Decentralized Charging Coordination with Battery Degradation Cost

Abstract

In Chap. 2, the decentralized charging method has been designed to effectively coordinate the charging behaviors of large-scale PEVs, like the valley-filling strategy, to minimize their impacts on the power grid. However high charging rates under the valley-filling strategy may result in high battery degradation cost. Consequently in this chapter, it formulates a class of PEV charging coordination problems which deals with the tradeoff between total generation cost and the accumulated battery degradation cost of PEV populations. Due to the autonomy of individual PEVs and the computational complexity of the system with large-scale PEVs, it is impractical to implement the solution in a centralized way. Alternatively in this part a decentralized method is proposed such that all of the individual PEVs simultaneously update their own best charging behaviors with respect to a common electricity price curve, which is updated as the generation marginal cost with respect to the aggregated charging behaviors of the PEV populations implemented at last step. The iteration procedure terminates in case the price curve does not update any longer. It has been shown that, by applying the proposed decentralized method and under certain mild conditions, the system can converge to a unique charging strategy which is nearly socially optimal or efficient. Simulation examples are studied to illustrate the results developed in this chapter.

3.1 Introduction

It has been widely studied, e.g., [1, 2], that some key characteristics of chemical batteries, like the state of health, the growth of resistance, the cycle life, etc., are effected by the charging behaviors. As a consequence, this chapter addresses the need for a charging coordination scheme which considers the tradeoffs between system-wide economic efficiency, distribution-level limitations and battery degradation concerns.

Charging behavior affects key battery characteristics, including the state of health, the resistance impedance growth and the cycle life, which are all strongly related to the energy capacity of a battery [1, 2]. Intermittent charging may also shorten the battery lifespan [3]. Optimal charging strategies that take into account both the total energy cost and the battery state of health have been studied for single PEVs [1, 4]. These ideas form the basis for the extension, undertaken in this chapter, to large-scale coordination. It analyzes the battery degradation cost of the $\text {LiFePO}_4$ battery with respect to the charging power.

In this part, it analyzes the battery degradation cost of the $\text {LiFePO}_4$ battery with respect to the charging power. The system considers the tradeoff between the total generation cost and the local costs associated with overloading and battery degradation. Furthermore, consider the charging flexibility of PEVs. PEVs can independently determine their total charging capacity according to their own characteristics and the environment in which they are charged. The optimal charging strategies are analyzed as the parameter varies. Some research work, e.g., [1, 4], studied the optimal charging behavior for a single PEV by taking into account both total the energy cost and the state of health of batteries.

Decentralized charging coordination methods are proposed for large-scale PEVs dealing with the total generation cost and the accumulated battery degradation cost over multi-time intervals. As the $\text {LiFePO}_4$ battery, a lithium-ion type battery, has been widely applied in the PEV market, e.g., GM Chevrolet Spark, Nissan Leaf, BYD (e6, F3DM, F6DM), Renault Clio, etc., the health model for $\text {LiFePO}_4$ cell units was specified in [5] based upon the evolution of battery cell characteristics developed in [6, 7]. By adopting the analysis given in [5], it analyzes the battery degradation cost for the $\text {LiFePO}_4$ battery with respect to the charging power.

In general it is challenging to achieve an optimal or near-optimal outcome by coordinating large-scale PEVs in a decentralized way. As discussed in [8], it is difficult to effectively fill the nigh-time valley based upon the time-based or fixed price; then in this chapter it will adopt a real-time price model which has been widely applied in the literature, e.g., [9, 10] for demand response management, and [11,12,13,14] for electric vehicle charging coordination. More specifically, the electricity price at an instant is determined by the total demand at that instant and represents the generation marginal cost.

In the decentralized approach to charging coordination proposed in this part, participating PEVs simultaneously determine their optimal charging strategy with respect to an energy price forecast. These proposed charging strategies are used to estimate the total demand over the charging horizon. An updated price forecast is obtained as a weighted average of the previous price forecast and the generation marginal cost evaluated at this latest demand forecast. The revised price is (re)broadcast to the PEVs, and the process repeats. This scheme is formalized in Sects. 3.3 and 3.4, for different energy demand requirements, where it is shown that convergence is guaranteed under mild conditions. Upon convergence, the price profile is coincident with the generation marginal cost over the charging horizon. As a consequence, the resulting collection of PEV charging strategies is efficient (socially optimal). Moreover, convergence is obtained without the need for artificial deviation costs to damp oscillations, as in [15, 16]. Cost terms introduced to mitigate the effects of local demand peaks and battery degradation play the same role as congestion pricing used for traffic control in communication networks [17], which has been adopted in [13] to schedule PEV charging.

It is worth to note that, in the decentralized framework proposed in this part, each of the individual PEVs deals with the tradeoff between the battery degradation cost and the electricity cost; then the system can improve the longevity of the batteries and may mitigate the oscillation caused by the greedy behavior for the cheap electricity resource by individual PEVs. As a result, the deviation cost artificially introduced to mitigate the oscillation in [15, 16] is no longer required. The battery degradation cost plays the same role as the congestion pricing which has been widely used in Internet traffic control, see [17], and has been adopted in [13] to schedule PEV charging strategies in the power grid.

The chapter is organized as follows. In Sect. 3.2, it formulates a class of charging coordination problems which deals with the tradeoff between total electricity cost and the accumulated battery degradation cost of all of the PEVs over the charging intervals. A decentralized charging coordination algorithm is presented in Sect. 3.3 and the convergence of the proposed method is analyzed. This method is updated and extended in Sect. 3.4 to the charging coordination problems with flexible energy demands. Simulations, given in Sect. 3.5, illustrate various characteristics of the algorithm. Section 3.6 concludes this chapter and discusses ongoing research.

3.2 Formulation of Charging Coordination with Battery Degradation Cost

3.2.1 Admissible Charging Strategies

In this section, it considers the charging coordination of a large population of PEVs with most of their specifications given in Sect. 2.2.1, except that, instead of the common charging horizon $\mathscr {T}$, the charging horizon of PEV n is $\mathscr {T}_n$ such that $\mathscr {T}_n \subset \mathscr {T}$.

Remark: The parameter $\mathscr {T}_n$ is determined by external factors such as driving style and vehicle type [18].

Hence, by (2.3), a charging strategy $\varvec{u}_n \equiv (u_{nt}; t \in \mathscr {T})$ is admissible if,

$$\begin{aligned}&u_{nt} {\left\{ \begin{array}{ll} \in [0, \varUpsilon ^+_n], &{} t \in \mathscr {T}_n \\ = 0, &{} t \in \mathscr {T}\setminus \mathscr {T}_n \end{array}\right. }, \end{aligned}$$

(3.1a)

$$\begin{aligned}&\Vert \varvec{u}_n\Vert _1 \equiv \sum _{t \in \mathscr {T}} u_{nt} = \frac{{\varGamma }_n}{\varsigma _n}(\mathrm {soc}_{n,\max } - {soc}_{n0}), \end{aligned}$$

(3.1b)

with $\varGamma _n$, $\varsigma _n$ and $\text {soc}_{n,\max }$ defined in Sect. 2.2.1.

Same as last chapter, the set of admissible charging strategies for PEV n, as specified in (3.1) above, is denoted by $\mathscr {U}_n$ as well.

Coordination of PEV charging across a large population has generally sought to minimize total generation cost over the charging horizon, see for example [15, 16, 19]. In contrast, the coordination strategies developed in this chapter seek to manage the tradeoff between total generation cost and local ones arising from high distribution-level demand and PEV battery degradation. These latter costs will now be discussed.

3.2.2 Analysis on Battery Degradation Cost

The $\text {LiFePO}_4$ battery, a lithium-ion type of battery, has been widely applied in PEVs, e.g., GM Chevrolet Spark, Nissan Leaf, BYD (e6, F3DM, F6DM), Renault Clio etc. Hence it will analyzes the battery degradation cost for the $\text {LiFePO}_4$ batteries with respect to the charging rates, based on the health model of $\text {LiFePO}_4$ cell units given in [5].

3.2.2.1 Evolution of Open Circuit Voltage of a Lithium-Ion Battery Cell w.r.t. Its SOC Value

It firstly reviews the relation between the open circuit voltage of a lithium-ion cell unit, denoted by $V_{cell}$, and its SOC value. As studied in the literature, e.g., [20, 21] and references therein, $V_{cell}$ evolves with respect to its SOC value, as displayed in Fig. 3.1.

Fig. 3.1

Evolution of the open circuit voltage of $\text {LiFePO}_4$ battery cell with respect to the SOC value

More specifically, as studied in [4, 22], as the SOC value of the cell unit varies from $0\%$ to a very small value, denoted by $soc_{min}$ with $soc_{min} > 0$, $V_{cell}$ increases rapidly from zero to its nominal value, denoted by $V_{norm}$, and stays near to $V_{norm}$ until the SOC of the cell unit varies beyond a high level of SOC value, denoted by $soc_{max}$.

As a summary, for the lithium-ion type of cell units, $V_{cell}$ approximately satisfies the following:

$$\begin{aligned} V_{cell} = V_{norm}, \quad \text {in case } SOC \in [soc_{min}, soc_{max}]. \end{aligned}$$

(3.2)

Hence following the above analysis and as considered in the literature, e.g., [15, 16, 23], in this chapter, it is supposed that $SOC \in [soc_{min}, soc_{max}]$, i.e., the battery is neither allowed to be over discharged to any SOC value below $soc_{min}$ nor allowed to be overcharged to any SOC value above $soc_{max}$.

3.2.2.2 Degradation Cost Modeling of $\text {LiFePO}_4$ Cell Units w.r.t. Charging Current and Voltage

In this work, it adopts the degradation modeling of the $\text {LiFePO}_4$ cell units, develops in [5], denoted by $h_{cell}(I,V)$, which measures the energy capacity loss per second (in $Amp \times Hour \times Sec^{-1}$) of a $\text {LiFePO}_4$ cell unit due to the resistance growth, see [1, 5], such that:

$$\begin{aligned} h_{cell} = {a}_1 + {a}_2 I + {a}_3 V + {a}_4 I^2 + {a}_5 V^2 + {a}_6 I V + {a}_7 V^3, \end{aligned}$$

(3.3)

with parameters ${a}_i$, with $i=1,\ldots ,7$, specified as $-1.148 \times 10^{-7}$, $3.9984 \times 10^{-8}$, $1.3158 \times 10^{-7}$, $5.5487 \times 10^{-10}$, $-4.968 \times 10^{-8}$, $-1.1166 \times 10^{-8}$, $6.1675 \times 10^{-9}$ respectively, see [5].

$V=V_{cell}$, i.e., the charging voltage V is nearly equal to the voltage of cell units; then by this together with (3.2), it can approximately set the charging voltage $V=V_{norm}$ in case $SOC \in [soc_{min}, soc_{max}]$. Moreover, by $u_{cell}=VI$ where $u_{cell}$ represents the charging power on a cell unit, the charging current I satisfies the following:

$$\begin{aligned} I(u_{cell}) = {u_{cell}}/{V_{norm}}, \text {in case } SOC \in [soc_{min}, soc_{max}]. \end{aligned}$$

Hence, based upon the above analysis, together with the battery degradation model specified in (3.3), it can obtain that, in case $SOC \in [soc_{min}, soc_{max}]$, the energy capacity degradation of a cell unit per second with respect to its charging power $u_{cell}$, denoted by $h_{cell}(u_{cell})$, is specified as:

$$\begin{aligned} h_{cell}(u_{cell})&= \mathfrak {a} u^2_{cell} + \mathfrak {b} u_{cell} + \mathfrak {c}, \end{aligned}$$

(3.4)

where $\mathfrak {a} = {{a}_4}/{V^2_{norm}}$, $\mathfrak {b} = {{a}_2}/{V_{norm}} + {a}_6$ and $\mathfrak {c} = {a}_1 + {a}_3 V_{norm} + {a}_5 V^2_{norm} + {a}_7 V^3_{norm}$.

Denote by $f_{cell}$ the degradation cost of a cell unit under a fixed charging power $u_{t,cell}$ over the charging interval t; then

$$\begin{aligned} f_{cell}(u_{t,cell})&= \, P_{cell} \cdot V_{norm} \cdot h_{cell}(u_{t,cell}) \cdot \triangle T, \end{aligned}$$

(3.5)

where $\triangle T$ denotes the length of charging interval t (in seconds), and $P_{cell}$ represents the price of a single energy unit of a battery cell.

Note: By (3.4) and (3.5), $f_{cell}(u_{cell})$ represents the monetary loss of a cell under a constant charging power $u_{cell}$ over t.

Suppose that the battery stored in each of the PEVs is composed of a collection of identical cell units, and for the analytical simplicity, it is also supposed that all of the cell units share the same SOC values and are charged with a common charging power. Denote by $\varPhi _n$ and $\varPhi _{cell}$ the aggregated battery size of PEV n and the energy size of the common cell units in PEV n respectively; then the number of cell units in PEV n, denoted by $M_n$, is valued as $M_n = {\varPhi _n}/{\varPhi _{cell}}$. Under the charging power $u_{nt}$ on the PEV n during interval t, the charging power on each of the cell units denoted by $u_{n,cell}$ is given as $u_{nt,cell} = 10^3 {u_{nt}}/{M_n}$ since $u_{nt}$ represents the power rate in kW.

Denote by $f_n(u_{nt})$ the battery degradation cost of PEV n during interval t under a charging power $u_{nt}$; then the following holds

$$\begin{aligned} f_n(u_{nt}) = \,&\mathfrak {a}_n u^2_{nt} + \mathfrak {b}_n u_{nt} + \mathfrak {c}_n \end{aligned}$$

(3.6)

with $\mathfrak {a}_n = \frac{10^6}{M_n} \cdot V_{norm} \cdot \triangle T \cdot P_{cell} \cdot \mathfrak {a}$, $\mathfrak {b}_n = 10^3 \cdot V_{norm} \cdot \triangle T \cdot P_{cell} \cdot \mathfrak {b}$, and $\mathfrak {c}_n = M_n \cdot V_{norm} \cdot \triangle T \cdot P_{cell} \cdot \mathfrak {c}$.

In [24], a quadratic form of $f_{n}(\cdot )$ is adopted as well.

Note: By (3.4) and (3.5), the degradation cost of the battery of PEV n, $f_n(u_{nt})$, as specified in (3.6), represents the total monetary losses of the battery package over interval t under the charging power $u_{nt}$.

For the purpose of demonstration, it is supposed that the battery package in the PEVs is composed of a collection of identical lithium battery cells, say ANR26650M1-B, from the A123 system, which has been widely applied in PEVs and grid stabilization energy storage systems. The nominal voltage and energy capacity of this type of cell units is 3.3 volts and 2.5 Ah (Amp $\times$ Hour) respectively, and the price of a single cell unit is about 15$.

As an example, it considers that the battery capacity in an individual PEV is 30 kWh; then it is straightforward to obtain that the battery is composed of about 3600 ANR26650M1-B units; then by (3.6), the degradation cost function, for the battery package specified above, is approximately given as below:

$$\begin{aligned} f_n(u_{nt}) = 0.004 u^2_{nt} + 0.075 u_{nt} + 0.003. \end{aligned}$$

(3.7)

Note: As discussed in Theorem 3.3, the best responses of individual PEVs are effected by the derivative of the degradation cost $f_n$, instead of $f_n$ itself, such that $f'_n(u_{nt}) = 2 \mathfrak {a}_n u_{nt} + \mathfrak {b}_n$ where the zero-order term $\mathfrak {c}_n$ is vanished. It implies that the small valued $\mathfrak {c}_n$ in (3.7) does not effect the results developed in this chapter.

