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Consider the undirected graph (N, A) with node set N, and weighted adja- cency matrix A = [a_ij]. The objective of the consensus protocol is to find the average of the |N |-dimensional vector h. Each node i holds an estimate of the average denoted by ˜hi, which is updated iteratively through commu- nication only with neighbours N_i = {j ∈ N :a_ij = 0}. The basic iterative, synchronous consensus protocol can be defined as follows over iterations k:

˜

h(_ik+1) = ˜h(_ik)+ξ

j∈Ni

(˜h(_jk)−˜h(_ik)), (6.15) whereξis the step size. Given a step sizeξ ∈(0,1/Δ] for Δ = max_i(_j=ia_ij) and given initial estimates ˜h(0)_i =hi, the synchronous consensus protocol will converge such that ˜h(_ik)= _N1 _jhj, ∀i∈ N as k → ∞[160].

Averaging consensus relies on maintaining the average value across the network after each iteration k. Specifically

i∈N ˜ h(_ik)= i∈N hi, ∀k. (6.16)

6. Asynch. Consensus for Dist. Primal Dual Sol. to the Smart Grid OPF Prob.

If (6.15) is performed asynchronously, for example if not all nodes are up- dated at each iteration, then the consensus result will not give the average of the initial condition and (6.16) will not be maintained [161]. Asynchronous, averaging consensus protocols have been presented in the literature and typ- ically employ a Symmetric Gossip strategy (a description of which is pre- sented in [166]). However existing implementations of this strategy require some form of local synchronization [163], explicit pairing (agreement between neighbours as to who controls communication) [167] or blocking [162]. Next we present a simple asynchronous averaging consensus protocol that avoids the need for these mechanisms and possesses the following properties:

• Distributed: No central controller or leader agent.

• Asynchronous: No inter-agent synchronization.

• Implicit pairing: No blocking or explicit pairing.

• Averaging: Average consensus is reached.

• Tracking: Average is tracked as network state changes.

If protocol (6.15) is applied for a single update, for example when a single node i performs an update at iteration k, then after the update is complete the sum of value estimates in (6.16) will be incorrect by a factor of

ξ_j∈N i(˜h

(k)

j −˜h(ik)). To maintain the average an amount can be subtracted from each neighbour’s estimate. This amount does not have to be subtracted immediately and can be queued by each neighbour and subtracted when it next performs an update. We specify the consensus correction variablewi to store this value and the resulting protocol is presented in algorithm 9.

Algorithm 9 maintains the average

i∈N

(˜h(_ik)+w(_ik)) = i∈N

hi, ∀k, (6.19)

even though the asynchronous nature of the algorithm, defined by the update set S(k)_{, implies some nodes may not be updated at iteration k.}

In practice, we can’t assume that updates forwi will not occur in parallel. Therefore all such requests should be queued. Additionally, since a single failure to updatewi will cause the average to be shifted and condition (6.19) to be breached, it is important that these updates require an acknowledge message from each neighbour.

Given non-empty update setS(k)_{, and ignoring constraint (6.19) such that}

w(_ik) = 0, ∀i, k, the update (7.20) can be synchronously defined in matrix

6. Asynch. Consensus for Dist. Primal Dual Sol. to the Smart Grid OPF Prob.

Algorithm 6 Asynchronous Consensus Protocol

Initialize ˜h(0)_i =h_i, w_i(0) = 0, ∀i∈ N.

For k = 1,2, . . .:

1. Choose the set of nodesS(k) _{⊆ N to update.}

2. For each i∈ S(k)_:

2.1 Update average estimate: ˜ h(_ik+1) = ˜h(_ik)+ξ j∈Ni (˜h(_jk)−˜h_i(k))−w(_ik). (6.17) 2.2 For each j ∈ N_i: w(_jk+1) =w_j(k)+ξ(˜h(_jk)−˜h(_ik)). (6.18) 2.3 Reset w_i(k+1) = 0. form as follows: ˜ h(k)= " _k $ l=0 P(l) # h, (6.20)

where P is the Perron matrix and is defined as

P(k) =I−ξD(k), D_ij(k) = ⎧ ⎨ ⎩ |Ni| j =i and i∈ S(k), −1 j ∈ N_i and i∈ S(k)_, 0 otherwise, (6.21)

where I is the identity matrix, and |N_i| is the cardinality of the neighbour set of node i.

Theorem 6.4.1. Assume there exists a positive constant m such that for the sequence {k, k+ 1, . . . , k+m} the graph associated with the matrix

%k+m

l=k P

(l) _{is fully connected for allk. Then under the iterations specified by}

algorithm 9, and given step size 0 < ξ < 1/Δ for Δ = maxi|Ni|, all local estimates ˜hi converge to the average of the initial valueshi as k → ∞, such that

˜

h(_ik) → _{|N |}1

i∈N

6. Asynch. Consensus for Dist. Primal Dual Sol. to the Smart Grid OPF Prob.

Remark. The assumption on the connectivity of the product of Peron ma- trices can be understood intuitively as there being a uniform upper bound on the number of iterations required before information from one agent can propagate through the network to any other agent.

The following proof is adapted for the asynchronous case from the syn- chronous convergence analysis provided by [160].

Proof. It can be seen from the updates in (6.20) that a consensus is reached if the limit lim_k→∞%k_l=+_kmP(l) _{exists. Having 0 < ξ < 1/Δ gives D}

ii < 1, and 0 < Pii<1, ∀i, and also 0< Pij, i=j. It then follows that the matrix

%k+m

l=k P

(l) _{has the follows properties:}

1. All diagonal elements are non-negative, 2. All off-diagonal elements are positive,

3. The digraph associated with the matrix is strongly connected.

Therefore the matrix is primitive and according to Lemma 4 from [160] we can say that limk→∞%kl=+kmP

(l) _{exists. It then follows that ˜h}(k)

i = ˜h(jk), ∀i, j ∈ N as k → ∞.

The average conservation variable wi can be considered a bias in the updates of (7.20), which does not affect the stability analysis of the algorithm [160]. Given the consensus of the variables ˜h(_ik)and since each node is updated such that w_i(k) = 0 at least every m iterations, it follows from (7.21) that

w(_ik) → 0 as k → ∞. Then (6.19) gives _i∈N ˜h(_ik) = _i∈N hi and from consensus ˜h(_ik) → _N1 _i∈Nhi, ∀i∈ N.

The preceding proof states that a bias does not affect the stability anal- ysis. This can be seen by following the progression of the average condition with biases added:

i∈N ˜ h(_ik) = i∈N " hi+ k−1 j=0 b(_ij) # . (6.23)

Therefore adding a bias is equivalent to modifying the initial state in terms of (6.16). It follows that the average condition will converge if_i∈Nk_j=0−1b(_ij) → C, for some C∈(−∞,∞).

For an analysis of the optimal choice of tuning parameter ξ the reader is referred to [168] where an eigenvalue analysis of the Laplacian provides both admissible and optimal values. This applies to the synchronous case, but extends to the asynchronous case by replacing the Laplacian with its expectation.

6. Asynch. Consensus for Dist. Primal Dual Sol. to the Smart Grid OPF Prob.