The existing state of the art distributed ED solutions in [120] [8] use similar structures. The Yang et al.’s ED protocol [8] is an iterative algorithm, uses incremental cost (IC) as the consensus variable and considers quadratic cost function.
8.2.1
Incremental Cost Criteria and Notations
Recall the ED problem defined in the Chapter 7, section 7.3. We consider, a group of generator nodes V = {1, . . . , n} jointly aims to solve the ED problem. At time step t, we assume, the estimated power production of the generator node i ∈ V is xi(t). We also consider, every generator node i ∈ V has a quadratic cost
function Ci. For instance, the cost of power production of node i at time step t
is Ci(xi(t)). The total customer demand is denoted by D, which the network V
must produce and supply. Moreover, Yang et al.’s ED protocol considers a local customer demand Di for initialization of node i such that D = P
n
i=1Di. The
incremental cost IC is the increase in total cost resulting from an increase in power generation. We define the IC of generator node i at time step t is λi(t).
The equal incremental cost criterion [121] met when every node has equal incremental cost, resulting in an optimal solution for the ED problem. The notation used in the chapter is given in Table 8.1. The incremental cost (IC) is used as the consensus variable and every nodes aims to reach the same IC after rounds of communication and computation.
8.2. YANG ET AL.’S CONSENSUS-BASED ED 77
8.2.2
System Model
In [8], authors assumed a strongly connected network topology as a directed graph G = (V, E). The set of vertices V = {1, . . . , n} represent the generator nodes (or utility providers) of the network and the set of edges E ⊆ V × V represent the communication structure between the nodes. Strongly connected means, there exists a path between any pair of two vertices in the directed graph. A direct edge from i to j is denoted by an ordered pair (i, j) ∈ E and means that a node j can receive information from i. The in-neighbors and out-neighbors of ith node are represented by N+
i = {j ∈ V |(j, i) ∈ E}
and Ni− = {j ∈ V |(i, j) ∈ E} respectively. A node can receive information from in-neighbors and send information to out-neighbors. As each node can know its own state information, each vertex belongs to both its in-neighbors and out-neighbors (i ∈ Ni+ and i ∈ Ni−).
8.2.3
Description of the Incremental Cost Consensus Al-
gorithm (Yang et al. [8])
Here, two matrices are defined as P, Q ∈ Rn∗n based on the topology of the
graph G. Let, pij and qij be the elements of matrices P and Q respectively. All
the elements of P and Q are public. They are defined as:
pij = ( 1 |N+ i| if j ∈ Ni+ 0 otherwise qij = ( 1 |Ni−| if i ∈ N − j 0 otherwise
The standard cost function used for ED calculation is quadratic as shown in (7.4). In [8], a slightly different quadratic cost function is being used (quadratic convex).
Ci(xi) =
(xi− αi)2
2βi
+ γi (8.1)
Where αi ≤ 0, βi > 0 and γi ≤ 0. However, the cost functions (7.4) and
(8.1) are basically equivalent if we replace αi = −(bi)/(2ai), βi = 1/2ai and
γi= ci−(b2i)/(4ai). The IC of node i at a discrete time index t can be formulated
as:
λi(t) =
xi(t) − αi
βi
– Initialization: Diis the local demand associated with the node i. Hence,
the total demand is D =P
i∈V Di.
Initially, ∀i ∈ V , we have:
xi(0) = xi, if xi< Di Di, if xi≤ Di≤ xi xi, if Di< xi
78 CHAPTER 8. PRIVACY-PRESERVING ED PROTOCOL I
λi(0) =
xi(0) − αi
βi
yi(0) = Di− xi(0)
– Main Algorithm: At round t + 1, the variables get updated as follows:
λi(t + 1) = X j∈Ni+ pijλj(t) + yi(t) xi(t + 1) = βiλi(t + 1) + αi yi(t + 1) = X j∈Ni+ qijyj(t) − (xi(t + 1) − xi(t))
At each round, every node updates its own IC λi(t) from the IC values
of the previous round received from its in-neighbors Ni+. Then, each node updates its output power xi(t) from the updated IC. Subsequently,
i calculates the mismatch yi(t) and forwards to its out-neighbors Ni− for
the next round. is a positive scalar and controls the convergence speed. Suppose, λ∗is the optimal value for IC and xi∗is the optimal power output
of node i for ED solution. Yang et al. claimed that if is sufficiently small the algorithm converges to the ED solution i.e.,
λi(t) → λ∗, xi(t) → xi∗, yi(t) → 0 as t → ∞, ∀i ∈ V [8, Thm. 2 and 3].
8.3
Attack on Yang et al.’s ED Protocol
In this section, we show an attack to the Yang et al.’s protocol described in the previous section.
8.3.1
Attacker Model
We assume the attacker i is a non-colluding semi-honest node in the network, i.e. i strictly follows the protocol but it may analyse the messages exchanged during the execution of the protocol to gain additional information. We also assume that the attacker knows the local demand Djof any other node j in the
network. This assumption is realistic as the local demand Dj is basically the
aggregated demand from the consumers of j and can be a public information. The privacy goals in distributed ED is given in section 7.4.2. An attack will be successful if the attacker achieves full knowledge about output power, generator constraints, and any cost function parameter corresponding to another node.
8.3.2
Privacy-sensitive Data Leakage
Let us assume that the messages are exchanged between two nodes, i (attacker) and j (another node) where i ∈ Nj− and Nj+⊆ Ni+.