3.3 Decentralized PEV Charging Coordination Method with Fixed Energy Demands

3.3.1 Charging Problems with Battery Degradation Costs

In this chapter, it is supposed that the system considers the tradeoff between the total electricity cost and the accumulated battery degradation cost formulated in Sect. 3.2.2 over the charging interval subject to the set of admissible charging strategies $\mathscr {U}$ of the PEV populations.

Denote, by ${J}(\varvec{u})$, the system cost under $\varvec{u}$ over the charging interval $\mathscr {T}$, such that

$$\begin{aligned} {J}(\varvec{u}) \triangleq \sum _{t \in \mathscr {T}} \left\{ \sum _{n \in \mathscr {N}} {f}_n({u}_{nt}) + \rho _t \cdot \left( d_t + \underset{n \in \mathscr {N}}{\sum } u_{nt} \right) \right\} , \end{aligned}$$

(3.8)

where $\rho _t$ represents the electricity price during interval t, and $d_t$ denotes the aggregated inelastic base demand during instant t in the power grid.

In this part, a real time price model is adopted, such that

$$\begin{aligned} \rho _t = p(D_t(\varvec{u}_t)), \quad \text { with } D_t(\varvec{u}_t) \triangleq d_t+ \underset{n \in \mathscr {N}}{\sum } u_{nt}, \end{aligned}$$

(3.9)

i.e., the electricity price at instant t is determined by the total demand at this instant.

The real-time price model has been widely applied in the literature, e.g., [9, 10] for demand response management, and [11,12,13, 15, 16] for PEV charging coordination problems.

Note: The real-time price represents the generation marginal cost, and hence is the derivative of the generation cost on the supply. Suppose the electricity generation cost, denoted by ${c}(\cdot )$, in a quadratic form on the supply, say ${c}(D_t) = \frac{1}{2} \mathsf {a} D^2_t + \mathsf {b} D_t + \mathsf {c}$, with properly valued parameters $\mathsf {a}$, $\mathsf {b}$ and $\mathsf {c}$, that has been widely considered, e.g., [25,26,27] and references therein; then the generation marginal cost evolves linearly with respect to the total demand, say $p_t(D_t) = \mathsf {a} D_t + \mathsf {b}$.

A class of centralized PEV charging coordination problems is formally formulated as follows:

Problem 3.1

$\min _{\varvec{u} \in \mathscr {U}} \, \left\{ J (\varvec{u}) \right\}$, that is to say, the objective of the PEV charging system is to implement a collection of socially optimal charging strategies, denoted by $\varvec{u}^{**}$, by minimizing the system cost (3.8).    $\blacksquare$

In this part, it considers the following assumption:

Assumption 3.1

$f_n(x)$, for all $n \in \mathscr {N}$, is monotonic increasing, strictly convex and differentiable on x.    $\blacksquare$

Note: Assumption 3.1 is consistent with the battery degradation cost formulated in (3.7).

Lemma 3.1

Under Assumption 3.1, the socially optimal strategy $\varvec{u}^{**}$ satisfies the following properties:

$$\begin{aligned} {u}_{nt} \ge {u}_{ns} \quad \text { and } \quad&d_t + \sum _{m \in \mathscr {N}} {u}_{mt} \le d_s + \sum _{m \in \mathscr {N}} {u}_{ms}, \end{aligned}$$

(3.10)

in case $D_{-n,t} \le D_{-n,s}$, for all $n \in \mathscr {N}$ and $t,s \in \mathscr {T}_n$, where $D_{-n,t} \equiv D_{-n,t}(\varvec{u}) \triangleq d_t + \underset{m \in \mathscr {N}/\{n\}}{\sum } {u}_{mt}$, i.e., $\varvec{D}_{-n}$ represents the aggregated demand trajectory composed of the base demand and the charging behaviors from all the others.

Proof

The socially optimal charging strategy is implemented by applying the method of Lagrange multiplier, see [28]. Denote, by $\varvec{\nu ^{op}} \equiv (\nu ^{op}_{nt}; n \in \mathscr {N},t \in \mathscr {T}_n)$, $\varvec{\lambda ^{op}} \equiv (\lambda ^{op}_{nt}; n \in \mathscr {N},t \in \mathscr {T}_n)$ and $\varvec{A^{op}} \equiv (A_n^{op}; n \in \mathscr {N})$, the Lagrange multipliers for the inequality constraints $\{ u_{nt} \ge 0; n \in \mathscr {N},t \in \mathscr {T}_n \}$, the inequality constraints $\{ u_{nt} \le \varUpsilon ^+_n; n \in \mathscr {N},t \in \mathscr {T}_n \}$, and the equality constraints $\{ \sum _{t \in \mathscr {T}} u_{nt} = \varGamma _n; n \in \mathscr {N} \}$, respectively; then the KKT conditions are given as below:

$$\begin{aligned}&\nu ^{op}_{nt} \ge 0, \, \nu ^{op}_{nt} u_{nt} = 0, \, \forall n \in \mathscr {N}, \forall t \in \mathscr {T}_n, \end{aligned}$$

(3.11a)

$$\begin{aligned}&\lambda ^{op}_{nt} \ge 0, \, \lambda ^{op}_{nt} (u_{nt} - \varUpsilon ^+_n) = 0, \, \forall n \in \mathscr {N}, \forall t \in \mathscr {T}_n, \end{aligned}$$

(3.11b)

$$\begin{aligned}&[\nabla J(\varvec{u})]_{nt} - \nu ^{op}_{nt} + \lambda ^{op}_{nt} - A_n^{op} = 0, \, \forall n \in \mathscr {N}, t \in \mathscr {T}_n, \end{aligned}$$

(3.11c)

together with the inequality and equality constraints defined in (3.1), where $\nabla J(\varvec{u})$ represents the gradient of $J(\varvec{u})$ and $[\nabla J(\varvec{u})]_{nt}$ denotes a component of $\nabla J(\varvec{u})$, such that

$$\begin{aligned}{}[\nabla J(\varvec{u})]_{nt} = f_n'(u_{nt}) + p'(D_t) \cdot D_t + p(D_t), \end{aligned}$$

with $D_t \equiv d_t + \sum _{n \in \mathscr {N}} u_{nt}$. Equations (3.11a) and (3.11c) are equivalent with the following:

$$\begin{aligned} \eta (\varvec{u}) \ge 0, \eta (\varvec{u}) u_{nt} = 0, \, \forall n \in \mathscr {N}, \forall t \in \mathscr {T}_n, \end{aligned}$$

(3.12)

with $\eta (\varvec{u}) \equiv f_n'(u_{nt}) + \mathsf {a} D_t + p(D_t) + \lambda ^{op}_{nt} - A_n^{op}$; then by (3.12) and (3.11b), it can derive that

$$\begin{aligned}&\eta (\varvec{u}) {\left\{ \begin{array}{ll} = 0, &{} \text { in case } u_{nt} > 0 \\ \ge 0, &{} \text { in case } u_{nt} = 0 \end{array}\right. }, \end{aligned}$$

(3.13a)

$$\begin{aligned}&u_{nt} {\left\{ \begin{array}{ll} = \varUpsilon ^+_n, &{} \text { in case } \lambda ^{op}_{nt} > 0 \\ \le \varUpsilon ^+_n, &{} \text { in case } \lambda ^{op}_{nt} = 0 \end{array}\right. }. \end{aligned}$$

(3.13b)

In the following the first part of (3.10) will be firstly verified using proof by contradiction.

Suppose that $u_{nt} < u_{ns}$ in case $D_{-n,t} \le D_{-n,s}$, with $t,s \in \mathscr {T}_n$; then $D_{-n,t} + u_{nt} < D_{-n,s} + u_{ns}$. It implies that $p \left( D_{-n,t} + u_{nt} \right) < p \left( D_{-n,s} + u^{**}_{ns} \right)$. By (3.13), $u_{nt} \ge u_{ns}$, which is contradicted with $u_{nt} < u_{ns}$.

The second part of (3.10) can be shown following the same technique applied above.    $\blacksquare$

Remark: By (3.10), it can be observed that the socially optimal charging coordination is different from the valley-filling strategy, denoted by $\varvec{u}^{\text {vf}}$, defined below:

$$\begin{aligned} \sum _{n \in \mathscr {N}} u^{\text {vf}}_{nt} {\left\{ \begin{array}{ll} = N p^{-1}(\nu ) - d_t, &{} \quad \text {in case } \displaystyle \sum _{n \in \mathscr {N}} u^{\text {vf}}_{nt} < \sum _{n \in \mathscr {N}} \varUpsilon ^+_n \\ = \sum _{n \in \mathscr {N}} \varUpsilon ^+_n, &{} \quad \text {in case } p \left( D_t \right) \le \nu \\ \end{array}\right. } \end{aligned}$$

(3.14)

for all $t \in \mathscr {T}$, with $D_t = \sum _{n \in \mathscr {N}} u^{\text {vf}}_{nt} + d_t$.

In the following, as illustrated, with a numerical simulation, the difference between the optimal solution $\varvec{u}^{**}$ for the underlying coordination problem and the valley-filling strategy $\varvec{u}^{\text {vf}}$ since $\varvec{u}^{**}$ deals with the tradeoff between the electricity cost and the aggregated battery degradation cost.

It specifies the charging coordination of a collection of PEVs with a population size of $N=10^4$ over a charging interval from 12:00 AM on one day to 12:00 AM on the next day, and considers that the length of each of the charging intervals, denoted by $\varDelta T$, is 1 h. Figure 3.2 illustrates a typical non-PEV base demand in a summer season.

Consider a specific real-time electricity price, $p_t = \mathsf {a} D_t + \mathsf {b}$ (in $\$$/kWh), with $\mathsf {a} = 3.8 \times 10^{-7}$ and $\mathsf {b} = 0.06$, which corresponds to the marginal cost of a quadratic generation cost.

For the purpose of demonstration, here it is simply supposed that $\mathscr {T}_n = \mathscr {T}$ for all n, and all of the PEVs share an identical battery capacity size, common minimum and maximum SOCs which are equal to 27 kWh, 15% and 90% respectively, and it also considers that all of the PEVs share a common initial SOC value of 15%, and a common charger efficiency 90%; then the total required charging energy over $\mathscr {T}$ is specified as $\varGamma _n = \frac{\varPhi _n}{\varsigma _n} (soc_{max} - soc_{n0})$, which is valued with 22.5 kWh, for all $n \in \mathscr {N}$.

Also suppose that all of the PEVs share a common degradation cost function specified in (3.7).

As illustrated in Fig. 3.2, the socially optimal charging strategy, which deals with the tradeoff between the electricity cost and the aggregated battery degradation cost, is different with the valley-filling strategy, since the valley-filling one may be penalized with high battery degradation cost.

Fig. 3.2

Socially optimal strategy versus valley-filling strategy

The socially optimal charging strategy can be effectively implemented in the case that the system has complete information and has a permission to directly coordinate the behaviors of all of the individual PEVs. However, in practice the PEVs may not be willing to share their private information with others, and the transmission of complete information may create heavy communication signals. Hence the centralized coordination method is usually computationally infeasible.

3.3.2 Decentralized Charging Coordination Algorithm

It proposes a decentralized charging coordination method for the PEV populations such that each of the PEVs autonomously implements its own best charging strategy which deals with the tradeoff between the charging cost and the battery degradation cost over the charging intervals. More specifically, it firstly analyzes the best charging behaviors of PEVs with respect to a given fixed price curve in Sect. 3.3.2.1; then in Sect. 3.3.2.2, a decentralized best response update mechanism is proposed. In Sects. 3.3.2.3 and 3.3.2.4, it studies the convergence and performance of the implemented equilibrium under the proposed method.

3.3.2.1 Best Response of Individual PEVs w.r.t. Fixed Price Curve

Define an individual cost of PEV n, denoted by $J_n(\varvec{u}_n; \varvec{\rho })$, under a charging strategy $\varvec{u}_n$ with a given fixed price curve $\varvec{\rho } \equiv (\rho _t; t \in \mathscr {T})$, such that

$$\begin{aligned} {J}_n(\varvec{u}_n; \varvec{\rho }) \triangleq \sum _{t \in \mathscr {T}} \Big \{ f_n(u_{nt}) + \rho _t u_{nt} \Big \}, \end{aligned}$$

(3.15)

i.e., PEV n implements its best charging strategy with respect to a presumed price curve $\varvec{\rho }$.

Denote by $\varvec{u}^*_n(\varvec{\rho })$ the best charging strategy of PEV n minimizing the cost function defined in (3.15) with a given $\varvec{\rho }$, that is $\varvec{u}^*_n(\varvec{\rho }) \triangleq \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {argmin}} \left\{ {J}_n(\varvec{u}_n; \varvec{\rho }) \right\}$, which is determined in Lemma 3.2 below.

Lemma 3.2

Under Assumption 3.1, the individual best strategy with respect to a given price curve $\varvec{\rho }$ is specified as $\varvec{u}_n(\varvec{\rho },A) \in \mathscr {U}_n$ such that

$$\begin{aligned} {u}_{nt}(\varvec{\rho },A) = {\left\{ \begin{array}{ll} \max \Big \{ 0, \min \big \{\varUpsilon ^+_n, [f'_n]^{-1}(A-\rho _t) \big \} \Big \}, &{} \text {if } t \in \mathscr {T}_n\\ 0, &{} \text {otherwise} \end{array}\right. }, \end{aligned}$$

(3.16)

where $[f'_n]^{-1}$ represents an inverse operator of $f'_n(\cdot )$.

Proof

The individual best charging strategy can be implemented with the method of Lagrange multiplier [28]. Denote, by $\varvec{\nu } \equiv (\nu _t, t \in \mathscr {T}_n)$, $\varvec{\lambda } \equiv (\lambda _t, t \in \mathscr {T}_n)$ and A, the Lagrange multipliers for the collection of inequality constraints $\{ u_{nt} \ge 0; t \in \mathscr {T}_n \}$, the inequality constraints $\{ u_{nt} \le \varUpsilon ^+_n; t \in \mathscr {T}_n \}$, and the equality constraint $\sum _{t \in \mathscr {T}} u_{nt} = \varGamma _n$ respectively. The KKT conditions for the optimization problem are given by:

$$\begin{aligned}&\nu _t \ge 0, \, \nu _t u_{nt} = 0, \, \forall t \in \mathscr {T}_n, \end{aligned}$$

(3.17a)

$$\begin{aligned}&\lambda _t \ge 0, \, \lambda _t (u_{nt} - \varUpsilon ^+_n) = 0, \, \forall t \in \mathscr {T}_n, \end{aligned}$$

(3.17b)

$$\begin{aligned}&[\nabla J_n(\varvec{u}_n; \rho )]_t - \nu _t + \lambda _t - A = 0, \, \forall t \in \mathscr {T}_n, \end{aligned}$$

(3.17c)

together with the inequality and equality constraints given in (3.1), where $\nabla J_n(\varvec{u}_n; \rho )$ represents the gradient of $J_n(\varvec{u}_n; \rho )$, and $[\nabla J_n(\varvec{u}_n; \rho )]_t = f_n'(u_{nt}) + \rho _t$ denotes the tth component of $\nabla J_n(\varvec{u}_n; \rho )$.

