nu }_{j,t}^1}=0. \end{aligned}$$

(A.3)

By inspection, optimality conditions (A.2) and (A.3) are identical. Hence, by Theorem 2, the Nash equilibrium of the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ market design, under High Charging Flexibility, is unique.

Consider the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ market design. The potential function $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}$ for the game with information-unaware EVs, where EV j has the cost function ${f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}$ in (32) and strategy set $S_j^{HF}$, is given by:

$$\begin{aligned} \overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}&=\sum \limits _{j,t}{\varDelta \lambda _{t} \left[ 1+m_{n,t}(\varSigma q_{j,t}^P) \right] q_{j,t}^{P}}+\delta (q_{j,t}^P)^2\nonumber \\&\quad -\sum \limits _{j,j'|j'>j,t}{\varDelta {{\lambda }_{t}}\gamma _n q_{j,t}^{P}q_{j',t}^{P}} -\frac{1}{2}\sum \limits _{j,t} \varDelta \lambda _t \gamma _n (q_{j,t}^P)^2. \end{aligned}$$

(A.4)

In addition, $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}$ is strictly convex in $q_{j,t}^P, \forall j$, and the Hessian of a single hour t of $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}$ in (A.4) is given by:

$$\begin{aligned} H_t=\begin{bmatrix} \varDelta \lambda _t \gamma _n +2\delta &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \dots &{}\quad \varDelta \lambda _t \gamma _n \\ \varDelta \lambda _t \gamma _n &{}\quad \varDelta \lambda _t \gamma _n +2\delta \, &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \dots &{}\quad \varDelta \lambda _t \gamma _n \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \varDelta \lambda _t \gamma _n &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \varDelta \lambda _t \gamma _n &{}\quad \dots &{}\quad \varDelta \lambda _t \gamma _n +2\delta \end{bmatrix}, \end{aligned}$$

where $\varDelta \lambda _t=\lambda _t^P-\lambda _R^t$. Given Assumption 2, the Hessian is positive definite. We can also show that the equilibrium conditions and optimality conditions of the equilibrium recovery problem where $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}$ is minimized over $S_j^{HF}, \ \forall j$ are identical. Hence, by Theorem 2, the Nash equilibrium of the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ market design, under High Charging Flexibility, is unique.

For the $\mathbf{EV^{agg}_{A}}$ market design, we note that the objective function of the load aggregator, $f^{\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}}(q_t^P)$, given by (33), under High Charging Flexibility, is strictly convex subject to linear constraints. Therefore, the optimal solution is unique.

We note that diagonal strict convexity of EV cost functions ${f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ and ${{f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}}$ can also be shown for both information-aware and information-unaware market designs, which is an equilibrium uniqueness condition provided by [32, Theorem 2].

1.6 A.6 Proof of Lemma 3 (Single-Commodity, Moderate/Low Charging Flexibility)

We illustrate the proof for the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ market design; the proofs for $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ and $\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}$ are similar. Assume the opposite is true, i.e., $\exists t$ s.t. $q_{j,t}^P<\frac{\overline{q}_j}{2}$. By the definition of Moderate Charging Flexibility, there should exist another hour $t^{\prime }{}$ s.t. $q_{j,t^{\prime }{}}^P>\frac{\overline{q}_j}{2}$. We then use the same contradiction argument as in Subsection A.4 in the opposite direction, and we show that the two expressions of $\underline{\zeta }_j$ cannot be identical. Therefore, $q_{j,t}^P\ge \frac{\overline{q}_j}{2}$, $\forall t$. Under Low Charging Flexibility, due to (29), $q_{j,t}^P>\frac{\overline{q}_j}{2}$, $\forall t$, regardless of Assumption 1 or 2.

1.7 A.7 Proof of Proposition 3 (Existence and Uniqueness, Moderate/Low Charging Flexibility)

Equilibrium existence for the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ and $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ market designs follows from Lemma A2 and Theorem 1.

Lemma A2

Consider the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ and $\mathbf{EV^{ind}_{A}}$ market designs. The feasible set $S_j^{MLF}$ of EV j in the EV self-scheduling problems, under Moderate or Low Charging Flexibility, is closed, bounded and convex. In addition, the cost functions ${f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ and ${{f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}}$ of EV j are both strictly convex and continuous in $q_{j}^P$.

