This section investigates the impact of control parameters (the gradient step size and the control timescale) on stability and convergence speed of the algorithm. We first study the conditions under which the proposed distributed control algorithm converges to the solution of the centralized optimization problem (4.4) in a static setting, i.e., no EVs arrive or depart and the change in the magnitude of uncontrollable loads is negligible. We then study the worst-case rate of convergence.
Algorithm 2: Rate adjustment at EV charger i input : pei,new congestion prices
while true do
λ ←new congestion prices path price ← Pl∈upstream nodesλl rate ← min {path price1 , pei}
Start charging the battery atrate
Wait until the nextmessage from parent end
4.6.1 Proof of Stability
Given that strong duality holds, the primal optimum is equal to the dual optimum.
Therefore, in the static setting, we only need to show that the distributed control algorithm converges to the solution of (4.6). We then verify convergence in a dynamic setting both by studying the worst-case change in home loads, and through extensive numerical simulations.
Let L be the length of the longest path from the substation to a charger; E be the maximum number of active EV chargers sharing a line or a transformer2; pmax := maxi∈E pei be the maximum charge power supported by EV chargers; and δ be the maximum communication delay between the root MCC node and a charger.
Theorem 1 Starting from any initial vector of feasible charge powers 0 pe pe and congestion prices λ 0, the distributed control algorithm converges to the primal-dual optimal values if the following conditions hold:
(1) τc≥ δ
(2) 0 < κ < κ∗ = 2
p2maxLE
Proof : The first condition guarantees that the control action at each MCC node, which leads to a price update, is taken only after all chargers have reacted to the previous control action. In this case, the continuous time system reduces to the discrete-time system studied in [59] and our theorem reduces to Theorem 1 proved in that work. The second condition maps directly to the necessary condition for Theorem 1 in [59].
2Assuming that the substation transformer is equipped with an MCC node, E would be the total number of EV chargers in the distribution network.
4.6.2 Convergence Speed in the Worst Case
We now prove that the control algorithm exhibits monotone convergence even in the worst case and use this to compute an upper bound on the convergence time.
The control algorithm converges most slowly when there is the greatest need for a decrease in EV charging demand, together with the least possible decrease in the charging rate in each iteration. Note that each iteration of the control algorithm reduces the EV load by a multiplicative factor, which corresponds to the sum of the non-negative congestion prices sent by upstream MCC nodes. Thus, the worst case is when home demands are initially zero and every EV charger is charging at its maximum rate pei. Subsequently, a single line or transformer becomes overloaded due to an increase in the aggregate home demand to its peak3. We assume that prior to the change in the aggregate home demand, control has stabilized, with the gradient step size equal to κ∗. To simplify the presentation, we also assume that EV chargers are identical; thus, pei = p for all i.
We first prove that system convergence after the described change in the aggregate home demand is monotone.
Monotonic Convergence
Suppose that the change in the aggregate home demand happens in the beginning of a time slot, denoted t0. Given that the system has been under-utilized before t0, all congestion prices are zero at time t0. We denote the line or transformer that becomes overloaded due to this change by l, and the aggregate charging demand that it supplies at time t by yl(t). Hence, yl(t+0) > cl as its loading exceeds its setpoint after t0. We now prove that yl converges monotonically to cl.
3If multiple lines and transformers become congested at the same time, EV charge powers reduce at a faster rate since they depend on the sum of congestion prices. Thus, it would not be the worst case.
The total EV charging load supplied by l is
where |E(l)| is the number of EV chargers downstream of l. Note that the second line of (4.19) is derived from the first line since l is the only congested line or transformer, and therefore the congestion price of other lines and transformers is zero.
We remark that yl remains constant for a few iterations after t0 until λlexceeds the threshold 1p, starting from zero. Let ts be the beginning of the first control interval in which the condition λl(t) > 1p holds, then yl starts decreasing when EV chargers receive the updated price of l. The following equation can be derived from (4.19) for t ≥ ts:
yl(t) − yl(t + 1) = |E(l)|( 1 The following theorem states that monotonic convergence of our control is guaranteed in the worst case.
Theorem 2 If, in a distribution network, the length of the longest path from the substation to an EV charger is at least 2, for all t > t0, yl(t) ≥ cl if yl(t0) ≥ cl.
Proof. We prove this theorem by contradiction. Suppose for some ˜t ≥ tswe have yl(˜t) ≥ cl, and yl(˜t + 1) < cl, or equivalently yl(˜t) − yl(˜t + 1) > yl(˜t) − cl. From (4.20) we have:
|E(l)| κ(yl(˜t) − cl)
λl(˜t)(λl(˜t) + κ(yl(˜t) − cl)) > yl(˜t) − cl
The following inequality is obtained by canceling out the yl(˜t) − cl term from both sides
The monotonic convergence of the control algorithm enables us to compute an upper bound on the time that it takes until the algorithm converges to the region specified by
±ρ around the setpoint.
Let tconv be the beginning of the first control interval in which the loading of l goes below the level cl+ ρ, and δconv be an upper bound on the time between ts and tconv. We
Figure 4.2: A one-line diagram of our test distribution network. An MCC node is depicted as a meter in this figure.
for ts ≤ t < tconv.
To compute δconv we approximate the decaying decrease rate of the line or transformer loading (after ts) with a constant value equal to κρ(c|E(l)|l+ρ)2:
δconv =
d|E(l)|yl(t0) − (cl+ ρ) κρ(cl+ ρ)2 e + 1
× τc
=
d(∆ − ρ)|E(l)|
κρ(cl+ ρ)2 e + 1
× τc (4.23)
Thus, δconv+ δs is an upper bound on the convergence time in the worst case.