Equations (3.17a) and (3.17c) are equivalent with the following:

$$\begin{aligned}&f_n'(u_{nt}) + \rho _t + \lambda _t - A \ge 0, (f_n'(u_{nt}) + \rho _t + \lambda _t - A) u_{nt} = 0, \end{aligned}$$

(3.18)

for all $t \in \mathscr {T}_n$; then by (3.18) and (3.17b), it can derive that

$$\begin{aligned}&f_n'(u_{nt}) + \rho _t + \lambda _t - A {\left\{ \begin{array}{ll} = 0, &{} \text { in case } u_{nt} > 0 \\ \ge 0, &{} \text { in case } u_{nt} = 0 \end{array}\right. }, \end{aligned}$$

(3.19a)

$$\begin{aligned}&u_{nt} {\left\{ \begin{array}{ll} = \varUpsilon ^+_n, &{} \text { in case } \lambda _t > 0 \\ \le \varUpsilon ^+_n, &{} \text { in case } \lambda _t = 0 \end{array}\right. }, \end{aligned}$$

(3.19b)

which is equivalent to the conclusion of (3.16).

Under Assumption 3.1, $J_n(\varvec{u}_n; \varvec{\rho })$ is convex with respect to $\varvec{u}_n$; then the best strategy defined in (3.16) is unique.    $\blacksquare$

The form of the dependence of ${u}_{nt}(\varvec{\rho },A)$ on A expressed in (3.16) ensures that, for any fixed $\varvec{\rho }$,

• There exists an $A_-$ such that for $A \le A_-$, $\sum _{t \in \mathscr {T}} u_{nt}(\varvec{\rho },A) = 0$.

• For $A>A_-$, $\sum _{t \in \mathscr {T}} u_{nt}(\varvec{\rho },A)$ is strictly increasing and continuous on A. Hence, $\sum _{t \in \mathscr {T}} u_{nt}(\varvec{\rho },A)$ is invertible.

Therefore a constraint $\sum _{t \in \mathscr {T}} \varvec{u}_{nt}(\varvec{\rho },A) = K > 0$ defines a unique $A>A_-$ for any fixed $\varvec{\rho }$, which may be written as $A(\varvec{\rho })$. The particular value of A, which ensures the constraint of $\sum _{t \in \mathscr {T}} {u}_{nt} = \varGamma _n$, shall be denoted by $A^*(\varvec{\rho })$. The resulting charging trajectory can be written as $\varvec{u}_n ( \varvec{\rho }, A^*(\varvec{\rho })) = \varvec{u}^*_n (\varvec{\rho })$.

3.3.2.2 Decentralized Charging Coordination Method

In Sect. 3.3.2.1, it studies the best response of PEV n, denoted by $\varvec{u}^*_n(\varvec{\rho })$, with respect to a given price curve $\varvec{\rho }$. By (3.9), the real-time electricity price is determined by the total demand. It implies that the presumed price curve has to satisfy the following fixed point equation:

$$\begin{aligned} \rho _t = p \left( d_t + \sum _{n \in \mathscr {N}} {u}^*_{nt}(\varvec{\rho }) \right) , \quad \text { for all } t \in \mathscr {T}. \end{aligned}$$

(3.20)

Since it can not presumably know the solution to (3.20), it will specify an off-line decentralized iterative price curve update procedure prior to the charging interval $\mathscr {T}$ in Algorithm 3.1 later. Essentially each of the PEVs updates its best strategy by minimizing the cost function (3.15) with respect to a common price curve $\varvec{\rho }$ which will be updated with $\widehat{\varvec{\rho }}$, such that $\widehat{\varvec{\rho }}_t = p \big ( d_t + \sum _{n \in \mathscr {N}} {u}^*_{nt}(\varvec{\rho }) \big )$, for all $t \in \mathscr {T}$.

A collection of decentralized charging strategies is implemented in case the price curve update procedure converges. Before it formally proposes the decentralized charging coordination method in Algorithm 3.1 below, it firstly gives the following discussions.

Suppose that each of the PEVs is greedy for the cheap electricity as considered in the literature, e.g., [9, 16, 29]; then charging intervals with high price at one iteration tend to induce low price at the following iteration, and vice versa. This occurs because PEVs move their charging requirements from expensive to inexpensive intervals; the resulting changes in demand reduce the generation marginal cost in the previously expensive intervals and raise the price in the previously inexpensive intervals. This results in an oscillatory pattern from one iteration to the next, preventing convergence to any price curve $\rho$. The deviation cost are artificially introduced to mitigate the oscillation, see [15, 16].

In this part, as specified in Sect. 3.3.2.1, each of individual PEVs deals with the tradeoff between battery degradation cost and electricity cost. As a consequence, the system can improve the longevity of the batteries and may mitigate the oscillation behavior caused by the greedy behaviors for the cheap electricity by individual PEVs.

Algorithm 3.1

(Decentralized charging coordination method)

• Set a presumed price curve $\varvec{\rho }^{(0)} \equiv (\rho ^{(0)}_t; t \in \mathscr {T})$;

• Set $k:=0$ and $\epsilon : = \epsilon _0$ for some $\epsilon _0>0$;

• While $\epsilon > 0$

  Obtain individual best response $\varvec{u}^{*,(k+1)}_n$ w.r.t. $\varvec{\rho }^{(k)}$, for all n, such that

  $$\begin{aligned} \varvec{u}^{*,(k+1)}_n := \underset{\varvec{u}_n \, \in \, \mathscr {U}_n}{\textsf {argmin}} \, \sum _{t \in \mathscr {T}} \left\{ f_n(u_{nt}) + \rho ^{(k)}_t \, u_{nt} \right\} ; \end{aligned}$$

  Set ${\rho }^{(k+1)}_t := p \Big ( d_t + \sum _{n \in \mathscr {N}}{u}^{*,(k+1)}_{nt} \Big )$, $\forall t \in \mathscr {T}$;

  Update $\epsilon := ||\varvec{\rho }^{(k+1)}-\varvec{\rho }^{(k)}||_1$;

  Update $k := k+1$;    $\blacksquare$

Note: $||.||_1$ represents the $l_1$-norm of a vector.

3.3.2.3 Convergence of the Proposed Method

The sufficient conditions will be specified in Theorem 3.1, such that the convergence of the decentralized algorithm is guaranteed. Before that it will first introduce additional notions which will be used in Lemma 3.3 later.

Recall that $\varvec{u}^*_n(\varvec{\rho }) \triangleq \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {argmin}} \, \left\{ J_{n}(\varvec{u}_n; \varvec{\rho }) \right\}$, i.e., $\varvec{u}^*_n(\varvec{\rho })$ represents the charging strategy that minimizes the individual cost function (3.15) with respect to a given $\varvec{\rho }$. Also by Lemma 3.2, $\varvec{u}^*_n(\varvec{\rho }) = \varvec{u}_{n}(\varvec{\rho }, A^*(\varvec{\rho }))$. Here it defines another individual charging strategy $\varvec{v}_n( \varvec{\rho }, \widehat{\varvec{\rho }})$ such that

$$\begin{aligned} {v}_{nt}( \varvec{\rho }, \widehat{\varvec{\rho }}) = {\left\{ \begin{array}{ll} {u}_{nt} (\widehat{\varvec{\rho }}, A^*(\varvec{\rho })), &{} \text { with } t \in \mathscr {T}_n \\ 0, &{} \text { otherwise} \end{array}\right. }. \end{aligned}$$

Remark: The $\varvec{\varvec{v}}_n( \varvec{\rho }, \widehat{\varvec{\rho }})$, specified above, describes an individual charging strategy satisfying (3.16) with respect to $\widehat{\varvec{\rho }}$ and $A^*(\varvec{\rho })$. There is no guarantee that $\varvec{\varvec{v}}_n( \varvec{\rho }, \widehat{\varvec{\rho }})$ is an admissible charging strategy or $\varSigma (\varvec{\varvec{v}}_n( \varvec{\rho }, \widehat{\varvec{\rho }}))=\varGamma _n$.

Lemma 3.3

$\varvec{u}^*_n(\varvec{\rho })$ and $\varvec{u}^*_n(\widehat{\varvec{\rho }})$ satisfy the following inequality:

$$\begin{aligned}&||\varvec{u}^*_{n}(\varvec{\rho }) - \varvec{u}^*_{n}(\widehat{\varvec{\rho }})||_1 \nonumber \\ \le \,&2 \sum _{t \in \mathscr {T}} \big |[f'_n]^{-1}(A^*(\varvec{\rho }) - \rho _t) - [f'_n]^{-1}(A^*(\varvec{\rho }) - \widehat{\rho }_t) \big |. \end{aligned}$$

(3.21)

Proof

(3.21) will be verified in (I) and (II) below.

• (I) First it will show that the following holds, for all $t \in \mathscr {T}$,

  $$\begin{aligned}&|u^*_{nt}(\varvec{\rho }) - {v}_{nt}(\varvec{\rho },\widehat{\varvec{\rho }})| \nonumber \\ \le \,&\big |[f'_n]^{-1}(A^*(\varvec{\rho }) - {\rho }_t) - [f'_n]^{-1}(A^*(\varvec{\rho }) - \widehat{\rho }_t) \big |. \end{aligned}$$

  (3.22)

  For notational simplicity, it considers $\varvec{\varvec{v}_n} \equiv \varvec{\varvec{v}_n}( \varvec{\rho }, \widehat{\varvec{\rho }})$. It is obvious that (3.22) holds for all $t \notin \mathscr {T}_n$, since by the specification of $u^*_{nt}(\varvec{\rho })$ and ${v}_{nt}(\varvec{\rho },\widehat{\varvec{\rho }})$, at each of these instants, $u^*_{nt}(\varvec{\rho }) = {v}_{nt}(\varvec{\rho },\widehat{\varvec{\rho }}) = 0$.

Show that (3.22) holds for all $t \in \mathscr {T}_n$ in (I.a)–(I.d) below:

(I.a):

${v}_{nt} = {u}^*_{nt}(\varvec{\rho }) = 0$. It follows immediately that ${v}_{nt} - {u}^*_{nt}(\varvec{\rho }) = 0$.

(I.b):

${v}_{nt}>0$ and ${u}^*_{nt}(\varvec{\rho }) = 0$. By (3.16), ${v}_{nt}>0$ implies $[f'_n]^{-1} ({A}^*(\varvec{\rho }) - \widehat{\rho }_t) > 0$, and ${u}^*_{nt}(\varvec{\rho }) = 0$ implies $[f'_n]^{-1} ({A}^*(\varvec{\rho }) - {\rho }_t) \le 0$; then

$$ 0 < {v}_{nt} - {u}^*_{nt}(\varvec{\rho }) \le [f'_n]^{-1} ({A}^*(\varvec{\rho }) - \widehat{\rho }_t) - [f'_n]^{-1} ({A}^*(\varvec{\rho }) - {\rho }_t). $$

(I.c):

${v}_{nt}=0$ and ${u}^*_{nt}(\varvec{\rho }) > 0$. Similar to (I.b), it can derive,

$$ 0 < {u}^*_{nt}(\varvec{\rho }) - {v}_{nt} \le [f'_n]^{-1} ({A}^*(\varvec{\rho }) - \rho _t) - [f'_n]^{-1} ({A}^*(\varvec{\rho }) - \widehat{\rho }_t). $$

(I.d):

${v}_{nt}>0$ and ${u}^*_{nt}(\varvec{\rho }) > 0$. By (3.16),

$$\begin{aligned}&|{v}_{nt} - {u}^*_{nt}(\varvec{\rho })| \le \big | [f'_n]^{-1} ({A}^*(\varvec{\rho }) - \widehat{\rho }_t) - [f'_n]^{-1} ({A}^*(\varvec{\rho }) - {\rho }_t) \big |. \end{aligned}$$

(II):

Verify the following inequality in (II.a)–(II.c)

$$\begin{aligned} ||\varvec{u}^*_n(\varvec{\rho }) - \varvec{u}^*_n(\widehat{\varvec{\rho }})||_1 \le 2 ||\varvec{u}^*_n(\varvec{\rho }) - \varvec{v}_n(\varvec{\rho },\widehat{\varvec{\rho }})||_1. \end{aligned}$$

For notational simplicity, in (II.a)–(II.c) below, it considers $\varSigma (\varvec{u}_n) \equiv \sum _{t \in \mathscr {T}} {u}_{nt}$.

(II.a):

$\varSigma (\varvec{v}_n)=\varSigma (\varvec{u}^*_n(\varvec{\rho }))$. This equality ensures $\varvec{v}_n \in \mathscr {U}_n(\varUpsilon ^+_n)$. Also, the strategy $\varvec{v}_n$ has the form (3.16) with $A={A}^*(\varvec{\rho })$. Therefore, by Lemma 3.2, $\varvec{v}_n$ is the individual best strategy with respect to $\widehat{\varvec{\rho }}$; then $\varvec{u}^*_n(\widehat{\varvec{\rho }}) = \varvec{v}_n$. Hence

$$\begin{aligned} ||\varvec{u}^*_n(\widehat{\varvec{\rho }}) - \varvec{u}^*_n(\varvec{\rho })||_1 = \,&||\varvec{v}_n (\varvec{\rho }, \widehat{\varvec{\rho }}) - \varvec{u}^*_n(\varvec{\rho })||_1 \\ \le \,&2 ||\varvec{v}_n (\varvec{\rho }, \widehat{\varvec{\rho }}) - \varvec{u}^*_n(\varvec{\rho })||_1. \end{aligned}$$

(II.b):

$\varSigma (\varvec{v}_n) > \varSigma (\varvec{u}^*_n(\varvec{\rho }))$. By (3.1), $\varSigma (\varvec{u}^*_n(\widehat{\varvec{\rho }})) = \varSigma (\varvec{u}^*_n(\varvec{\rho }))=\varUpsilon ^+_n$. Therefore $\varSigma (\varvec{u}^*_n(\widehat{\varvec{\rho }})) < \varSigma (\varvec{v}_n)$, together with (3.16) and the definitions of $\varvec{u}^*_n(\widehat{\varvec{\rho }})$ and $\varvec{v}_n$, implies, ${A}^*(\widehat{\varvec{\rho }}) < {A}^*(\varvec{\rho })$, and $u^*_{nt}(\widehat{\varvec{\rho }}) \le {v}_{nt}$, for all t. Hence

$$\begin{aligned} 0 \le&||\varvec{v}_n - \varvec{u}^*_n(\widehat{\varvec{\rho }})||_1 = \varSigma (\varvec{v}_n) - \varSigma (\varvec{u}^*_n(\widehat{\varvec{\rho }})) \\ =&\varSigma (\varvec{v}_n) - \varSigma (\varvec{u}^*_n(\varvec{\rho })) \le ||\varvec{v}_n - \varvec{u}^*_n(\varvec{\rho })||_1 \end{aligned}$$

where the last inequality is a consequence of the triangle inequality for norms, taking into account that $\varSigma (\cdot ) = || \cdot ||_1$ for all admissible non-negative charging trajectories; then

$$\begin{aligned} ||\varvec{u}^*_n(\widehat{\varvec{\rho }}) - \varvec{u}^*_n(\varvec{\rho })||_1 \le&||\varvec{v}_n - \varvec{u}^*_n(\varvec{\rho })||_1 + ||\varvec{v}_n - \varvec{u}^*_n(\widehat{\varvec{\rho }})||_1 \nonumber \\ \le&2 |\varvec{v}_n - \varvec{u}^*_n(\varvec{\rho })|_1. \end{aligned}$$

(3.23)

(II.c):

$\varSigma (\varvec{v}_n) < \varSigma (\varvec{u}^*_n(\varvec{\rho }))$. A similar argument to (II.b) can be applied to show that (3.23) holds in this case.

The conclusion of (3.21) can be obtained following (I) and (II).    $\blacksquare$

Based upon Lemma 3.3, it will show, in Theorem 3.1 below, that by adopting the method proposed in Algorithm 3.1, the system converges to a unique collection of strategies under certain mild conditions.

