Chapter II: Load-Side Frequency Control in Single-Machine Systems
2.3 Decentralized load control: algorithm and convergence
In this section, we introduce a method to mitigate the inconsistencies ofpi between different loads, and describe formally the decentralized, frequency-based algorithm that solves the optimal load control problem OLC. Then, we discuss the communi- cation architecture that supports this algorithm. We also present convergence results of the proposed algorithm.
Decentralized load control algorithm
The decentralized algorithm was informally discussed in Section 2.1. The dual variable update in (2.10) requires estimating u locally. As shown in Section 2.2, there may be inconsistencies between local estimates ofu, and hence between pi, for different loadsi.
We use neighborhood communication between the loads to mitigate such incon- sistencies. The information flow of such communication can be regarded as an undirected graph, since the communication is in two ways. In this graph, denote the set of neighbors of loadiat timet asN(i,t). Loadiis assigned a weightri j(t) for all j ∈ N(i,t), and a weightrii(t)for itself. Note that if j ∈ N(i,t)theni ∈ N(j,t).
We makeri j(t) =rji(t), and can always find the weights that satisfy X
j=i,j∈N(i,t)
ri j(t)= 1, X
j=i,j∈N(i,t)
rji(t)= 1. (2.16) Other conditions on the weights will be discussed later in this section. Through neighborhood communication, loadi receives the values pj(t) of the dual variable
from all j ∈ N(i,t), and calculates their average value, denoted byqi(t), as qi(t) = X
j=i,j∈N(i,t)
ri j(t)pj(t). (2.17) This averaging procedure is typically used in consensus algorithms [95]. Consensus, in our problem, means that the loads seek agreement on the values of the dual variable p. In (2.17),qi is an auxiliary variable which denotes a local average of the values of the dual variable across loadiand its neighbors. As the algorithm iterates, this local averaging propagates to a global agreement on the values of the dual variable throughout the network. Combining such a consensus procedure with the estimation of u in Section 2.2, we have the following decentralized algorithm to solve OLC (2.4) and its dual (2.9).
Algorithm 2.1. Decentralized load-side frequency control for single-machine sys- tems
At time t = 0, the following information is known to all loadsi = 1, . . . ,N: the matrices A, B andC in system model (2.2), the lower bound p and upper bound p defined in Section 2.1, and a sequence of positive stepsizes {γ(t),t = 1,2, . . .} which is the same for all the loads. Each loadi starts from an arbitrary initial state estimate ˆxi(1|0)and an initial value of dual variableqi(0).
At time instantst = 1,2, . . ., every loadi:
1. Measures the frequency deviation∆ωi(t), and calculates ˆui(t−1|t)using the input estimation (2.11).
2. Updates the value of dual variable according to pi(t) = max(
min{qi(t−1)+γ(t)uˆi(t−1|t),p},p)
(2.18) and transmitspi(t)to all of its neighbors j ∈ N(i,t).
3. Receives thepj(t)from all j ∈ N(i,t), and calculatesqi(t)as (2.17).
4. Computes load reduction∆di(t) =∆di(qi(t))where∆di(·)is defined in (2.8).
Before proving the convergence of Algorithm 2.1, we first introduce the neigh- borhood communication supporting the information exchange in (2.17), and other conditions on ri j(t) besides (2.16), which are necessary for the convergence of Algorithm 2.1.
Neighborhood area communication
As an example, we take the smart grid communication architecture proposed by Trilliant, Inc. [96] shown in Fig. 2.2.
Wide Area Network
Neighborhood Area Network
Home Area Network
Figure 2.2: The home area network (HAN) supports the communication between appliances and smart meters. The neighborhood area network (NAN), which is used in our load control, aids the communication between utilities and smart meters. The wide area network (WAN) aids the long range communication between substations.
This figure is a slightly modified version of a similar figure in [96].
The load control scheme given by Algorithm 2.1 does not rely on communication between all the loads and a control center. Instead, it uses communication between each load and its neighbors. This neighborhood communication uses mainly a neighborhood area network (NAN). In NAN, reliable, scalable, fast responding and cost-effective communication technologies such as 802.2.15.4/ZigBee are widely used to facilitate the implementation of the decentralized load control.
For the convergence proof of Algorithm 2.1, we make the following assumption on the weightsri j(t)in (2.17).
Assumption 2.4. There exists a scalar 0 < η < 1 such that for alli = 1, . . . ,N and allt ≥ 0, we haveri j(t) ≥ ηif j =ior j ∈ N(i,t), andri j(t) =0 otherwise.