Proof

Since $\lambda _t^P+\lambda _t^R>0$, ${f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ and ${{f}_{j}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}}$ are both convex. The convexity, closedness and boundedness of $S_j^{MLF}$ are shown similarly to the proof of Lemma A1. In this case, we define $\overline{S}_j^t$ as $\overline{S}_j^t=[\frac{\overline{q}_j}{2},\overline{q}_j]$. $\square$

Consider the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ market design. The potential function $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$ for the game with information-aware EVs, where EV j solves (34) subject to $S_j^{MLF}$ is:

$$\begin{aligned} \overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}= & {} \,\,\,\sum \limits _{j,t}( 1+m_{n,t}(\varSigma q_{j,t}^P))({{\lambda }_{t}}^P q_{j,t}^{P}-\lambda _{t}^R(\overline{q}_j-q_{j,t}^P)+\delta (q_{j,t}^P)^2, \nonumber \\&-\sum \limits _{j,j'|j'>j,t}({{\lambda }_{t}^P}+\lambda _{t}^R)\gamma q_{j,t}^{P}q_{j',t}^{P}+\sum _{j,j'|j'\ne j,t}\lambda _{t}^R\gamma q_{j,t}^P \overline{q}_{j'}. \end{aligned}$$

(A.5)

In addition, $\overline{P}$ is strictly convex in $q_{j,t}^P, \forall j$. The Hessian of the potential function in (A.5) has the following form:

$$\begin{aligned} H_t=\begin{bmatrix} 2\varSigma \lambda _t \gamma _n +2\delta &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \dots &{}\quad \varSigma \lambda _t \gamma _n \\ \varSigma \lambda _t \gamma _n &{}\quad 2\varSigma \lambda _t \gamma _n +2\delta \, &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \dots &{}\quad \varSigma \lambda _t \gamma _n \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \varSigma \lambda _t \gamma _n &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \dots &{}\quad 2\varSigma \lambda _t \gamma _n +2\delta \end{bmatrix}, \end{aligned}$$

where $\varSigma \lambda _{t}=\lambda _{t}^P + \lambda _{t}^R$. Due to positivity of $\varSigma \lambda _{t}$, the above Hessian is positive definite and the objective function is strictly convex. Uniqueness of Nash equilibrium is then shown by identical optimality conditions. The optimality condition of the EV self-scheduling problem (Step 2 of Algorithm 3) w.r.t. $q_{j,t}^P$ is given by:

$$\begin{aligned}&(\lambda _{t}^P+\lambda _{t}^R)(1+m_{n,t}+\gamma q_{j,t}^P)-\gamma \lambda _{t}^R\overline{q}_{j}+2\delta q_{j,t}^P-\underline{\zeta }_j +\nu _{j,t}^1=0. \end{aligned}$$

(A.6)

One can show that the optimality condition of the potential function $\overline{P}$ in (A.5) w.r.t. $q_{j,t}^P$ is identical to (A.6).

Consider the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$ market design. The potential function $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}$ for the game with information-unaware EVs, where EV j minimizes the cost function in (35) is given by:

$$\begin{aligned}&\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}=\sum \limits _{j,t}( 1+m_{n,t})({{\lambda }_{t}}^P q_{j,t}^{P}-\lambda _{t}^R(\overline{q}_j-q_{j,t}^P)+\delta (q_{j,t}^P)^2 \nonumber \\&\quad -\sum \limits _{j,j'|j'>j,t}({{\lambda }_{t}^P}+\lambda _{t}^R)\gamma q_{j,t}^{P}q_{j',t}^{P}+\sum _{j,j',t}\lambda _{t}^R\gamma q_{j,t}^P \overline{q}_{j'}-\frac{1}{2} \sum \limits _{j,t} (\lambda _t^P+\lambda _t^R) \gamma (q_{j,t}^P)^2. \end{aligned}$$

(A.7)

and strictly convex. The Hessian of the potential function in (A.7) has the following form:

$$\begin{aligned} H_t=\begin{bmatrix} \varSigma \lambda _t \gamma _n +2\delta &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \dots &{}\quad \varSigma \lambda _t \gamma _n \\ \varSigma \lambda _t \gamma _n &{}\quad \varSigma \lambda _t \gamma _n +2\delta \, &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \dots &{}\quad \varSigma \lambda _t \gamma _n \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \varSigma \lambda _t \gamma _n &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \varSigma \lambda _t \gamma _n &{}\quad \dots &{}\quad \varSigma \lambda _t \gamma _n +2\delta \end{bmatrix}, \end{aligned}$$

where $\varSigma \lambda _t=\lambda _t^P+\lambda _R^t$, and is positive definite. In addition, it is straightforward to show that the optimality conditions of the EV self-scheduling problem (Step 2 of Algorithm 2) and the optimality conditions of the problem where $\overline{P}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}$ is minimized over $S_j^{MLF}$ are identical. Hence, uniqueness follows from Theorem 2.

For the $\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}$ market design, we note that the objective function of the load aggregator, $f^{\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}}(q_t^P)$, given by (36), under Moderate or Low Charging Flexibility, is strictly convex subject to linear constraints. Therefore, the optimal solution is unique.