Theorem 3.1

(Convergence of Algorithm) Suppose that the generation marginal cost p(.) is strictly increasing and consider Assumption 3.1, such that

$$\begin{aligned} |f'_n(x) - f'_n(\widehat{x})| \ge \xi |x-\widehat{x}|, |p(D) - p(\widehat{D})| \le \eta |D-\widehat{D}|, \end{aligned}$$

(3.24)

with $\xi> 2\eta N > 0$; then, by adopting Algorithm 3.1, the system converges to a unique collection of charging strategies.

Proof

Under Assumption 3.1, $[f'_n]^{-1}$, the inverse of $f'_n$, exists; moreover by adopting $|f'_n(x) - f'_n(\widehat{x})| \ge \xi |x-\widehat{x}|$, the following holds $|[f'_n]^{-1}(z) -[f'_n]^{-1}(\widehat{z})| \le \frac{1}{\xi } |z-\widehat{z}|$; then it can obtain

$$\begin{aligned}&\big |[f'_n]^{-1}(A^*(\varvec{\rho }) - {\rho }_t) - [f'_n]^{-1}(A^*(\varvec{\rho }) - \widehat{\rho }_t) \big | \nonumber \\ \le&\, |(A^*(\varvec{\rho }) - {\rho }_t) - (A^*(\varvec{\rho }) - \widehat{\rho }_t)| / {\xi } = \, |{\rho }_t - \widehat{\rho }_t| / {\xi }; \end{aligned}$$

(3.25)

by which together with (3.21),

$$\begin{aligned} || \varvec{u}^*_{n}(\varvec{\rho }) - \varvec{u}^*_{n}(\widehat{\varvec{\rho }}) ||_1 \le {2} || \varvec{\rho } - \widehat{\varvec{\rho }} ||_1 / {\xi }. \end{aligned}$$

(3.26)

Also by (3.24), it obtains that

$$\begin{aligned}&\Big | p \Big ( d_t + \sum _{n \in \mathscr {N}} {u}^*_{nt}(\varvec{\rho }) \Big ) - p \Big ( d_t + \sum _{n \in \mathscr {N}} \varvec{u}^*_{nt}(\widehat{\varvec{\rho }}) \Big ) \Big | \nonumber \\ \le&\eta \sum _{n \in \mathscr {N}} \left| {u}^*_{nt}(\varvec{\rho }) - {u}^*_{nt}(\widehat{\varvec{\rho }}) \right| \end{aligned}$$

(3.27)

Denote by $\varvec{p}(\varvec{\rho }) \equiv ({p}_t(\varvec{\rho }); t \in \mathscr {T})$ with $p_t(\varvec{\rho }) \equiv p \big ( d_t + \sum _{n \in \mathscr {N}} {u}^*_{nt}(\varvec{\rho }) \big )$; then by (3.27),

$$\begin{aligned} ||\varvec{p} (\varvec{\rho }) - \varvec{p}(\widehat{\varvec{\rho }})||_1 \le \eta \sum _{n \in \mathscr {N}} || \varvec{u}^*_{n}(\varvec{\rho }) - \varvec{u}^*_{n}(\widehat{\varvec{\rho }}) ||_1. \end{aligned}$$

(3.28)

Hence by (3.26) and (3.28),

$$\begin{aligned} ||\varvec{p} (\varvec{\rho }) - \varvec{p}(\widehat{\varvec{\rho }})||_1 \le {2\eta N} ||\varvec{\rho } - \widehat{\varvec{\rho }}||_1 / {\xi }; \end{aligned}$$

(3.29)

then by applying the contraction mapping theorem, [30], the system can converge to a unique price curve $\varvec{\rho }^*$. In summary, by applying the proposed decentralized algorithm, the system converges to a unique collection of charging strategies $(\varvec{u}^*_n(\varvec{\rho }^*); n \in \mathscr {N})$ under the constraints of (3.24).    $\blacksquare$

3.3.2.4 Performance of the Proposed Method

Theorem 3.2

(Performance of Implemented Strategy) Suppose that the generation marginal cost $p(\cdot )$ is strictly increasing, and $\varvec{u}^* \equiv \{ \varvec{u}^*_n; n \in \mathscr {N} \}$ is the implemented strategy by applying Algorithm 3.1; then $\varvec{u}^*$ satisfies (3.10) under Assumption 3.1.

Remark. By Lemma 3.1, Theorems 3.1 and 3.2, it can obtain that, by applying the proposed method, the system can converge to a collection of unique charging strategies which is nearly socially optimal.

Proof of Theorem 3.2. Consider a collection of strategies $\varvec{u}^{\dagger } \equiv (\varvec{u}^{\dagger }_n; n \in \mathscr {N})$, and denote by $\varvec{\rho }(\varvec{u}^{\dagger })$ the updated price curve under $\varvec{u}^{\dagger }$; then $\rho _t = p \Big ( d_{t} + \sum _{m \in \mathscr {N}} {u}^{\dagger }_{mt} \Big )$. Hence, at the convergence, the updated best strategy with respect to the price curve $\varvec{\rho }(\varvec{u}^{\dagger })$, denoted by $\varvec{u}^* \equiv \varvec{u}^* (\varvec{\rho }(\varvec{u}^{\dagger }))$, satisfies the following:

$$\begin{aligned} \varvec{u}^*_n = \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {argmin}} \sum _{t \in \mathscr {T}} \Big \{ f_n(u_{nt}) + \rho _t \cdot u_{nt} \Big \}, \quad \text { and } \quad \varvec{u}^*_n = \varvec{u}^{\dagger }_n. \end{aligned}$$

(i). Proof of the 1st part of (3.10): The proof by contradiction will be used. Suppose that $u^{\dagger }_{nt} < u^{\dagger }_{ns}$ in case $D^{\dagger }_{-n,t} \le D^{\dagger }_{-n,s}$, with $t,s \in \mathscr {T}_n$, where $D^{\dagger }_{-n,t} \equiv d_t + \underset{m \in \mathscr {N}/\{n\}}{\sum } {u}^{\dagger }_{mt}$; then $D^{\dagger }_{-n,t} + u^{\dagger }_{nt} < D^{\dagger }_{-n,s} + u^{\dagger }_{ns}$; hence $p ( D^{\dagger }_{-n,t} + u^{\dagger }_{nt} ) < p ( D^{\dagger }_{-n,s} + u^{\dagger }_{ns} )$. By (3.19) in Lemma 3.2, ${u}^*_{nt} \ge {u}^*_{ns}$ which, together with the assumed $u^{\dagger }_{nt} < u^{\dagger }_{ns}$, implies $\varvec{u}^*_n \ne \varvec{u}^{\dagger }_n$ which is contradicted with $\varvec{u}^*_n = \varvec{u}^{\dagger }_n$ since the $\varvec{u}^*_n$ is assumed to be an implemented strategy of PEV n.

(ii). Proof of the 2nd part of (3.10): The proof by contradiction will be used again. Suppose that $d_t + \sum _{n \in \mathscr {N}} {u}^{\dagger }_{nt} > d_s + \sum _{n \in \mathscr {N}} {u}^{\dagger }_{ns}$ in case $D^{\dagger }_{-n,t} \le D^{\dagger }_{-n,s}$, with $t,s \in \mathscr {T}_n$; then it implies that $p(d_t + \sum _{n \in \mathscr {N}} {u}^{\dagger }_{nt}) > p(d_s + \sum _{n \in \mathscr {N}} {u}^{\dagger }_{ns})$. By (3.19) in Lemma 3.2, ${u}^*_{nt} < {u}^*_{ns}$ which is contradicted with the 1st part of (3.10) verified in (i).    $\blacksquare$

In Corollary 3.1 below, it will study the performance of the implemented strategy $\varvec{u}^*$ in case $\mathscr {T}_n = \mathscr {T}$ for all n.

Corollary 3.1

Consider $\mathscr {T}_n = \mathscr {T}$ for all n; then

$$\begin{aligned} {u}^*_{nt} \ge {u}^*_{ns}, \, \text { and } \, d_t + \sum _{n \in \mathscr {N}} {u}^*_{nt} \le d_s + \sum _{n \in \mathscr {N}} {u}^*_{ns}, \end{aligned}$$

(3.30)

in case $d_t \le d_s$, for all $n \in \mathscr {N}$ and for any pair of distinct instants $t,s \in \mathscr {T}$.

Proof by contradiction. Suppose that there exists a PEV n satisfies $u_{nt}^* < u_{ns}^*$; then by (3.19) in Lemma 3.2 and the convexity of $f_n$, $p(D_t) > p(D_s)$, by which together with the increasing property of p(D), it obtains that $d_t + \sum _{n \in \mathscr {N}} {u}_{nt}^* > d_s + \sum _{n \in \mathscr {N}} {u}_{ns}^*$. Hence there exists an $m \in \mathscr {N}/\{n\}$, such that $u_{mt}^* > u_{ms}^*$. By (3.19) in Lemma 3.2, $p(D_t) < p(D_s)$ which is contradicted with $p(D_t) > p(D_s)$ in case $d_t \le d_s$. By $u_{nt}^* \ge u_{ns}^*$, $\forall n \in \mathscr {N}$, and (3.19) stated in Lemma 3.2, $d_t + \sum _{n \in \mathscr {N}} {u}_{nt}^* \le d_s + \sum _{n \in \mathscr {N}} {u}_{ns}^*$.    $\blacksquare$

3.3.3 Numerical Examples

In this section, unless specified, it adopts the parameters considered in the example in Sect. 3.3.1.

Consider the price $p(\cdot )$ (in $\$$/kWh) specified in Sect. 3.3.1, $|p(D) - p(\widehat{D})| = \mathsf {a} |D - \widehat{D}|$, with $\mathsf {a} = 3.8 \times 10^{-7}$. Also by the battery degradation cost $f_n$ given in (3.7) for the PEVs specified in Sect. 3.3.1, it obtains that $|f_n'(x_1) - f_n'(x_2)| = 2 \mathfrak {a} |x_1 - x_2|$, with $\mathfrak {a} = 0.004$; then it is straightforward to verify that $f_n$ and $p_t$ satisfy (3.24).

As illustrated in Fig. 3.3, the system converges to a nearly socially optimal charging strategy by adopting the method proposed in Algorithm 3.1. This is consistent with Theorem 3.1 and Corollary 3.1. Moreover it can be observed that the performance under the decentralized method is better than that under the valley-filling strategy.

Fig. 3.3

A converged best response update procedure of PEVs in case $f_n$ and p satisfy the condition of (3.24)

The iteration updates of the associated price curve are displayed in Fig. 3.4.

Fig. 3.4

A converged price curve update procedure in case $f_n$ and p satisfy the condition of (3.24)

The above simulation verifies the results developed in this chapter. Nevertheless the system may still converge in case the sufficient condition (3.24) is not satisfied. For example, suppose that the battery degradation cost for some other PEVs is specified as $f_n = 0.0027 u^2_{nt} + 0.05 u_{nt}$; then though it can verify that (3.24) does not hold any longer, as illustrated in Fig. 3.5, the system still converges.

Fig. 3.5

A converged iteration update procedure in case the sufficient condition stated in Theorem 3.1 is unsatisfied

However if it is supposed that the value of $\mathfrak {a}$, the coefficient in battery degradation cost $f_n$, is much smaller, e.g., $\mathfrak {a} = 0.001$; then as illustrated in Fig. 3.6, the iterative update procedure does not converge any longer.

Fig. 3.6

An unconverged iteration update procedure in case the sufficient condition stated in Theorem 3.1 is unsatisfied

For approaching a realistic situation of PEV populations, suppose that the initial SOC values of PEV populations, $\{soc_{n0}; n \in \mathscr {N}\}$, approximately satisfy a Gaussian distribution, $\text {N}(\hat{\mu },\hat{\gamma })$, see [31, 32]. For the purpose of demonstration, it considers $\hat{\mu } = 0.5$ and $\hat{\gamma } = 0.1$. Figure 3.7 illustrates the iteration updates of the decentralized strategies following Algorithm 3.1 such that the system converges in a few of iteration steps.

Fig. 3.7

An illustration of the equilibrium of the two-stage market

3.4 Decentralized Methods with Flexible Energy Demands

In last section, it studies the charging coordination problems with degradation cost function $f_n(\cdot )$ and a fixed charging energy demand $\varGamma _n$.

Besides the individual degradation cost, due to the distribution-level impacts of PEV charging including line and transformer overloading, low voltages and increased losses, all these effects are a consequence of coincident high charger power demand $u_{nt}$. Therefore undesirable distribution-grid effects can be minimized by encouraging PEVs to charge at lower power levels. This can be achieved by introducing a demand charge,

$$\begin{aligned} \textit{Cost}_{\textit{demand},nt} = f_{\textit{demand},nt}(u_{nt}) \end{aligned}$$

(3.31)

whereby PEVs incur a higher cost as their charging power increases, i.e. $f_{\textit{demand},nt}(\cdot )$ is a strictly increasing function. This charge is in addition to the cost of the energy delivered to the battery, and is consistent with existing tariff structures for larger consumers [33].

Moreover, it may allow the individual PEVs to consider the tradeoff between the charging cost and the total charged energy during the charing interval.

Consequently, in this section, it is worth to design the decentralized method to solve the PEV charging problems with flexible charging energy demands, and involving another individual cost for each PEV which is specified below.

3.4.1 Centralized PEV Charging Coordination

The coordination problem of interest considers the tradeoff between the total cost of supplying energy to the PEV population and the benefit derived from doing so. The total cost is composed of the generation cost, the demand charge discussed earlier in Sect. 3.4, and the PEV battery degradation cost formulated in Sect. 3.2.2. Coordination must ensure that all charging strategies are admissible, $\varvec{u}_n \in \mathscr {U}_n$ for all $n \in \mathscr {N}$.

Given a collection of admissible charging strategies $\varvec{u} \in \mathscr {U}$, the system cost function can be expressed as,

$$\begin{aligned} J(\varvec{u})&\triangleq \sum _{t \in \mathscr {T}} \left\{ c \left( d_t + \sum _{n \in \mathscr {N}} u_{nt} \right) + \sum _{n \in \mathscr {N}} g_{nt}(u_{nt}) \right\} - \sum _{n \in \mathscr {N}} \Big \{ h_n \left( \Vert \varvec{u}_n\Vert _1 \right) \Big \}, \end{aligned}$$

(3.32)

where:

• ${c}(\cdot )$ gives the generation cost with respect to the total demand $d_t + \sum _{n \in \mathscr {N}} u_{nt}$, and $d_t$ denotes the aggregate inelastic base demand at time t;

• $g_{nt}(u_{nt}) = f_{\textit{demand},nt}(u_{nt}) + f_n(u_{nt})$ captures the demand charge (3.31) and battery degradation cost (3.6) of PEV n; and,

• $h_n \left( \Vert \varvec{u}_n\Vert _1 \right)$ denotes the benefit function of PEV n with respect to the total energy delivered over the charging horizon. In [34], this function has the quadratic form,

  $$\begin{aligned} h_n \left( \Vert \varvec{u}_n\Vert _1 \right) = - \delta _n (\Vert \varvec{u}_n\Vert _1 - \varGamma _n)^2, \end{aligned}$$

  (3.33)

  with the factor $\delta _n$ reflecting the relative importance of delivering the full charge to the PEV over the charging horizon.

The utility function of PEV n, for a charging strategy $\varvec{u}_n \in \mathscr {U}_n$, can be written,

$$\begin{aligned} v_n(\varvec{u}_n) \triangleq h_n (\Vert \varvec{u}_n\Vert _1) - \sum _{t \in \mathscr {T}} \, g_{nt}(u_{nt}). \end{aligned}$$

(3.34)

The system cost $J(\varvec{u})$ given by (3.32) can then be rewritten:

$$\begin{aligned} J(\varvec{u}) = \sum _{t \in \mathscr {T}} c \left( d_t + \sum _{n \in \mathscr {N}} u_{nt} \right) - \sum _{n \in \mathscr {N}} v_n(\varvec{u}_n). \end{aligned}$$

(3.35)

The individual utility function (3.34) is similar to that specified in [13], where a decentralized PEV charging algorithm ia developed based on congestion pricing concepts from internet traffic control [17].

Remarks on generation cost $c(\cdot )$: ${c}(\cdot )$ is briefly discussed on the paragraph following Assumption 2.1. In Assumption 3.2 below, it gives a formal consideration on $c(\cdot )$.

Assumption 3.2

${c}(\cdot )$ is monotonically increasing, strictly convex and differentiable.

   $\blacksquare$

Actually, as stated earlier in Chap. 2, it is commonly assumed, see [25,26,27] and references therein, that the electricity generation cost can be approximated by the quadratic form,

$$\begin{aligned} {c}({D}_t) = \frac{1}{2} \mathsf {a} {D}^2_t + \mathsf {b} {D}_t + \mathsf {c}, \end{aligned}$$

(3.36)

with parameters $\mathsf {a}$, $\mathsf {b}$ and $\mathsf {c}$ which reflect system conditions. The marginal generation cost, which is the derivative of generation cost, therefore varies linearly with the total demand, $p({D}_t) \triangleq {c}'({D}_t) = \mathsf {a} {D}_t + \mathsf {b}$.

Centralized PEV charging coordination is formulated as the optimization problem below:

Problem 3.2

$$\begin{aligned} \min _\mathbf{{u} \in \mathscr {U}} \; J (\varvec{u}). \end{aligned}$$

(3.37)

The objective is to implement a socially optimal collection of charging strategies for all PEVs, denoted by $\varvec{u}^{**}$, that minimizes the system cost (3.32) or its equivalent form (3.35).    $\blacksquare$

The following assumption will apply throughout the chapter for $g_{nt}(\cdot )$:

Assumption 3.3

$g_{nt}(\cdot )$, for all $n \in \mathscr {N}$, $t \in \mathscr {T}$, is monotonically increasing, strictly convex and differentiable.    $\blacksquare$

When the benefit function takes the form (3.33), the solution obtained by minimizing $J(\varvec{u})$ subject only to (3.1a) always satisfies (3.1b), and hence is also the solution for Optimization Problem 3.2. To see this, define the set of charging strategies that satisfy (3.1a) for PEV n,

$$ \mathscr {S}_n \triangleq \big \{ \varvec{u}_n \equiv (u_{nt}; t \in \mathscr {T}); \text { s.t. constraint (3.1a)} \big \}, $$

and let $\mathscr {S}$ denote the collection of such sets for all $n \in \mathscr {N}$. Consider the optimization problem $\min _{\varvec{u} \in \mathscr {S}} J (\varvec{u})$ rather than (3.37).

Based on Assumptions 3.2, 3.1 and 3.3, the efficient (socially optimal) charging behavior is unique and can be characterized by its associated KKT conditions [28]. The optimal solution $\varvec{u}^{**}$ is therefore given by:

$$\begin{aligned} \frac{\partial }{\partial u_{nt}} J(\varvec{u}) \ge 0, \quad \; u_{nt} \ge 0, \quad \; \frac{\partial }{\partial u_{nt}} J(\varvec{u}) u_{nt} = 0, \end{aligned}$$

(3.38)

for all $n \in \mathscr {N}$ and $t \in \mathscr {T}_n$, where:

$$ \frac{\partial }{\partial u_{nt}} J(\varvec{u}) = {c}' \left( d_t + \sum _{n \in \mathscr {N}} u_{nt} \right) - \frac{\partial }{\partial u_{nt}} v_n(\varvec{u}_n). $$

It follows from (3.38) that the efficient charging behavior $\varvec{u}^{**}$ is uniquely specified by:

$$\begin{aligned} p^{**}_t {\left\{ \begin{array}{ll} = \frac{\partial }{\partial u_{nt}} v_n(\varvec{u}^{**}_n), &{} \text {when } u^{**}_{nt} > 0, \\ \ge \frac{\partial }{\partial u_{nt}} v_n(\varvec{u}^{**}_n), &{} \text {when } u^{**}_{nt} = 0, \end{array}\right. } \end{aligned}$$

(3.39)

where $p^{**}_t = {c}' \big (d_t + \sum _{n \in \mathscr {N}} u_{nt}^{**} \big )$ is the generation marginal cost over the charging horizon with respect to the efficient allocation $\varvec{u}^{**}$.

Notice that if $\Vert \varvec{u}_n^{**}\Vert _1 \ge \varGamma _n$ then (3.33) together with Assumptions 3.1 and 3.3, ensure that $\frac{\partial }{\partial u_{nt}} v_n(\varvec{u}^{**}_n) <0$. But $p_t^{**}>0$ according to Assumption 3.2, so (3.39) implies that $\varvec{u}_n^{**}=\varvec{0}$. Hence a contradiction. Accordingly, $\Vert \varvec{u}_n^{**}\Vert _1 < \varGamma _n$ and (3.1b) is always satisfied.

3.4.2 Numerical Examples

This example considers coordinated charging of a population of 5000 PEVs over a common charging interval from noon on one day to noon on the next. In accordance with (3.36), the generation cost function has the quadratic form,

$$\begin{aligned} {c}({D}_t) = 2.9 \times 10^{-7} {D}^2_t + 0.06 {D}_t, \end{aligned}$$

(3.40)

where ${D}_t = d_t+ \sum _{n \in \mathscr {N}} u_{nt}$. The base demand $\varvec{d}$, which is shown in Fig. 3.8, is representative of a typical hot summer day.

Fig. 3.8

Aggregate demand due to the efficient charging strategies

The battery pack of each PEV is composed of $\text {LiFePO}_4$ lithium-ion cells which have a nominal voltage of 3.3 V and energy capacity of 2.5 Ah (Amp$\times$Hour). These are typical values for batteries that are used in PEVs. Assume the price of battery cell capacity is $10/Wh. Furthermore, let all PEVs have battery capacity of 40 kWh. Then the battery degradation cost (3.6) for each PEV is given (approximately) by,

$$\begin{aligned} f_n(u_{nt}) = 0.0012 u^2_{nt} + 0.11 u_{nt} - 0.02. \end{aligned}$$

Each PEV is subject to a quadratic demand charge $f_{demand,nt}(u_{nt}) = 0.0018 u_{nt}^2$. Thus, the local costs incurred by each PEV at time t amount to,

$$\begin{aligned} g_{nt}(u_{nt}) = 0.003 u^2_{nt} + 0.11 u_{nt} - 0.02. \end{aligned}$$

(3.41)

Initially all PEVs use the same weighting factor $\delta _n = 0.03$ in their utility function $v_n(\varvec{u}_n)$ given by (3.33) and (3.34). This will be relaxed in later investigations.

For simplicity, assume all PEVs have identical minimum and maximum SOC, with $soc_{min} = 15\%$ and $soc_{max} = 90\%$. The upper limit (3.1b) on the energy that can be delivered to each PEV is given by,

$$ \varGamma _n = 40 (soc_{max} - soc_{n0}), $$

which equals 30 kWh if $soc_{n0} = soc_{min} = 15\%$ for all n.

Fig. 3.9

Evolution of the SOC of an individual PEV

The efficient (socially optimal) charging strategies given by (3.37), $\varvec{u}_n^{**}$ for $n \in \mathscr {N}$, result in the aggregate demand shown in Fig. 3.8. The evolution of the SOC of one of the PEVs is shown in Fig. 3.9. As a comparison, the aggregate demand of the valley-filling strategy $\varvec{u}^\textit{vf}$ given by [15] is also shown in Fig. 3.8. Note that the algorithm developed in [15] enforces an equality constraint on the energy delivered, $\Vert \varvec{u}_n^\textit{vf}\Vert _1 = \varGamma _n^\textit{vf}$, rather than incorporating a benefit function of the form (3.33). Therefore, to ensure a meaningful comparison, the total charge requirement in the valley-filling case is set equal to the energy delivered in the socially optimal solution, $\varGamma _n^\textit{vf} = \Vert \varvec{u}_n^{**}\Vert _1$.

The socially optimal charging strategy given by (3.37) establishes a tradeoff between the total generation cost of the system and the local costs (demand charge and battery degradation cost) of the PEV population. As illustrated in Fig. 3.8, this results in an outcome that differs quite considerably from the valley-filling strategy which is solely concerned with minimizing total generation cost. This distortion away from valley-filling increases as higher weighting is given to the local costs of the PEV population.

This difference between the socially optimal and valley-filling strategies can be quantified by considering,

$$\begin{aligned} \sum _{t \in \mathscr {T}} c \left( d_t+ \sum _{n \in \mathscr {N}} u^{**}_{nt} \right) - \sum _{t \in \mathscr {T}} c \left( d_t+ \sum _{n \in \mathscr {N}} u^\textit{vf}_{nt} \right)&= 211.7 \\ \sum _{n \in \mathscr {N}} \sum _{t \in \mathscr {T}} g_{nt} \left( u^{**}_{nt} \right) - \sum _{n \in \mathscr {N}} \sum _{t \in \mathscr {T}} g_{nt} \left( u^{\textit{vf}}_{nt} \right)&= - 671.3. \end{aligned}$$

where the constraint $\varGamma _n^\textit{vf} = \Vert \varvec{u}_n^{**}\Vert _1$ has been taken into account. It can be seen that incorporating the local costs resulted in an increase in generation cost of $211.7/day. However this is more than offset by a reduction in the local costs of $671.3/day, resulting in an overall saving of $459.6/day. As a further comparison, if $\varGamma _n^\textit{vf} = \varGamma _n$, i.e. PEVs must fully charge rather than settle for the reduced energy delivery of $\Vert \varvec{u}_n^{**}\Vert _1$, then adopting $\varvec{u}^{**}$ instead of $\varvec{u}^{\textit{vf}}$ would result in a cost saving of $1640.2/day.

The following investigations consider the effects of variations in the battery degradation cost $f_n$ and the benefit function $h_n$ on the optimal charging strategies of a population of PEVs.

Figure 3.10 shows the evolution of the (efficient) aggregate demand as the battery price $P_{cell}$ is varied. For this study, the demand charge (3.31) is set to zero and $\delta _n = 0.03$ in all cases. It can be seen that the efficient aggregate demand approaches valley-filling as $P_{cell}$ decreases, and becomes exactly valley-filling when $P_{cell}=0$. Figure 3.11 shows that the total delivered energy $\Vert \varvec{u}_n\Vert _1$ decreases as $P_{cell}$ increases.

Fig. 3.10

Aggregate demand for efficient charging strategies as battery price $P_{cell}$ varies

Fig. 3.11

Total delivered energy $\Vert \varvec{u}_n\Vert _1$ for an individual PEV as battery price $P_{cell}$ varies

Figure 3.12 shows the variation in the efficient aggregate demand as the benefit function parameter $\delta _n$ is varied. The general shape of the aggregate demand remains largely unchanged. It is shown in Fig. 3.13 that the total delivered energy $\Vert \varvec{u}_n\Vert _1$ increases with $\delta _n$, approaching the energy capacity limit $\varGamma _n$.

Fig. 3.12

Aggregate demand for efficient charging strategies as the benefit function parameter $\delta _n$ varies

Fig. 3.13

Total delivered energy $\Vert \varvec{u}_n\Vert _1$ for an individual PEV as the benefit function parameter $\delta _n$ varies

Centralized coordination is only possible when the system operator has complete information, including the characteristics of PEV batteries and the valuation functions of individual PEVs. It is unlikely, however, that individuals would be willing to share such private information. Also, for a large population, centralized coordination may be computationally infeasible. Thus, the remainder of the chapter is devoted to the development of a decentralized coordination process where each PEV updates its charging strategy with respect to a common electricity price profile, and then the price profile is updated based on the latest charging strategies for the population.

Decentralized coordination of PEV charging can be achieved using an algorithm of the form:

(S1):

Each PEV autonomously determines its optimal charging strategy with respect to a given electricity price profile $\varvec{p}\equiv (p_t, t \in \mathscr {T})$. This optimal strategy takes into account the tradeoff between the electricity cost and local (demand and battery degradation) costs over the entire charging horizon.

(S2):

The electricity price profile $\varvec{p}$ is updated to reflect the latest charging strategies determined by the PEV population in (S1).

(S3):

Steps (S1) and (S2) are repeated until the change in the price profile at (S2) is negligible.

Section 3.4.3 establishes the optimal charging strategy $\varvec{u}^*_n(\varvec{p})$ of each PEV, $n \in \mathscr {N}$, with respect to a given price profile $\varvec{p}$. A mechanism for updating the electricity price profile is designed in Sect. 3.4.4. Section 3.4.5 then formalizes the algorithm (S1)–(S3), establishes convergence properties, and shows that decentralized coordination gives the socially optimal (economically efficient) charging strategy.

3.4.3 Optimal Response of Each PEV w.r.t. a Fixed Price Profile

The individual cost function of PEV n, under charging strategy $\varvec{u}_n \in \mathscr {U}_n$ and with respect to the price profile $\varvec{p}$, can be written:

$$\begin{aligned} J_n(\varvec{u}_n;\varvec{p}) \triangleq \sum _{t \in \mathscr {T}} p_t u_{nt} - v_n(\varvec{u}_n). \end{aligned}$$

(3.42)

Alternatively, using (3.34), this cost function can be expressed in the form:

$$ J_n(\varvec{u}_n; \varvec{p}) = \sum _{t \in \mathscr {T}} \Big \{ p_t u_{nt} + g_{nt}(u_{nt}) \Big \} - h_n \left( \sum _{t \in \mathscr {T}} u_{nt} \right) , $$

where it becomes clear that the cost is composed of the total electricity cost $\sum _{t \in \mathscr {T}} p_t u_{nt}$, the total local cost $\sum _{t \in \mathscr {T}} g_{nt}(u_{nt})$, and the benefit derived from the total energy delivered over the charging horizon $h_n \left( \sum _{t \in \mathscr {T}} u_{nt} \right)$.

The optimal charging strategy of PEV n, with respect to $\varvec{p}$, is obtained by minimizing the cost function (3.42),

$$\begin{aligned} \varvec{u}^*_n(\varvec{p}) = \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {argmin}} \, J_n(\varvec{u}_n;\varvec{p}). \end{aligned}$$

(3.43)

It will be shown in Theorem 3.3 that the optimal response of PEV n has the form:

$$\begin{aligned} u_{nt}(\varvec{p}, A_n) = {\left\{ \begin{array}{ll} \max \big \{ 0, [g'_{nt}]^{-1} (A_n-p_t) \big \}, &{} t \in \mathscr {T}_n \\ 0, &{} t \in \mathscr {T}\setminus \mathscr {T}_n \end{array}\right. } \end{aligned}$$

(3.44)

for some $A_n$, where $g'_{nt}$ is the derivative of $g_{nt}$, and $[g'_{nt}]^{-1}$ is the corresponding inverse function. Since the total charging energy is elastic, the value of $A_n$ is dependent upon the PEV specifications and the price $\varvec{p}$, with this dependence established in Theorem 3.3.

Determining the optimal charging strategy (3.43) proceeds as follows. Lemma 3.4 addresses the restricted problem of finding the optimal charging strategy when the total delivered energy takes a specified value, $\Vert \varvec{u}_n\Vert _1=\omega$, where $0 \le \omega \le \varGamma _n$. Lemma 3.5 considers the charging strategy in the form (3.44) and establishes the relationship between the value of $A_n$ and the total energy delivered $\Vert \varvec{u}_n(\varvec{p},A_n)\Vert _1$. Lemma 3.6 shows that the sensitivity of charging cost to changes in $\omega$ is given by $A_n$. Finally, Theorem 3.3 brings all the results together and determines the unique value of $A_n$ that ensures (3.44) is optimal in the sense of (3.43).

