We have proposed a distributed algorithm that schedules EV charging to optimally fill the valley in electricity demand. The EV charging scheduling problem has been formulated as an optimal control problem, and a gradient projection type distributed algorithm is proposed accordingly to solve the problem. The algorithm is iterative. In each iteration, each EV updates its own charging profile according to the control signal broadcast by a centralized coordinator, and the coordinator guides their updates by altering the control signal. We have proved that the algorithm converges to optimal charging profiles irrespective of the specifications of EVs.
Chapter 3
Real-Time Distributed Load Control
Real-time load control has the potential to compensate for the random fluctuations of renewable generation, by reducing or shifting the power consumption of electric loads in response to generation fluctuations. In this chapter, a real-time distributed algorithm that schedules deferrable loads to reduce the deviation of aggregate load (load minus renewable generation) from some externally specified target is proposed. At every time step, the algorithm minimizes the expected deviation to go with updated predictions on renewable generation and deferrable load arrivals. We prove that suboptimality of the algorithm vanishes quickly as prediction horizon expands. Further, we evaluate the algorithm via trace-based simulations.
Literature A number of distributed deferrable load control algorithms have been proposed in the literature. Some of these algorithms are evaluated based on simulations [4,55,77], while some others have theoretical performance guarantees [41, 75]. In particular, the algorithm proposed in [75] is optimal if electric vehicles are identical, and the algorithm proposed in [41] achieves optimality even if electric vehicles are not identical.
However, the algorithms proposed in [4, 41, 55, 75, 77] do not consider the uncertainties in renew- able generation and deferrable load arrivals. In practice, only predictions of these quantities are known ahead of time, and the impact of prediction errors can be dramatic, e.g., see Figure 3.3.
Algorithms that consider the uncertainties in renewable generation or/and deferrable load arrivals have also been proposed in the literature. Most of these algorithms are evaluated with simulation- based studies, e.g., [22,31,34], while some are provided with analytic performance guarantees [28,72, 97]. For example, the algorithm proposed in [28] proposes an algorithm that achieves the optimal competitive ratio in the case where renewable generation is precisely known (and constant). The algorithm proposed in [72] also has certain worst-case performance guarantees.
While the algorithms proposed in [28, 72, 97] are analyzed with a “worst-case” perspective, this
chapter focuses on the “average-case” perspective to highlight the value of prediction.
Summary The goal of this chapter is to propose a real-time distributed deferrable load control al- gorithm that incorporates uncertain predictions on deferrable load arrivals and renewable generation.
In particular, contributions of the chapter are threefold.
First, wemodel renewable generation prediction evolution as a Wiener filtering process[106] (see Section 3.1.1), that is able to model any zero-mean and stationary prediction error.
Second, wepropose a real-time distributed algorithm (Algorithm 2) for deferrable load control in the presence of uncertainties (in Section 3.2), that reduces the deviation of aggregate load (load minus renewable generation) from some externally specified target profile. At every time step, Algorithm 2 minimizes the expected deviation to go with up-to-date predictions on deferrable load arrivals and renewable generation. A key technique is the introduction of apseudo load, that is simulated at the centralized coordinator to represent future deferrable load arrivals.
Third, weanalyze the expected deviation achieved by Algorithm 2 and provide trace-based simu- lations. In particular, the theorems in Section 3.3 characterize the impact of prediction inaccuracies on the expected deviation. As time horizon expands, the expected deviation approaches the optimal value (Corollary 3.7), and the performance gain of Algorithm 2 increases over the optimal open-loop control (Corollary 3.10, 3.11). Trace-based simulations in real-world settings are provided in Section 3.4 to validate the analytic results, highlighting that Algorithm 2 obtains a small suboptimality even under high uncertainties, and improves significantly over the optimal open-loop control.
3.1 Model Overview and Notation
This chapter studies the design and analysis of real-time deferrable load control algorithms to com- pensate for the random fluctuations of renewable generation. In the following we present a model for this scenario that includes renewable generation, non-deferrable loads, and deferrable loads, which are described in turn.
Throughout, we consider a discrete-time model over a finite time horizon. The time horizon is divided into T time slots of equal length and indexed 1, . . . , T. In practice, the time horizon could be one day and the length of a time slot could be 10 minutes.
3.1.1 Renewable Generation and Non-Deferrable Load
We aggregate renewable generation and non-deferrable load into a single process, termed thebase load b = {b(τ)}Tτ=1, that is defined as the difference between non-deferrable load and renewable generation. Renewable generation like wind and solar randomly fluctuates and is difficult to predict, thereforebis a stochastic process.
Figure 3.1: Diagram of the notation and structure of the model for base load, i.e., non-deferrable load minus renewable generation.
To model the uncertainty of base load, we use a causal filter based model described as follows, and illustrated in Figure 3.1. In particular, the base load at timeτis modeled as a random deviation δb={δb(τ)}Tτ=1 around its expectation ¯b={¯b(τ)}Tτ=1. The process ¯b is specified externally to the model, e.g., from historical data and weather report, and the processδb(τ) is further modeled as an uncorrelated sequence of identically distributed random variables e={e(τ)}Tτ=1 with mean 0 and varianceσ2, passing through a causal filter. Specifically, let f ={f(τ)}∞τ=−∞ denote the impulse response of this causal filter and assume thatf(0) = 1, thenf(τ) = 0 forτ <0 and
δb(τ) = XT s=1
e(s)f(τ−s), τ= 1, . . . , T.
