6.2 Demand Response Model
6.2.4 Information exchange models
Consumption behavior of other individuals at time h ljh can provide valuable infor-
mation about the consumption preferences gh in that time slot. This information
is of use to the consumer i in estimating consumption for the next time slot h+ 1
if the preferences of the users do not change in that time slot, that is, gh = gh+1. Otherwise, the information is not helpful in estimating behavior of others for time
slot h+ 1 because the change in the preference distribution is assumed to be in-
the assumption that the preferences remain the same for a given amount of time
starting from time h and lasting until there is a change in the consumption prefer-
ences, that is, gh =g0 := [g10, . . . , gN0] with prior distribution Pg0 for the time zone
T ={h∈ H:gh =g0}. If there is a change in the preference distribution we restart
the information exchange process. The prediction of renewable source term Pωh is
allowed to vary for h ∈ T. We use IΩ
ih to denote the set of information available to
consumer i at time slot h∈ T for the information exchange model Ω.
Private. The information specific to consumers is the merest possible when it consists
of the private preference gi0, that is, IihP ={gi0} for h∈ T.
Action Sharing. Power control schedulers are interconnected via a communication
network represented by a graph G(N,E) with its nodes representing the customers
N = {1, . . . , N} and edges belonging to the set E indicating possibility of commu-
nication. Customer i observes consumption levels of its neighbors in the network
Ni :={j ∈ N : (j, i)∈ E} after each time slot. The vector of i’s d(i) := #Ni neigh-
bors is denoted by [i1, . . . , id(i)]. Given the communication setup, the information of customeriat time slot h∈ T contains his self-preferencegi0 and the consumption of his neighbors up to time h−1, that is, IAS
ih ={gi0,{lNit}t=0,...,h−1} where we define
the actions of i’s neighbors at time t by lNit := [li1t, . . . , lid(i)t]. We assume that the
power consumption schedulers keep the information received from neighbors private
and that the schedulers know the network structure G.
SO Broadcast. The SO collects all the individual consumption behavior at each
time h and broadcasts the total consumption to all the customers, that is, IB
ih =
Consumption behavior model, i.e., selfish (S), altruistic (U), or welfare (W) max- imizer, and the information exchange model, i.e., private (P), action sharing (AS) or
SO broadcast (B) determine the consumption decisions of useri. We remark that in
Chapter 5, the consumption behavior model is Γ = S and the information exchange model is Ω = P.
In the next section, we define the consumer rational behavior using the solution concept Bayesian Nash equilibrium. The game and the solution concept presented in this chapter is equivalent to the BNG presented in Chapter 1. Moreover, the information structure is Gaussian, hence the consumer behavior model is a Gaussian quadratic network games to which we defined and analyzed in Chapter 2. In partic- ular, in the action sharing model agents can use the QNG filter to behave optimally as we show in Section 6.4. The redundant presentation of these concepts here is because of the different notation adopted for the demand response model in Part II. We draw the connections with the BNG and QNG filter where they are relevant.
6.3
Bayesian Nash equilibria
Useri’s load consumption at time h∈ T is determined by hisbelief qihand strategy
sih. The belief of i is a conditional probability distribution on g0 given IihΩ, qih(·) :=
Pg0(·|I Ω ih). We useE Ω ih[·] :=Eg0[·|I Ω
ih] to indicate conditional expectation with respect
to belief of qih. In order to second-guess the consumption of other customers, user
i forms beliefs on preferences given the common prior Pg0 and its information I
Ω
ih.
User i’s load consumption at time h ∈ T is determined by its strategy which is a
complete contingency plan that maps any possible local observation that it may have to its consumption, that is,sih:IihΩ 7→Rfor anyIihΩ. In particular, for user i, its best
the strategies of other customerss−ih :={sjh}j6=i, BRΓ(IihΩ;s−ih) = arg max lih Eωh EihΩ uΓih(lih,s−ih;gi0, ωh) . (6.7)
Before we define the BNE solution concept, we state the following lemma that characterizes the general form of the best response function for all the consumer models Γ = {S, U, W}.
Lemma 6.1. The best response strategy for the consumer behavior models Γ∈ {S,
U, W} has the following general form
BRΓ(IihΩ;s−ih) = gi0−µΓhω¯h−λΓh P j6=iEihΩ[sjh] 2(τΓ h +αh) (6.8) where λS h = µSh = τhS = γh, λUh = 2γh, µUh = τhU = γh, and λWh = 2κh, µWh = 0, τhW =κh.
The proof follows by taking the derivative of the corresponding utility with re- spect i’s consumption lih, equating to zero and solving the equality for lih. Note
that when ¯ω= 0 and γh =κh then aggregate utility maximizers have the same best
response function as the welfare maximizers. A BNE strategy profile is a strategy in which each customer maximizes expected utility with respect to its own belief given that other customers play with respect to BNE strategy.
Definition 6.2. A Bayesian Nash equilibrium (BNE) strategy sΓ := {sΓ
ih}i∈N,h∈T
for the consumer behavior model Γ ∈ {S, U, W} is such that for all i ∈ N, h ∈ T, and {IihΩ}i∈N,h∈T, Eωh EihΩuΓih(sΓih,sΓ−ih;gi0, ωh) ≥Eωh EihΩuΓih(sih,sΓ−ih;gi0, ωh) . (6.9)
A BNE strategy (6.9) is computed using beliefs formed according to Bayes’ rule. Note that BNE strategy profile is defined for all time slots, that is, no user at any given point in time has a profitable deviation to another strategy. In (6.9), consumers keep beliefs on consumption behavior of others, which is a function of their beliefs and strategies, to respond optimally.
Equivalently, a BNE strategy is one in which users play best response strategy given their individual beliefs as per (6.7) to best response strategies of other users – see [34, 62, 64] for similar notions of equilibrium. As a result, the BNE strategy is defined with the following fixed point equations
sΓih(IihΩ) = BR(IihΩ;sΓ−ih) (6.10)
for all i ∈ N, h ∈ T, and IihΩ. We denote i’s realized load consumption from the equilibrium strategysΓ
ihand informationIihΩ withlΓih:=sΓih(IihΩ). Using the definition
in (6.10), we characterize the unique linear BNE strategy in the next section for any information exchange and consumer behavior model.