To begin with, consider the cost for PEV n, but excluding $h_n$. This can be written,

$$\begin{aligned} F_n(\varvec{u}_n; \varvec{p}) \triangleq \sum _{t \in \mathscr {T}} \Big \{ p_t u_{nt} + g_{nt}(u_{nt}) \Big \}. \end{aligned}$$

(3.45)

This function will be used to examine scenarios where the total charging is constant, $\Vert \varvec{u}_n\Vert _1 = \omega$. In such cases, the cost $h_n(\Vert \varvec{u}_n\Vert _1)$ can be neglected since it is equal across all such scenarios. Accordingly, define the set of charging strategies where total energy $\omega$ is delivered to PEV n as,

$$\begin{aligned} \mathscr {U}_n(\omega ) \triangleq \Big \{ \varvec{u}_n \in \mathscr {U}_n; \text { s.t. } \Vert \varvec{u}_n\Vert _1 = \omega \Big \}. \end{aligned}$$

(3.46)

Lemma 3.4

Consider a fixed $\omega$, with $0 \le \omega \le \varGamma _n$, and a fixed $\varvec{p}$. Then $\varvec{u}_n(\varvec{p},A_n)$, defined in (3.44), is the unique charging strategy minimizing the cost function (3.45) subject to the set of admissible charging strategies $\mathscr {U}_n(\omega )$.

Proof

Define the Lagrangian function,

$$\begin{aligned} L_n(\varvec{u}_n, A_n ; \varvec{p}) \triangleq F_n(\varvec{u}_n; \varvec{p}) + A_n ( \omega -\Vert \varvec{u}_n\Vert _1 ), \end{aligned}$$

(3.47)

with $u_{nt} \ge 0$ for $t \in \mathscr {T}$, and $A_n$ the Lagrangian multiplier associated with the constraint on total delivered energy (3.46). The desired optimal strategy must satisfy the KKT conditions [28]:

• (i) $\frac{\partial L_n}{\partial A_n} = 0$.

• (ii) $\frac{\partial L_n}{\partial u_{nt}} \ge 0$, ${u}_{nt} \ge 0$, with complementary slackness.

The equality (i) recovers the constraint on total energy $\Vert \varvec{u}_n\Vert _1 = \omega$, while the inequalities given in (ii) can be expressed in the form,

$$\begin{aligned} p_t + g'_{nt}(u_{nt}) - A_n {\left\{ \begin{array}{ll} = 0, &{} \text { when } {u}_{nt} > 0 \\ \ge 0, &{} \text { otherwise,} \end{array}\right. } \end{aligned}$$

(3.48)

which is equivalent to (3.44). Moreover, since $F_n(\varvec{u}_n; \varvec{p})$ is convex with respect to $\varvec{u}_n$, the optimal charging strategy defined by (3.44) must be unique for a given $\mathscr {U}_n(\omega )$.

   $\blacksquare$

This lemma establishes the minimum (3.43) when $\varvec{u}_n$ is restricted to $\mathscr {U}_n(\omega )$, for a specified value of $\omega$. Relaxing that restriction, by allowing $\varvec{u}_n \in \mathscr {U}_n$, is achieved in Theorem 3.3. Before reaching that point, it is necessary to establish some notation and intermediate results.

For any PEV, $n \in \mathscr {N}$, the pair of values $A^{-}_{n}(\varvec{p})$ and $A^{+}_{n}(\varvec{p})$ are defined as:

$$\begin{aligned} A^{-}_{n}(\varvec{p})&= \max \big \{ A, \, \text { such that } \Vert \varvec{u}_n(\varvec{p},A)\Vert _1 = 0 \big \} \end{aligned}$$

(3.49)

$$\begin{aligned} A^{+}_{n}(\varvec{p})&= A, \text { such that } \Vert \varvec{u}_n(\varvec{p},A)\Vert _1 = \varGamma _n. \end{aligned}$$

(3.50)

Note that the subscript n is included on $A_n^-$ for consistency, though it is independent of n.

The following lemma establishes a few basic relationships between the Lagrangian multiplier $A_n$ and the minimizing charging strategy.

Lemma 3.5

Consider a fixed price profile $\varvec{p}$. Then:

(i):

Every ${u}_{nt}(\varvec{p},A_n)$, $t \in \mathscr {T}$, is non-decreasing with $A_n \in \mathbb {R}$, and hence $\Vert \varvec{u}_n(\varvec{p},A_n)\Vert _1$ is non-decreasing with $A_n$. Furthermore, $\Vert \varvec{u}_n(\varvec{p},A_n)\Vert _1$ is strictly increasing for $A_n \ge A^{-}_{n}(\varvec{p})$.

(ii):

$\varvec{u}_n(\varvec{p},A_n)$ is admissible for $A^{-}_{n}(\varvec{p}) \le A_n \le A^{+}_{n}(\varvec{p})$, but not admissible for any $A_n > A^{+}_{n}(\varvec{p})$.

Proof

Property (i) holds by the specification of ${u}_{nt}(\varvec{p},A_n)$ given in (3.44) and verified by Lemma 3.4, keeping in mind Assumptions 3.1 and 3.3. From (3.49) and (3.50), $\Vert \varvec{u}_n(\varvec{p},A^{-}_{n}(\varvec{p}))\Vert _1= 0$ and $\Vert \varvec{u}_n(\varvec{p},A^{+}_{n}(\varvec{p}))\Vert _1 = \varGamma _n$. Therefore, since $\Vert \varvec{u}_n(\varvec{p},A_n)\Vert _1$ is strictly increasing for $A_n \ge A^{-}_{n}(\varvec{p})$, it follows that:

• ${u}_{nt}(\varvec{p},A_n) \ge 0$ for all $t \in \mathscr {T}$, and $\Vert \varvec{u}_{n}(\varvec{p},A_n)\Vert _1 \le \varGamma _n$ for $A^{-}_{n}(\varvec{p}) \le A_n \le A^{+}_{n}(\varvec{p})$, so $\varvec{u}_n(\varvec{p},A_n)$ is admissible.

• $\Vert \varvec{u}_{n}(\varvec{p},A_n)\Vert _1 > \varGamma _n$ when $A_n > A^{+}_{n}(\varvec{p})$, so $\varvec{u}_n(\varvec{p},A_n)$ is not admissible.

This establishes property (ii).    $\blacksquare$

Lemma 3.5 guarantees that for a fixed $\varvec{p}$, $\Vert \varvec{u}_n(\varvec{p},A_n)\Vert _1$ strictly increases from 0 to $\varGamma _n$ on the interval $A_n \in [A^{-}_{n}(\varvec{p}),A^{+}_{n}(\varvec{p})]$. This implies that $\Vert \varvec{u}_n(\varvec{p},A)\Vert _1$ is invertible on $[A^{-}_{n}(\varvec{p}),A^{+}_{n}(\varvec{p})]$, with the inverse denoted:

$$\begin{aligned} \mathscr {A}_n(\varvec{p},\cdot ): [0,\varGamma _n] \rightarrow [A^{-}_{n}(\varvec{p}),A^{+}_{n}(\varvec{p})]. \end{aligned}$$

(3.51)

It follows that $\mathscr {A}_n(\varvec{p},\omega )$ is strictly increasing with $\omega$ and,

$$\begin{aligned} \mathscr {A}_n(\varvec{p},{\omega }) = A_n \quad \Longleftrightarrow \quad \Vert \varvec{u}_n(\varvec{p},A_n)\Vert _1={\omega }. \end{aligned}$$

(3.52)

The charging strategy that satisfies (3.44) and delivers total energy of ${\omega }$ will be denoted by $\varvec{u}_n(\varvec{p},\mathscr {A}_n(\varvec{p},{\omega }))$, and therefore $\Vert \varvec{u}_n(\varvec{p},\mathscr {A}_n(\varvec{p},\omega ))\Vert _1 = \omega$.

Because of the non-negativity constraint on $u_{nt}$ and the corresponding complementary slackness requirement from Lemma 3.4, it is not straightforward to determine a closed form expression for the function $\mathscr {A}_n(\varvec{p},\omega )$. A valuable property of $\mathscr {A}_n(\varvec{p},\omega )$ is, however, established in the following lemma.

Lemma 3.6

For any fixed price profile $\varvec{p}$,

$$\begin{aligned} \frac{d}{d\omega } {F}^*_{n}(\varvec{p}, \omega ) = \mathscr {A}_n(\varvec{p},\omega ), \quad \text { with } \omega \in [0,\varGamma _n], \end{aligned}$$

(3.53)

where,

$$\begin{aligned} F^*_{n}(\varvec{p}, \omega ) \triangleq \min _{\varvec{u}_n \in \mathscr {U}_n(\omega )} F_n(\varvec{u}_n; \varvec{p}). \end{aligned}$$

(3.54)

Proof

From (3.47), $A_n$ is the Lagrangian multiplier associated with the constraint $\Vert \varvec{u}_n\Vert _1=\omega$. Based on duality theory [28], the sensitivity of the minimum value $F^*_{n}(\varvec{p}, \omega )$ with respect to changes in $\omega$ is therefore given by $A_n$. The result follows from (3.52).    $\blacksquare$

It is now possible to establish the optimal charging strategy for a given $\varvec{p}$. This is achieved in the following theorem, which implicitly determines the optimal value for $\omega$ in the process.

Theorem 3.3

Assume $h_n(\omega )$ is continuously differentiable, increasing and concave on $0 \le \omega \le \varGamma _n$. Define,

$$\begin{aligned} \widehat{h}_n(\varvec{p},\omega ) \triangleq \mathscr {A}_n(\varvec{p},\omega ) - h'_n(\omega ) \end{aligned}$$

(3.55)

with $\mathscr {A}_n(\varvec{p},\cdot )$ given by (3.51), and

$$\begin{aligned} A^*_n(\varvec{p}) = {\left\{ \begin{array}{ll} \mathscr {A}_n(\varvec{p},\varGamma _n), &{} \text { if } \widehat{h}_n(\varvec{p},\varGamma _n) \le 0 \\ \mathscr {A}_n(\varvec{p},0), &{} \text { if } \widehat{h}_n(\varvec{p},0) \ge 0\\ \mathscr {A}_n(\varvec{p},\omega ^*), &{} \text { if } \widehat{h}_n(\varvec{p},\omega ^*) =0 \end{array}\right. } \end{aligned}$$

(3.56)

where $0<\omega ^{*}<\varGamma _n$. Then the charging strategy $\varvec{u}_n(\varvec{p},A^*_n(\varvec{p}))$ defined in (3.44) uniquely minimizes the cost function (3.42) with respect to a given $\varvec{p}$, i.e. $\varvec{u}_n^*(\varvec{p}) = \varvec{u}_n(\varvec{p},A^*_n(\varvec{p}))$.

Proof

Recall that

$$ J_{n}(\varvec{u}_n;\varvec{p}) = F_n (\varvec{u}_n; \varvec{p}) - h_n(\omega ), $$

for all $\varvec{u}_n \in \mathscr {U}_n(\omega )$. Then,

$$\begin{aligned} \min _{\varvec{u}_n \in \mathscr {U}_n} J_{n}(\varvec{u}_n;\varvec{p})&= \min _{\varvec{u}_n \in \mathscr {U}_n} \big \{ F_n (\varvec{u}_n; \varvec{p}) - h_n(\Vert \varvec{u}_n\Vert _1) \big \} \\&= F^*_{n} (\varvec{p},\omega ^*) - h_n(\omega ^*), \end{aligned}$$

where $\omega ^*$ is the total charging energy that minimizes the cost function $J_{n}(\cdot ;\varvec{p})$. Note that $\omega ^*$ is constrained to $0\le \omega ^* \le \varGamma _n$. For $0< \omega ^* < \varGamma _n$, the optimal energy demand $\omega ^*$ is implicitly defined by the stationarity condition,

$$\begin{aligned} \big . \frac{d}{d\omega }\big (F^*_{n} (\varvec{p},\omega ) - h_n(\omega ) \big )\big |_{\omega =\omega ^*} = \widehat{h}_n(\varvec{p},\omega ^*) =0 \end{aligned}$$

(3.57)

where Lemma 3.6 has been used to establish the first equality. Moreover, $h'_n(\omega )$ decreases on $\omega$, since $h_n$ is assumed to be concave, and $\mathscr {A}_n(\varvec{p},\omega )$ is strictly increasing with $\omega$. Therefore, if a solution for (3.57) exists over $0< \omega ^* < \varGamma _n$, then it must be unique.

If (3.57) cannot be satisfied for $0< \omega < \varGamma _n$ then no stationary point exists over that open interval. Consequently, the cost $F^*_{n} (\varvec{p},\omega ) - h_n(\omega )$ must exhibit monotonic behavior over $0< \omega < \varGamma _n$. If the cost strictly increases with $\omega$, so $\mathscr {A}_n(\varvec{p}, \omega ) - h'_n(\omega ) = \widehat{h}_n(\varvec{p},\omega ) > 0$ for $0< \omega < \varGamma _n$, then the minimum cost solution will occur at the lower end, $\omega ^* = 0$. Similarly, if the cost is strictly decreasing with $\omega$, so the derivative $\widehat{h}_n(\varvec{p},\omega ) < 0$ over $0< \omega < \varGamma _n$, then the minimum cost solution will occur at the upper end, $\omega ^* = \varGamma _n$.    $\blacksquare$

3.4.4 Price Profile Update Mechanism

If the price profile $\varvec{p}$ is equal to the optimal (efficient) generation marginal cost $\varvec{p}^{**}$ given by (3.39), then the collection of PEV charging strategies $\varvec{u}^*(\varvec{p}) \equiv \big ( \varvec{u}_n(\varvec{p},A_n^*(\varvec{p})), n \in \mathscr {N} \big )$ given by (3.44) and (3.56) would be efficient. However, this optimal price $\varvec{p}^{**}$ cannot be determined a priori. Hence there is a need for an update mechanism that guarantees convergence of the price profile to the efficient marginal cost $\varvec{p}^{**}$. Consider the scheme,

$$\begin{aligned} p^+_t(\varvec{p}) = p_t + \eta \left( {c}' \left( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{p}) \right) - p_t \right) , \quad t \in \mathscr {T} \end{aligned}$$

(3.58)

where $\eta >0$ is a fixed parameter, and $\varvec{u}^*_n (\varvec{p})$, defined in (3.43), is the optimal charging strategy for PEV n with respect to $\varvec{p}$.

Given a system price profile $\varvec{p}$ over the charging horizon $\mathscr {T}$, if $p_t$ is lower than the generation marginal cost ${c}' \big ( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{p}) \big )$ at time t, the system will set a higher price $p^+_t$ to encourage PEVs to reduce their charging demand at that time. Likewise, if $p_t$ is higher than the marginal cost ${c}'$, the system will set a lower system price $p^+_t$ to encourage PEVs to increase their charging demand at that time.

Notice that the price update mechanism (3.58) can be written in the form,

$$ \varvec{p}^+ = (1 - \eta ) \varvec{p} + \eta \mathscr {P}(\varvec{p}), $$

where

$$ \mathscr {P}_t(\varvec{p}) = {c}' \left( d_t + \sum _{n \in \mathscr {N}} u_{nt}^*(\varvec{p}) \right) , \quad t \in \mathscr {T}. $$

This price update iteration takes the form of the Krasnoselskij iteration [35, 36], and is therefore guaranteed to converge to a fixed point of $\mathscr {P}(\cdot )$ for any $\eta \in (0, 1)$ if $\mathscr {P}(\cdot )$ is non-expansive. Corollary 3.2 establishes a more general sufficient condition under which the system converges to the unique price profile $\varvec{p}^{**}$ which is the efficient marginal cost.