With Assumption 2.4, equation (2.17) simplifies to qi(t) =
N
X
j=1
ri j(t)pj(t). (2.19)
Moreover, in order to make the information at loadjaffect loadiinfinitely often, we assume that within any fixed period of time, the set of communication links which have appeared form a connected, undirected graph. DefineEt := {(i,j)|ri j(t) > 0}
to be the set of undirected links at timet. The connectivity requirement above is formally stated in the following assumption.
Assumption2.5. There exists a integerQ ≥ 1 such that the graph(V, S
τ=1,...,QEt+τ−1) is connected for allt.
In reality, the NAN may have specific topologies, e.g., bus, ring, star, linear topol- ogy, or mixed topologies, as discussed in [97], [98]. All these topologies satisfy Assumption 2.5. However, the convergence analysis does not require any additional assumptions on the topology beyond Assumption 2.5. We will consider a realistic topology in case studies in Section 2.5.
DefineR(t)to be the matrix with(i,j)-th entryri j(t), and defineΦ(t,s) := R(t)R(t− 1). . .R(s+1). The following result given by [95, Lemma 3.2] will be used in the convergence proof of Algorithm 2.1:
[Φ(t,s)]i j − 1 N
≤ θ βt−s, (2.20)
where
θ = 1− η
4N2 −2
, β =
1− η 4N2
Q1
. (2.21)
Convergence of Algorithm 2.1
Now we present results regarding the convergence of Algorithm 2.1. We first consider the case where the sequence {γ(t),t = 1,2, . . .}of stepsizes converges to some nonnegative constant. Theorem 2.1 gives a bound on the difference between the maximal expected value of the dual objective functionΨand the optimal value of Dual OLC, denoted byΨ∗.
Theorem 2.1. Suppose Assumptions 2.1–2.5 hold. If lim
t→∞γ(t) = γ ≥ 0 and
∞
P
t=1γ(t)= ∞, then, for alli= 1, . . . ,N, lim sup
t→∞ E[Ψ(pi(t))]≥ Ψ∗− γ(G2+σ2)
2 −γG(αN L+G) 2+ Nθ β 1− β
!
, (2.22) where G := max
(
N
P
i=1
di−∆g
,|∆g|
)
, σ is the bound on input estimate error in Corollary 2.1, α is defined in Assumption 2.2, N is the number of loads, and L := p−p.
Proof. See Appendix 2.B.
Taking γ = 0 in (2.22), we have the following corollary, which is straightforward from Theorem 2.1.
Corollary2.2. Suppose Assumptions 2.1–2.5 hold. If lim
t→∞γ(t) = 0 and
∞
P
t=1
γ(t) =
∞, then, for alli= 1, . . . ,N, lim sup
t→∞ EΨ pi(t) = Ψ∗.
Define ∆d(t) = [∆d1(t), . . . ,∆dN(t)]T. With further restrictions on the stepsize γ(t), the sequence {∆d(t), t = 1,2, . . .} produced by Algorithm 2.1 converges almost surely to the optimal point of OLC, as stated in Theorem 2.2.
Theorem 2.2. Suppose Assumptions 2.1–2.5 hold,
∞
P
t=1γ(t) =∞, and
∞
P
t=1γ(t)2< ∞.
Then, for alli =1, . . . ,N, the sequence{qi(t)}converges to the same optimal point of Dual OLC with probability 1 and in mean square. Moreover, the sequence{∆d(t)}
converges to the optimal point of OLC with probability 1.
Proof. See Appendix 2.C.
In Algorithm 2.1, neighborhood communication is used to mitigate the effect of measurement noise. Now we consider a special case where the process disturbance ζ and the measurement noise ξi for all i = 1, . . . ,N are zero. In this case, the following theorem shows that OLC can be solved by using a simplified version of Algorithm 2.1 which does not need neighborhood communication.
Theorem 2.3. Suppose Assumptions 2.1–2.3 hold, and the following conditions are satisfied:
1. ζ(t) =0 andξi(t) =0 for alli= 1, . . . ,N andt ≥ 0.
2. In Algorithm 2.1, for alli =1, . . . ,N, ˆxi(1|0)= x(1), andqi(0)are the same.
3. For alli =1, . . . ,N and allt ≥0,rii(t)= 1, andN(i,t) =∅.
4. Constant stepsizeγ(t) = γ, whereγ satisfies 0< γ < 2/(αN), is used.
Then, for all i = 1, . . . ,N, any limit point (at least one exists) of the sequence { ∆d(t),qi(t), t = 1,2, . . .}is primal-dual optimal for OLC and Dual OLC.
Proof. See Appendix 2.D.
The algorithm and convergence proof above are based on an underlying assumption that all the loads simultaneously and synchronously measure their local frequency deviations and take control actions. In practice, this is obviously not the case.
In the next section we will investigate convergence of the proposed scheme under asynchronous measurements and actuations with bounded time delays.