Given the model above, at timet= 1, . . . , T, a prediction algorithm can estimate the sequencee(s) fors= 1, . . . , t, and predictsbas1
bt(τ) = ¯b(τ) + Xt s=1
e(s)f(τ−s), τ = 1, . . . , T. (3.1)
Note thatbt(τ) =b(τ) forτ= 1, . . . , tsincef is causal.
This model allows for non-stationary base load through the specification of ¯band a broad class of models for uncertainty in the base load viaf and e. In particular, two specific filtersf that we consider in detail later in the paper are:
Example 3.1. A filter with finite but flat impulse response, i.e., there exists ∆∈(0, T) such that
f(t) =
1 if 0≤t <∆ 0 otherwise;
Example 3.2. A filter with an infinite and exponentially decaying impulse response, i.e., there exists a∈(0,1) such that
f(t) =
at if t≥0 0 otherwise.
1This prediction algorithm is a Wiener filter [106].
These two filters provide simple but informative examples for our discussion in Section 3.3.
3.1.2 Deferrable Load
To model deferrable loads we consider a setting wherendeferrable loads arrive over the time horizon, each requiring a certain amount of electricity by a given deadline. Further, a real-time algorithm has imperfect information about the arrival times and sizes of these deferrable loads.
More specifically, we assume a total of n deferrable loads and label them in increasing order of their arrival times by 1, . . . , n, i.e., load i arrives no later than load i+ 1 for i = 1, . . . , n−1.
Further, letn(t) denote the number of loads that arrive before (or at) timet for t= 1, . . . , T and fixn(0) := 0. Thus, load 1, . . . , n(t) arrive before or at timetfort= 1, . . . , T andn(T) =n.
For each deferrable load, its arrival time and deadline, as well as other constraints on its power consumption, are captured via upper and lower bounds on its possible power consumption during each time. Specifically, the power consumption of deferrable loadiat timet,pi(t), must be between given lower and upper boundspi(t) andpi(t), i.e.,
pi(t)≤pi(t)≤pi(t), i= 1, . . . , n, t= 1, . . . , T. (3.2) These are specified externally to the model. For example, if an electric vehicle plugs in with Level II charging then its power consumption must be within [0,3.3]kW. However, if it is not plugged in (has either not arrived yet or has already departed) then its power consumption is 0kW, i.e., within [0,0]kW. Further, we assume that a deferrable load i must withdraw a fixed amount of energyPi
by its deadline, i.e.,
XT t=1
pi(t) =Pi, i= 1, . . . , n. (3.3) Finally, thendeferrable loads arrive randomly throughout the time horizon. Define
a(t) :=
n(t)X
i=n(t−1)+1
Pi (3.4)
as the total energy request of all deferrable loads that arrive at timetfort= 1, . . . , T. We assume that {a(t)}Tt=1 is a sequence of independent identically distributed random variables with mean λ and variances2. Further, let
A(t) :=
XT τ=t+1
a(τ) (3.5)
denote the total energy requested after timetfort= 1, . . . , T.
In summary, at timet= 1, . . . , T, a real-time algorithm has full information about the deferrable loads that have arrived, i.e.,pi,pi, andPi fori= 1, . . . , n(t), and knows the expectation of future
deferrable load total energy requestE(A(t)). However, a real-time algorithm has no other knowledge about deferrable loads that arrive after timet.
3.1.3 The Deferrable Load Control Problem
Recall that the objective of real-time deferrable load control is to compensate for the random fluc- tuations in renewable generation and non-deferrable load in order to “flatten” the aggregate load p0={p0(t)}Tt=1, which is defined as
p0(t) =b(t) + Xn i=1
pi(t), t= 1, . . . , T. (3.6)
In this chapter, we focus on minimizing thevariance of the aggregate loadp0,V(p0), as a measure of “flatness”, that is defined as
V(p0) = 1 T
XT t=1
p0(t)− 1 T
XT τ=1
p0(τ)
!2
. (3.7)
To summarize, the optimal deferrable load control (ODLC) problem is as follows. Let T :=
{1, . . . , T}, N :={0, . . . , n}, and defineN+:=N \{0}.
ODLC: min XT t=1
p0(t)− 1 T
XT τ=1
p0(τ)
!2
over pi(t), i∈ N, t∈ T s.t. p0(t) =b(t) +
Xn i=1
pi(t), t∈ T; (3.8a)
pi(t)≤pi(t)≤pi(t), i∈ N, t∈ T; (3.8b) XT
t=1
pi(t) =Pi, i∈ N. (3.8c)
In the above ODLC problem (3.8), the objective is simplyT times the variance of aggregate load, V(p0), and the constraints correspond to equations (3.6), (3.2), and (3.3), respectively. We chose V(p0) as the objective for ODLC because of its significance for microgrid operators [52]. However, additionally, it is proved in Chapter 2 that the optimal solution does not change if the objective function in (3.8) is replaced byf(p0) =P
t∈T U(p0(t)) whereU :R→Ris strictly convex. Hence, we can useV(p0) without loss of generality.