3.4.5 Decentralized Coordination of PEV Charging

It is now possible to formalize a decentralized coordination algorithm for determining the optimal charging strategy for a population of PEVs.

Algorithm 3.2

(Decentralized coordination method)

• Specify the aggregate base demand $\varvec{d}$;

• Define an $\varepsilon _\mathrm{stop}$ to terminate iterations;

• Initialize $\varepsilon > \varepsilon _\mathrm{stop}$ and an initial price profile $\varvec{p}^{(0)}$;

• Set $k=0$;

• While $\varepsilon > \varepsilon _\mathrm{stop}$

  • Determine the optimal charging profile $\varvec{u}^{(k+1)}_n$ w.r.t. $\varvec{p}^{(k)}$ for all $n \in \mathscr {N}$ PEVs simultaneously by minimizing the individual cost function (3.42),

    $$ \varvec{u}^{(k+1)}_n ( \varvec{p}^{(k)} ) \triangleq \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {argmin}} \, \left\{ \sum _{t \in \mathscr {T}} p_t^{(k)} u_{nt} - v_n(\varvec{u}_n) \right\} ; $$

  • Determine $\varvec{p}^{(k+1)}$ from $\varvec{p}^{(k)}$ and $\varvec{u}^{(k+1)} \big ( \varvec{p}^{(k)} \big )$ using (3.58),

    $$ p^{(k+1)}_t = {p}^{(k)}_t + \eta \left( {c}' \left( d_t + \sum _{n \in \mathscr {N}} u^{(k+1)}_{nt} \right) - p^{(k)}_t \right) , $$

    for all $t \in \mathscr {T}$;

  • Update $\varepsilon := \Vert \varvec{p}^{(k+1)} - \varvec{p}^{(k)} \Vert _1$;

  • Update $k := k+1$.    $\blacksquare$

If Algorithm 3.2 converges, this decentralized process will achieve the efficient solution. Iterations could, however, be oscillatory or even divergent. In order to establish convergence, it is useful to define $\nu _{nt}$ as the Lipschitz constant for the function $[g'_{nt}]^{-1} (\cdot )$ over the interval $[g'_{nt}(0), g'_{nt}(\varGamma _n)]$, with

$$\begin{aligned} \nu = \underset{n \in \mathscr {N},t \in \mathscr {T}}{\max } \; \nu _{nt}, \end{aligned}$$

(3.59)

and to define $\kappa$ as the Lipschitz constant for ${c}'(\cdot )$ over the typical range in the total demand. The following intermediate result is also required.

Lemma 3.7

Assume the terminal valuation function $h_n$ is increasing and strictly concave. Then,

$$\begin{aligned} \Vert \varvec{u}^*_n(\varvec{p}) - \varvec{u}^*_n(\varvec{\rho })\Vert _1 \le 2 \nu \Vert \varvec{p} - \varvec{\rho }\Vert _1 \end{aligned}$$

(3.60)

where $\Vert .\Vert _1$ denotes the $l_1$ norm of the associated vector.

The proof of Lemma 3.7 is given in Appendix 3.7.1. $\blacksquare$

It is now possible to establish the convergence properties of Algorithm 3.2, and hence of the decentralized coordination process.

Corollary 3.2

(Convergence of Algorithm 3.2) Suppose $\alpha \equiv |1-\eta | + 2 N \kappa \nu \eta < 1$ and consider any initial charging price $\varvec{p}^{(0)}$. Then Algorithm 3.2 converges to the efficient solution $\varvec{u}^{**}$ which is specified in (3.39). Moreover, for any $\varepsilon > 0$, the system converges to a price profile $\varvec{p}$, such that $\Vert \varvec{p} - \varvec{p}^{**}\Vert _1 \le \varepsilon$, in $K(\varepsilon )$ iterations, with

$$\begin{aligned} K(\varepsilon ) = \displaystyle \left\lceil \frac{1}{\mathrm {ln}(\alpha )} \Big ( \mathrm {ln}(\varepsilon ) - \mathrm {ln}(T) - \mathrm {ln}(\varrho _{max}) \Big ) \right\rceil , \end{aligned}$$

(3.61)

where $\varrho _{max}$ denotes the maximum possible price, and $\lceil x \rceil$ represents the minimal integer value larger than or equal to x.

Remarks:

• (i) In practice, convergence to the desired tolerance $\varepsilon$ requires many fewer iterations than the upper bound established in (3.61). Typical convergence behavior is illustrated in Sect. 3.5.

• (ii) The upper bound on the iteration count, $K(\varepsilon )$, is of order $O(|\mathrm {ln}(\varepsilon )|)$, and is independent of the size of the PEV population. The choice of $\alpha$ is influenced by the PEV population size, with the condition $\alpha < 1$ requiring that $N < \frac{1}{2 \kappa \nu }$. Notice from (3.36), though, that $\kappa (N) \in O(\frac{1}{N^2})$ because the generation cost ${c}(\cdot )$ must remain finite as the PEV population N grows. Therefore this necessary condition on N is not restrictive.

Proof of Corollary 3.2.

Consider a pair of price profiles $\varvec{p}$ and $\varvec{\rho }$, and the respective updated price profiles $\varvec{p}^+$ and $\varvec{\rho }^+$ given by (3.58). Then,

$$\begin{aligned} \Vert&\varvec{p}^+ - \varvec{\rho }^+\Vert _1 \nonumber \\&= \sum _{t \in \mathscr {T}} \left| \left\{ p_t + \eta \left( {c}' \left( d_t+\sum _{n\in \mathscr {N}} u^*_{nt}(\varvec{p}) \right) - p_t \right) \right\} \right. \nonumber \\&\qquad \left. - \left\{ {\varrho }_t + \eta \left( {c}' \left( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{\rho }) \right) - {\varrho }_t \right) \right\} \right| \nonumber \\&= \sum _{t \in \mathscr {T}} \left| \eta \left( {c}' \left( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{p}) \right) - {c}' \left( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{\rho }) \right) \right) \right. \nonumber \\&\qquad + (1-\eta ) ({p}_t - {\varrho }_t) \Bigg | \nonumber \\&\le \eta \sum _{t \in \mathscr {T}} \left| {c}' \left( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{p}) \right) - {c}' \left( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{\rho }) \right) \right| \nonumber \\&\qquad + |1-\eta | \times \Vert \varvec{p} - \varvec{\rho }\Vert _1 . \end{aligned}$$

(3.62)

Also, given the definition of $\kappa$, it follows that,

$$\begin{aligned} \sum _{t \in \mathscr {T}} \Bigg | {c}' \Bigg ( d_t&+ \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{p}) \Bigg ) - {c}' \Bigg (d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{\rho }) \Bigg ) \Bigg | \nonumber \\&\le \kappa \sum _{t \in \mathscr {T}} \Big | \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{p}) - \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{\rho }) \Big | \nonumber \\&\le \kappa \sum _{t \in \mathscr {T}} \sum _{n \in \mathscr {N}} \left| {u}^*_{nt}(\varvec{p}) - {u}^*_{nt}(\varvec{\rho }) \right| \nonumber \\&= \kappa \sum _{n \in \mathscr {N}} \sum _{t \in \mathscr {T}} \left| {u}^*_{nt}(\varvec{p}) - {u}^*_{nt}(\varvec{\rho }) \right| \nonumber \\&= \kappa \sum _{n \in \mathscr {N}} \Vert \varvec{u}^*_{n}(\varvec{p}) - \varvec{u}^*_{n}(\varvec{\rho }) \Vert _1 \nonumber \\&\le 2 \kappa \nu \sum _{n \in \mathscr {N}} \Vert \varvec{p} - \varvec{\rho } \Vert _1 \end{aligned}$$

(3.63)

$$\begin{aligned}&= 2 N \kappa \nu \Vert \varvec{p} - \varvec{\rho } \Vert _1 \end{aligned}$$

(3.64)

where the inequality (3.63) invokes Lemma 3.7. Inequalities (3.62) and (3.64) together imply,

$$\begin{aligned} \Vert \varvec{p}^+ - \varvec{\rho }^+\Vert _1 \le (|1-\eta | + 2 N \kappa \nu \eta ) \Vert \varvec{p} - \varvec{\rho }\Vert _1. \end{aligned}$$

(3.65)

If $|1-\eta | + 2 N \kappa \nu \eta < 1$, then,

$$\begin{aligned} \Vert \varvec{p}^+ - \varvec{\rho }^+\Vert _1 < \Vert \varvec{p} - \varvec{\rho }\Vert _1, \end{aligned}$$

(3.66)

so the price update operator $\varvec{p}^+(\varvec{p})$ specified in (3.58) is a contraction map. Hence by the contraction mapping theorem [30], the system price $\varvec{p}^{(k)}$ converges to a unique price profile $\varvec{p}^*$ from any initial price profile $\varvec{p}^{(0)}$. At the converged price, $\varvec{p}^+(\varvec{p}^*) = \varvec{p}^*$, and so (3.58) implies,

$$\begin{aligned} {c}'\left( d_t + \sum _{n \in \mathscr {N}} u^*_{nt}(\varvec{p}^*) \right) = {p}^*_t, \quad \text { for all } t \in \mathscr {T}. \end{aligned}$$

(3.67)

The price converges to the generation marginal cost over the charging horizon.

It will now be shown that $\varvec{u}^*(\varvec{p}^*)$ is the efficient (socially optimal) charging strategy that minimizes the central optimization problem (3.37). Let $\omega ^* = \Vert \varvec{u}^*(\varvec{p}^*)\Vert _1$. If $0< \omega ^* < \varGamma _n$ then by Theorem 3.3, $A_n^*(\varvec{p}^*) = h'_n(\omega ^*)$. According to Lemma 3.4,

$$ p^*_t + g'_{nt}(u^*_{nt}(\varvec{p}^*)) - h'_n(\omega ^*) {\left\{ \begin{array}{ll} = 0, &{} \text {if } {u}^*_{nt}(\varvec{p}^*) > 0 \\ \ge 0, &{} \text {otherwise } \end{array}\right. } $$

so (3.39) is satisfied.

If $\omega ^*=0$ then by Theorem 3.3, $A_n^*(\varvec{p}^*) \ge h'_n(\omega ^*)$. Also, in this case, $u_{nt}^*(\varvec{p}^*)=0$ for all $t \in \mathscr {T}$. Therefore, from Lemma 3.4,

$$ p^*_t + g'_{nt}(u^*_{nt}(\varvec{p}^*)) \ge A_n^*(\varvec{p}^*) \ge h'_n(\omega ^*) $$

which again satisfies (3.39).

Suppose $\omega ^* = \varGamma _n$, then by Theorem 3.3, $A_n^*(\varvec{p}^*) \le h'_n(\omega ^*) = 0$. There must exist at least one instant $t \in \mathscr {T}_n$ where $u_{nt}^* > 0$. At such an instant, Lemma 3.4 implies,

$$\begin{aligned} p^*_t + g'_{nt}(u^*_{nt}(\varvec{p}^*)) - A_n^*(\varvec{p}^*)&=0 \\ \Rightarrow \quad p^*_t + g'_{nt}(u^*_{nt}(\varvec{p}^*))&\le 0 \end{aligned}$$

and so $p_t^*<0$ because $g'_{nt}(u^*_{nt}(\varvec{p}^*))>0$ according to Assumptions 3.1 and 3.3. However this is not possible, as Assumption 3.2 ensures that $p_t^*>0$ for all $t \in \mathscr {T}$. Hence, $\omega ^*<\varGamma _n$, which is consistent with the efficient solution.

Therefore the collection of optimal PEV charging strategies is efficient, $\varvec{u}^* = \varvec{u}^{**}$, with respect to the converged price $\varvec{p}^{*} = \varvec{p}^{**}$.

Given $\alpha \equiv |1-\eta | + 2 N \kappa \nu \eta < 1$, (3.65) implies,

$$ \Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1 \le \alpha ^k \Vert \varvec{p}^{(0)}-\varvec{p}^{**}\Vert _1. $$

With ${p}^{(0)}_t, p^{**}_t \in [0, \varrho _{max}]$ for all $t \in \mathscr {T}$, this gives,

$$ \Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1 \le \alpha ^k T \varrho _{max}, $$

which implies that $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1 \le \varepsilon$ for k satisfying (3.61). $\blacksquare$

3.5 Numerical Illustrations

3.5.1 Convergence

This section provides an illustration of Algorithm 3.2 using parameter values that match those of Sect. 3.4.2. For the generation cost function (3.40), the marginal cost is given by,

$$ p({D}_t) = {c}'({D}_t) = 5.8 \times 10^{-7} {D}_t + 0.06. $$

The corresponding Lipschitz constant is $\kappa = 5.8 \times 10^{-7}$. From (3.41), the Lipschitz constant for $[g'_{nt}]^{-1}(\cdot )$ is $\nu = 1/0.006 = 166.7$. Given these values,

$$ \alpha \equiv |1-\eta | + 2 N \kappa \nu \eta = |1 - \eta | + 0.967 \eta , $$

which is less than 1 whenever $0< \eta < 1.017$. Thus by Corollary 3.2, the system is guaranteed to converge to the efficient solution for all $\eta \in (0, 1.017)$. It is straightforward to show that $\alpha$ is a minimum when $\eta = 1$, giving $\alpha = 0.967 < 1$.

With $\alpha = 0.967$, $T=24$ and $\varrho _{max}=0.3$, Algorithm 3.2 will converge to a price profile $\varvec{p}^*$, such that $\Vert \varvec{p}^*-\varvec{p}^{**}\Vert _1 < \varepsilon = 0.0001$, in less than $K=334$ iterations, according to (3.61). Figure 3.14 shows the evolution of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$, with $\eta =1$, from an initial price profile ${p}^{(0)}_t = {c}'(d_t)$ for all $t \in \mathscr {T}$, i.e. zero charging load. It can be seen that convergence to the desired tolerance is achieved in about 10 iterations, which is much less than the theoretical upper bound of 334.

Fig. 3.14

Convergence of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$ for Algorithm 3.2, with $\eta =1$

As $\eta$ increases over the range $0 < \eta \le 1$, the value of $\alpha$ decreases, with (3.65) suggesting faster convergence of Algorithm 3.2. Figure 3.15 shows this to be the case. Further increasing $\eta$ results in $\alpha$ increasing, with convergence only guaranteed while $\eta < 1.017$. Nevertheless, as shown in Fig. 3.16, the algorithm may still converge even when this sufficient condition is violated. It is also apparent, however, that larger values of $\eta$ result in non-convergence, with oscillations occurring when $\eta =2$.

Fig. 3.15

Evolution of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$ for values of $\eta$ that satisfy the sufficient condition of Corollary 3.2

Fig. 3.16

Evolution of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$, with the sufficient condition of Corollary 3.2 not satisfied for $\eta > 1.017$

The price profile updates achieved by Algorithm 3.2 are shown in Fig. 3.17, while Fig. 3.18 shows the corresponding total aggregate demand at each iteration. Note that the converged case in Fig. 3.18 corresponds exactly to the socially optimal strategy in Fig. 3.8.

Fig. 3.17

Price profile updates achieved by Algorithm 3.2

Fig. 3.18

Total demand at each iteration of Algorithm 3.2

3.5.2 Heterogeneity

To consider the effect of heterogeneity in the PEV population, suppose that the initial value of each PEV’s SoC, $soc_{n0}$, satisfies a Gaussian distribution $N(\hat{\mu },\hat{\gamma })$ where $\hat{\mu } = 50\%$ and $\hat{\gamma } = 0.1$, which is consistent with [31, 32]. The updates of total aggregate demands are shown in Figs. 3.19 and 3.20 for charging interval in summer and spring seasons respectively.