1.8 A.8 Proof of Proposition 4 (Stability of N.E. Under Information Awareness)

In the 2-h model, the cost function of information-aware EV j, $f_j^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}(q_{j,1}^P)$, given in (31), satisfies the following condition for $J\le 3$:

$$\begin{aligned} \sum \limits _{j|j\ne j'} \left| \frac{\partial ^2 f_{j'}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}}{\partial q_{j',1}^P \partial q_{j,1}^P}(q^P)\right| < \left| \frac{\partial ^2 f_{j'}^{\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}}{\partial ^2 q_{j',1}^P}(q^P)\right| , \ \ \ \forall j'\, , q_{j',1}^P \in [0,\underline{s}_j]. \end{aligned}$$

(A.8)

Therefore, by [23, Theorem 4], the Nash equilibrium in the $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ market design is stable.

1.9 A.9 Proof of Proposition 6 (Closed-Form Equilibria Expressions for Simplified 24-h Problem)

We provide the proofs for $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ and $\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}$ market designs; the proof is similar for $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}$. The first-order optimality conditions of $\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}$ market design in (33) subject to only the minimum charging demand constraint (8) with J identical EVs can be written as:

$$\begin{aligned}&q_{j,t}^{P,\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}}=\text {max}\left\{ 0,\frac{\underline{\zeta }-\varDelta \lambda _t(1+\gamma d_t^{f})}{2J\gamma \varDelta \lambda _t+2\delta }\right\} , \end{aligned}$$

(A.9)

$$\begin{aligned}&\left( -\sum \limits _t q_{j,t}^P+\underline{s}\right) \underline{\zeta }=0. \end{aligned}$$

(A.10)

Since EVs are identical, $\underline{\zeta }_j=\underline{\zeta }$, $\underline{s}_j=\underline{s}$, and $q_{j,t}^{P}=q_t^P, \, \forall {j}$. Combining (A.9) and (A.10), we can write: $\underline{\zeta }= \left[ \sum \limits _{t'}\frac{\varDelta \lambda _{t'}(1+\gamma d_t^{f})}{2J\gamma \varDelta \lambda _{t'} +2\delta }+\underline{s}\right] / \left( \sum \limits _{t'} \frac{1}{2J\gamma \varDelta \lambda _{t'} +2\delta }\right) ,$ where $t'=\{t|q_{j,t}^{P}>0\}$, and substituting in (A.9), we obtain:

$$\begin{aligned} q_{j,t}^{P,\mathbf{EV}^{\mathbf{agg}}_{\mathbf{A}}}=\frac{\sum \limits _{t'}\frac{\varDelta \lambda _{t'}(1+\gamma d_{t'}^{f})}{2J\gamma \varDelta \lambda _{t'} +2\delta }+\underline{s}}{\sum \limits _{t'} \frac{2J\gamma \varDelta \lambda _t+2\delta }{2J\gamma \varDelta \lambda _{t'} +2\delta }}-\frac{\varDelta \lambda _t(1+\gamma d_t^{f})}{2J\gamma \varDelta \lambda _t+2\delta }. \end{aligned}$$

(A.11)

Note that (A.11) matches (41) if EVs are connected only for 2 h.

The first-order optimality conditions of $\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}$ in (31) subject to only (8) are: $q_{j,t}^{P,\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}} = \text {max}\left\{ 0,\frac{\underline{\zeta }-\varDelta \lambda _t(1+\gamma d_{t'}^{f})}{(J+1)\gamma \varDelta \lambda _t+2\delta }\right\}$, and $(-\sum \limits _t q_{j,t}^P+\underline{s})\underline{\zeta }=0$, yielding:

$$\begin{aligned} q_{j,t}^{P,\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}=\frac{\sum \limits _{t'}\frac{\varDelta \lambda _{t'}(1+\gamma d_{t'}^{f})}{(J+1)\gamma \varDelta \lambda _{t'} +2\delta }+\underline{s}}{\sum \limits _{t'} \frac{(J+1)\gamma \varDelta \lambda _t+2\delta }{(J+1)\gamma \varDelta \lambda _{t'} +2\delta }}-\frac{\varDelta \lambda _t(1+\gamma d_t^{f})}{(J+1)\gamma \varDelta \lambda _t+2\delta }. \end{aligned}$$

1.10 A.10 Proof of Proposition 7 (Asymptotic Coincidence of Individual Self-scheduling Information-Aware and Information-Unaware EVs)

We illustrate this on the 2-h equilibria given by (42) and (43). As $J \rightarrow \infty$, $\frac{J+1}{J} \rightarrow 1$ rendering $q_{j,1}^{P,\mathbf{EV}^{\mathbf{ind}}_{\mathbf{Un}}}=q_{j,1}^{P,\mathbf{EV}^{\mathbf{ind}}_{\mathbf{A}}}$.