Monotonicity property for a two-node network

In this section, we consider a network with one feeder and two load nodes. We show the monotonicity property (see Definition 5.6.1) for the power allocation function

5.12. Monotonicity property for a two-node network 173

p(·). We focus on a single type of EVs and without loss of generality, we remove the polyhedral constraintspi ≤cmax. Recall thatF(z)is the feasible set of (5.6) for any

pointzand denote byΦthe feasible set of (5.7). Note thatΦis independent of the pointzandF(z)andΦare convex sets.

We start our analysis with the following definition.

Definition 5.12.1(Normal cone). Let S⊆Rn _{be a convex set. Its normal cone at point}

x∈S is defined as follows:

NS(x):={y∈Rn: for anys∈S, yT(s−x)≤0}.

Consider the vectorsz= (z1,z2)∈(0,∞)2andy= (y1,y2)∈(0,∞)2, and denote

the optimal solutions of (5.6) byp(z) = (p1(z),p2(z))andp(y) = (p1(y),p2(y)),

respectively. Note that the optimal solutions of (5.7) are given by Λ(z) = (Λ₁(z),Λ2(z)) = (z1p1(z),z2p2(z))

and

Λ(y) = (Λ1(y),Λ2(y)) = (y1p1(y),y2p2(y)).

We derive the monotonicity result by contradiction. Assume thatz ≤ ywith

p1(z) ≥ p1(y)and p2(z) < p2(y). Observe thatΛ2(z) < Λ2(y), which yields

Λ1(z)>Λ1(y), otherwise the pointΛ(z)can not be optimal. By the optimality of

the pointp(z), we have that

(z1u01(p1(z)),z2u20(p2(z)))∈ NF(z)(p(z)).

That is, for anys∈F(z),

z1u01(p1(z))(s1−p1(z)) +z2u02(p2(z))(s2−p2(z))≤0,

which can be equivalently written as

u0₁(p1(z))(S1−Λ1(z)) +u02(p2(z))(S2−Λ2(z))≤0,

whereS= (S1,S2):= (z1s1,z2s2)∈Φ. In other words,

(u0₁(p1(z)),u02(p2(z)))∈ NΦ(Λ(z)).

By the optimality of pointΛ(y), we have thatΛ(y)∈Φand hence

u0₁(p1(z))(Λ1(y)−Λ1(z)) +u02(p2(z))(Λ2(y)−Λ2(z))≤0,

which can be written as

u01(p1(z))(Λ1(z)−Λ1(y)) +u02(p2(z))(Λ2(z)−Λ2(y))≥0. (5.68)

By the properties of the utility functions, we have thatu0_i(·)is a strictly decreasing function. That is,u0₁(p1(z))≤u01(p1(y))andu02(p2(z))>u02(p2(y)). Applying the

last inequalities in (5.68), we have that

u0₁(p1(z))(Λ1(z)−Λ1(y))≥u02(p2(z))(Λ2(y)−Λ2(z)) >u0₂(p2(y))(Λ2(y)−Λ2(z)),

which yields

u0₁(p1(z))(Λ1(z)−Λ1(z)) +u02(p2(y))(Λ2(z)−Λ2(y))>0. (5.69)

Using the fact thatu0₁(p1(z))≤u01(q1),(Λ1(z)−Λ1(y))>0, and (5.69), we obtain u0₁(q1)(Λ1(z)−Λ1(y)) +u20(q2)(Λ2(z)−Λ2(y))≥u01(p1(z))(Λ1(z)−Λ1(y))

+u0₂(p2(z))(Λ2(z)−Λ2(y))>0.

In other words,(u0₁(p1(y)),u02(p2(y)))∈ N/ Φ(Λ(y))and this contradicts the optimal-

ity ofp(y). Using similar arguments, we derive the same contradiction if we assume

p1(z)<p1(y)andp2(z)≥ p2(y).

Remark 5.4. We make a comment on how the previous proof can be extended to the case of a line network with arbitrary number of nodes. First, it is enough to show the monotonicity property for the casey =z+e_i. The key property we need to show in order to apply the previous idea is:Λk(z)≥Λk(y), for k6=i.

Chapter 6

A Stochastic Network for Electric

Vehicle Charging: Explicit Results and

Case Study

Based on:A. Aveklouris, M. Vlasiou and B. Zwart. A stochastic resource-sharing network for electric vehicle charging.Accepted for publication in IEEE Transactions on Control of Network Systems, vol. 6, no 3 (2019), pp. 1050–1061. In this chapter, we continue our work on modeling a distribution grid used to charge electric vehicles. We show appealing properties of the stochastic model and we apply our results to design a control rule that maximizes the fraction of successful charges. Last, we complement our findings with a case study using real bus networks and with several special cases that allow for explicit computations.

6.1

Introduction

In this chapter, we continue our work on modeling a distribution grid used to charge electric vehicles. We work on the same model as introduced in Chapter 5 and we use exactly the same notation. The goal of the present section is to present a number of explicit examples, at the expense of making additional assumptions. Moreover, we present a case study based on the SCE 47-bus and SCE 56-bus networks where we explore results for both the aggregate system and the individual nodes.

When we replace the AC load flow model with the simpler linearized Distflow model [63], we obtain more explicit results, as the capacity set becomes polyhedral. For the class of weighted proportional fairness controls where the weights are chosen as function of the line resistances, we derive a fairness property. In this case, all EVs are charged at the same rate, independent of their location in the network, while keeping voltage drops bounded. When the weights are instead chosen equally, we can even derive an explicit formula for the invariant distribution of the original stochastic process. Specifically, we show that under certain assumptions, our network behaves like a multiclass processor-sharing queue. Such properties have proven quite fruitful in other areas of engineering, particularly in the analysis of computer systems [179], communication networks [168], and wireless networks [182].

Subsequently, we design control schemes in order to optimize the behavior of the system. By focusing on the weighted proportional fairness allocation and single type of EVs, we describe a procedure of choosing the weights such that the log-run fraction of EVs that gets successfully charged is maximized. Additionally, we illustrate our finding with numerical examples.

Last, in the remainder of this chapter we consider Poisson arrival processes with constant rates. This seems to be a reasonable assumption in the setting of EVs; for validations we refer to [183, 184]. Furthermore, methods to reduce the case of time-varying arrival rates to fixed arrival rates are explored in [185].

The chapter is organized as follows. In Section 6.2, we show how the weighted proportional fairness control mechanism leads to a product form property for the stochastic model. Further, we present a specific class of controls which lead to a fairness property. Section 6.3 presents a procedure to design a control rule that maximizes the fraction of successful charges. A case study using the SCE 47-bus (mostly) and SCE 56-bus networks is presented in Section 6.4. We present additional numerical results in Section 6.5 and all proofs are gathered in Section 6.6.

Notational Conventions. In the remainder of this chapter, we assume that the arrival rates areλij >0 and the lower bound of the voltage drops is the same for all

the nodes, i.e.,υi =υfor alli. Moreover, recall thatP(i)is the unique path from node ito feeder and defineγij:=

λij

ρi(ρi∧Ki),ri:=∑els∈P(i)rls, andδ=

W00−υ

2 .

6.2

Explicit results under the linearized Distflow model and a line