Fig. 3.19

Total demand at each iteration of Algorithm 3.2 for a heterogeneous PEV population

Fig. 3.20

Total demand at each iteration of Algorithm 3.2 for a heterogeneous PEV population with base demand typical of a spring day

By adopting the proposed decentralized algorithm, the process converges to the efficient solution in a few iterations. Moreover, Fig. 3.21 illustrates the converged charging strategies for a sample of the heterogeneous PEVs.

Fig. 3.21

Optimal charging strategies for individuals within a heterogeneous PEV population

3.5.3 Comparison with Other Methods

The relative performance of Algorithm 3.2 will be illustrated through a comparison with the optimal decentralized charging algorithm of [16] (referred to as GTL). The optimal charging strategy attained by the GTL algorithm is valley filling since the objective is to minimize the electricity cost over the charging horizon, and there are no battery degradation or demand costs involved. Also, GTL assumes that all PEVs are fully charged by the end of the charging period.

In order to provide a meaningful comparison with Algorithm 3.2, the GTL algorithm must be modified to take into account PEV utility (3.34). Accordingly, the optimal charging profile of each PEV is given by,

$$\begin{aligned} \varvec{u}^{(k+1)}_n ( \varvec{u}_n^{(k)}, \varvec{p}^{(k)} )&\triangleq \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {argmin}} \sum _{t \in \mathscr {T}} \Big \{ p_t^{(k)} u_{nt} + g_{nt}(u_{nt}) \Big \} \nonumber \\&\quad + \frac{1}{2} \Vert \varvec{u}_n - \varvec{u}_n^{(k)}\Vert ^2 - h_n \left( \sum _{t \in \mathscr {T}} u_{nt} \right) . \end{aligned}$$

(3.68)

The parameter values from Sect. 3.5.1 have again been used for generation cost and PEV charging characteristics.

Fig. 3.22

Evolution of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$ for the GTL algorithm for different values of $\gamma$

Fig. 3.23

Evolution of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$ for the GTL algorithm without battery and demand costs

The GTL algorithm defines a parameter $\gamma$ which governs the update process at each iteration,

$$ p^{(k+1)}_t = \gamma {c}' \Big ( d_t + \sum _{n \in \mathscr {N}} u^{(k+1)}_{nt} \Big ). $$

Figure 3.22 shows the evolution of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$ for different values of $\gamma$. Compared with Fig. 3.15, Algorithm 3.2 provides faster convergence than the GTL algorithm.

The analysis in [16] shows that convergence to the optimal strategy is guaranteed as the number of iterations approaches infinity. In contrast, Corollary 3.2 guarantees that Algorithm 3.2 will converge to an $\varepsilon$-optimal strategy in no more than $K(\varepsilon )$ iterations.

Next, the influence of the battery and demand charges on convergence performance will be demonstrated. This is achieved by setting $g_{nt}(\cdot )$ to zero in (3.68). Figure 3.23 shows the evolution of $\Vert \varvec{p}^{(k)}-\varvec{p}^{**}\Vert _1$ for the GTL algorithm in this case. Comparison with Fig. 3.22 suggests that inclusion of the battery and demand costs tends to improve the convergence performance.

3.6 Conclusions

With the increasing of PEVs, they may have significant impacts on the power grid. And some key characteristics of chemical batteries, like the state of health, the growth of resistance, and the cycle life, are effected by the charging behaviors. In this chapter, the cost function underpinning the strategy establishes a tradeoff between the cost of energy and costs associated with battery degradation. It also introduces a charge that penalizes high demand, thereby mitigating occurrences of high coincident charging on local distribution grids.

A price-based strategy has been formulated to coordinate the charging strategies of large-scale PEVs in a decentralized way with the consideration of the battery degradation cost. The optimal charging strategy avoids the damage caused by the excessive charging power of the valley filling strategy to the grid and the batteries of PEVs. A decentralized scheme is proposed where all PEVs simultaneously update their optimal charging strategies with respect to a common price profile, and then the price profile is updated using these latest proposed charging strategies. The chapter establishes sufficient conditions that ensure this iterative process converges to the unique efficient collection of charging strategies. At convergence, the price profile coincides with the generator marginal cost. Simulation examples are studied to illustrate the results developed in this chapter.

3.7 Appendices

3.7.1 Proof of Lemma 3.7

Recall that $\varvec{u}^*_n(\varvec{p}), \varvec{u}^*_n(\varvec{\rho }) \in \mathscr {U}_n$ represent the optimal response of PEV n with respect to price profiles $\varvec{p}$ and $\varvec{\rho }$ respectively,

$$ \varvec{u}^*_n(\varvec{p}) \triangleq \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {min}} \, J_{n}(\varvec{u}_n; \varvec{p}), \quad \varvec{u}^*_n(\varvec{\rho }) \triangleq \underset{\varvec{u}_n \in \mathscr {U}_n}{\text {min}} \, J_{n}(\varvec{u}_n; \varvec{\rho }). $$

Also, by Theorem 3.3,

$$ \varvec{u}^*_n(\varvec{p}) = \varvec{u}_{n}(\varvec{p}, A^*_n(\varvec{p})), \quad \varvec{u}^*_n(\varvec{\rho }) = \varvec{u}_{n}(\varvec{\rho }, A^*_n(\varvec{\rho })). $$

For later analysis, it is useful to define another charging strategy for PEV n,

$$\begin{aligned} \varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) \equiv \varvec{u}_n(\varvec{\rho }, A^*_n(\varvec{p})), \end{aligned}$$

(3.69)

which is the charging strategy satisfying (3.44) with respect to $\varvec{\rho }$ and $A^*_n(\varvec{p})$. Note that $\varvec{\vartheta }_n(\varvec{p},\varvec{\rho })$ may not satisfy the admissibility constraint (3.1a). However, $\varvec{\vartheta }_n(\varvec{p},\varvec{\rho })$ will only be used as a medium to establish a relationship between the pair of admissible charging strategies $\varvec{u}^*_n(\varvec{p})$ and $\varvec{u}^*_n(\varvec{\rho })$.

It is sufficient to show (3.60) if the following inequalities hold:

$$\begin{aligned}&|{u}^*_{nt}(\varvec{p}) - \vartheta _{nt}(\varvec{p},\varvec{\rho })| \le \nu |p_t - {\varrho }_t|, \quad \forall t \in \mathscr {T}, \end{aligned}$$

(3.70)

$$\begin{aligned}&\Vert \varvec{u}^*_n(\varvec{p}) - \varvec{u}^*_n(\varvec{\rho })\Vert _1 \le 2 \Vert \varvec{u}^*_n(\varvec{p}) - \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1. \end{aligned}$$

(3.71)

These relationships will be verified below respectively.

Verifying (3.70)

Firstly, consider the case where ${u}^*_{nt}(\varvec{p}), {\vartheta }_{nt}(\varvec{p},\varvec{\rho }) > 0$. It follows from (3.44), Theorem 3.3 and (3.69) that:

$$\begin{aligned} |{u}^*_{nt}(\varvec{p})&- {\vartheta }_{nt}(\varvec{p},\varvec{\rho })| \\&= \big | [g'_{nt}]^{-1} (A^*_n(\varvec{p})-p_t) - [g'_{nt}]^{-1} (A^*_n(\varvec{p})-{\varrho }_t) \big | \\&\le \nu |p_t - {\varrho }_t|, \end{aligned}$$

where the inequality holds by the specification of $\nu$ given in (3.59). Following similar analysis, (3.70) holds for the other cases.

Verifying (3.71)

The proof of (3.71) will be established by considering three cases.

Case 1: $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 = \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$

Let $\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = \bar{\omega }$ so that,

$$ \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 = \Vert \varvec{u}_n(\varvec{\rho }, A^*_n(\varvec{p}))\Vert _1 = \bar{\omega }. $$

Then (3.52) implies $\mathscr {A}(\varvec{\rho },\bar{\omega }) = A_n^*(\varvec{p})$. From (3.55),

$$\begin{aligned} \widehat{h}_n(\varvec{\rho },\bar{\omega }) = \mathscr {A}(\varvec{\rho }, \bar{\omega }) - h'_n(\bar{\omega }) = A_n^*(\varvec{p}) - h'_n(\bar{\omega }) = \widehat{h}_n(\varvec{p},\bar{\omega }). \end{aligned}$$

It may be concluded from (3.56) that $A_n^*(\varvec{\rho }) = A_n^*(\varvec{p})$, and so $\varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) = \varvec{u}_n(\varvec{\rho }, A^*_n(\varvec{\rho })) = \varvec{u}_n^*(\varvec{\rho })$. Consequently,

$$\begin{aligned} \Vert \varvec{u}^*_n(\varvec{p}) - \varvec{u}^*_n(\varvec{\rho })\Vert _1&= \Vert \varvec{u}^*_n(\varvec{p}) - \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 \le 2 \Vert \varvec{u}^*_n(\varvec{p}) - \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1. \end{aligned}$$

Case 2: $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 > \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$

The first step is to show that,

$$\begin{aligned} \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{p})\Vert _1. \end{aligned}$$

(3.72)

This is achieved using proof by contradiction. Three subcases will be considered: (2A) $\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = \varGamma _n$, (2B) $0< \Vert \varvec{u}^*_n(\varvec{p})\Vert _1 < \varGamma _n$, and (2C) $\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = 0$.

 

Case (2A): :

$\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = \varGamma _n$. The charging strategy $\varvec{\vartheta }_n(\varvec{p},\varvec{\rho })$ is not admissible since the total charging demand $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 > \varGamma _n$. Because $\varvec{u}^*_n(\varvec{\rho })$ must be admissible, $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1$ cannot exceed $\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = \varGamma _n$. Therefore $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 < \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1$ which implies, by Lemma 3.5, that

$$\begin{aligned} {A}^*_n(\varvec{\rho }) < {A}^*_n(\varvec{p}). \end{aligned}$$

(3.73)

To show that $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 = \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$, assume $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 < \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$. Then by (3.56) and the assumed condition $\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = \varGamma _n$, it follows that,

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1}&\le {A}^*_n(\varvec{\rho }) \end{aligned}$$

(3.74a)

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1}&\ge {A}^*_n(\varvec{p}). \end{aligned}$$

(3.74b)

From (3.73) and (3.74),

$$ h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1} < h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1}, $$

which implies, by the concavity of $h_n$, that $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 > \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$. However this contradicts the assumption $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 < \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$. Hence,

$$ \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 > \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 = \Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = \varGamma _n. $$

Case (2B): :

$0< \Vert \varvec{u}^*_n(\varvec{p})\Vert _1 < \varGamma _n$. The desired result will be achieved by showing $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1$ in (2B.1), and $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$ in (2B.2).

(2B.1):

Suppose that,

$$\begin{aligned} \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 < \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1. \end{aligned}$$

(3.75)

Then $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1>0$ because $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 > 0$. Also, Lemma 3.5 implies ${A}^*_n(\varvec{p}) < {A}^*_n(\varvec{\rho })$. Together with $0< \Vert \varvec{u}^*_n(\varvec{p})\Vert _1 < \varGamma _n$ and (3.56), this gives,

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1}&= {A}^*_n(\varvec{p}) \nonumber \\&< A^*_n(\varvec{\rho }) \le h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1}. \end{aligned}$$

(3.76)

However, (3.75) and $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 > \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$ together imply $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 > |\varvec{u}^*_n(\varvec{p})\Vert _1$, and so,

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1} \le h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1} \end{aligned}$$

(3.77)

by the concavity of $h_n$. This contradicts (3.76), indicating that the original assumption (3.75) is incorrect. Hence, $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1$.

(2B.2):

Suppose that

$$\begin{aligned} \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 < \Vert \varvec{u}^*_n(\varvec{p})\Vert _1. \end{aligned}$$

(3.78)

Then $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1<\varGamma _n$, and $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 < \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1$ because $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 > \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$. Hence, by Lemma 3.5, ${A}^*_n(\varvec{\rho }) < {A}^*_n(\varvec{p})$. Together with $0<\Vert \varvec{u}^*_n(\varvec{p})\Vert _1<\varGamma _n$ and (3.56), this gives,

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1}&\le {A}^*_n(\varvec{\rho }) \nonumber \\&< {A}^*_n(\varvec{p}) = h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1}. \end{aligned}$$

(3.79)

However, given the concavity of $h_n$, (3.78) implies,

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1} > h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1}, \end{aligned}$$

(3.80)

which contradicts (3.79). Hence, $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$.

Case (2C): :

$\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = 0$. Because $\varvec{u}^*_n(\varvec{\rho })$ is an admissible strategy, $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{p})\Vert _1=0$. Suppose that,

$$\begin{aligned} \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 < \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1. \end{aligned}$$

(3.81)

Then $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1>0$. Also, Lemma 3.5 implies ${A}^*_n(\varvec{p}) < {A}^*_n(\varvec{\rho })$. Together with $\Vert \varvec{u}^*_n(\varvec{p})\Vert _1 = 0$ and (3.56), this gives,

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1}&\le {A}^*_n(\varvec{p}) \nonumber \\&< {A}^*_n(\varvec{\rho }) \le h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1}. \end{aligned}$$

(3.82)

However, (3.81) together with $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 > \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$ imply $\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1 > \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$, giving,

$$\begin{aligned} h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{p})\Vert _1} \ge h'_n(\omega )|_{\omega =\Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1}, \end{aligned}$$

(3.83)

by the concavity of $h_n$. This contradicts (3.82), indicating that the assumption (3.81) is incorrect. Hence, $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1$.

 

Case 2 summary: It has been shown in Cases (2A)–(2C) that (3.72) holds. Because $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 \ge \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1$, Lemma 3.5 implies that ${A}^*_n(\varvec{\rho }) \le {A}^*_n(\varvec{p})$, and therefore that,

$$\begin{aligned} 0 \le u^*_{nt}(\varvec{\rho }) \le {\vartheta }_{nt}(\varvec{p},\varvec{\rho }), \end{aligned}$$

(3.84)

for all $t \in \mathscr {T}$. Hence,

$$\begin{aligned} 0&\le \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) - \varvec{u}^*_n(\varvec{\rho })\Vert _1 \nonumber \\&= \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 - \Vert \varvec{u}^*_n(\varvec{\rho })\Vert _1&\qquad \text {by (3.84)} \nonumber \\&\le \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 - \Vert \varvec{u}^*_n(\varvec{p})\Vert _1&\qquad \text {by (3.72)} \nonumber \\&\le \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) - \varvec{u}^*_n(\varvec{p})\Vert _1 \end{aligned}$$

(3.85)

where the final inequality is due to the reverse triangle inequality. Therefore,

$$\begin{aligned} \Vert \varvec{u}^*_n(\varvec{p})&- \varvec{u}^*_n(\varvec{\rho })\Vert _1 \\&\le \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) - \varvec{u}^*_n(\varvec{p})\Vert _1 + \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) - \varvec{u}^*_n(\varvec{\rho })\Vert _1 \\&\le \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) - \varvec{u}^*_n(\varvec{p})\Vert _1 + \Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho }) - \varvec{u}^*_n(\varvec{p})\Vert _1 \\&= 2 \Vert \varvec{u}^*_n(\varvec{p}) - \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1, \end{aligned}$$

where the first inequality is due to the triangle inequality and the second is from (3.85).

Case 3: $\Vert \varvec{\vartheta }_n(\varvec{p},\varvec{\rho })\Vert _1 < \Vert \varvec{u}^*_n(\varvec{p})\Vert _1$

Following a similar approach to that adopted for Case 2, it can again be shown that (3.71) holds. $\blacksquare$