Page 474 - Decision Making Applications in Modern Power Systems

Multistage and decentralized operations of Chapter | 16 433

Objective
X N EV
min c a p a 1 γU c n ðp n Þ
ðÞ
n51
p a ;p n
Subject to
X N EV
p a 5 p n
n51
EV constraints

aggregator constraints
where c a p a denotes the aggregator’s cost function, and c n ðp n Þ is the cost
ðÞ
function of each vehicle n. As the aggregator consists of all the EV agents,
we have p a 5 P N EV p n as an additional constraint, that is, the aggregator
n51
load profile should be the summation of the load from all EVs. γ is the
weight factor for EV agent’s cost and is set to 1 in this case study. If we
model the aggregator as one additional agent together with all the EV agents,
the total number of agents, N, is equal to N EV 1 1. According to Refs.
[27,28], this problem can be modeled and rewritten as the exchange problem
using ADMM, which is as follows:
Objective

X N
min c n ðx n Þ
n51
p a ;p n
Subject to
X N EV
p a 5 p n
n51
EV constraints
aggregator constraints
Following this approach, each agent is able to compute the optimal solu-
tion of its own and exchange limited amount of information for each itera-
tion. The optimization problem at each stage is as follows:
ρ k
k 2
k
p k11 5 min c n p n 1 U:p n 2p 1P 1u : ð16:34Þ
ðÞ
n n 2
p n 2
k
k
where p is the optimal power profile for EV agent n at iteration k, and P
n
denotes the averaged power profiles of all EV agents at iteration k. ρ denotes
the augmented Lagrangian parameter. For the aggregator the cost func-
tion is modified to minimize the difference between the real-world
aggregator profile and the one from the day-ahead planning, that is,
D:¼½P 1ðÞ; P 2ðÞ; ... ; PTðފ. Thus the updated optimization problem for the
aggregator is as follows: