Privacy-preserving ED (PPED) Protocol

– Leakage of Output Power (xj): At t = 0, i receives λj(0) and yj(0).

As i knows the local power demand Dj, i can simply get the value of xj(0).

Then, i can find out the output power estimate xj(t) at every round as i

can getP

k∈N_j+qjkyk(t) from each k in Nj+at round t and yj(t + 1) from

j at round t + 1: xj(t + 1) = X k∈N+ j qjkyk(t) − yj(t + 1) + xj(t)

If the protocol ends at round T , xj(T ) is the final power output estimate

for node j. This can be easily estimated by node i.

– Leakage of Cost Function Parameters (aj, bj): i knows values of

λj(t) and xj(t), from two rounds t and t + 1:

xj(t) = βjλj(t) + αj

xj(t + 1) = βjλj(t + 1) + αj

Solving two linearly independent equations, node i can find the values of αj and βj which will give values of aj and bj.

– Leakage of Generator Constraint Parameters (xj and xj): At

initial round t = 0, i observes xj(0), so i might get one of the values of xj

or xj.

8.4

Privacy-preserving ED (PPED) Protocol

To prevent the privacy leakage, we present our PPED protocol which uses a pri- vacy layer in each round using a secure sum protocol. We can use the consensus- based algorithm proposed by Yang et al. [8] described in section 8.2.3 as the basis. The secure sum protocol used in PPED is similar to the n − private protocol for summation presented by Benaloh in [41] but uses (n − 1) partitions instead of n.

8.4.1

System Model

We consider a complete synchronous network G = (V, E) with n nodes and a secure and reliable point to point communication channel between every node (no eavesdropping). As G is a complete graph, every element in P and Q matrix will be 1_n.

A practical assumption would be that all numerical values we want to compute are fixed point values for ED algorithms. We can multiply any fixed point elements with a suitable constant and convert them into integers. In Step 2 (next section), we convert λi(t) and yi(t) to integer and in Step 7 we convert

80 CHAPTER 8. PRIVACY-PRESERVING ED PROTOCOL I

Figure 8.1: PPED protocol for n = 4

8.4.2

PPED protocol

The diagram of our PPED protocol with 4 nodes is illustrated in figure 8.1. The protocol is outlined as follows:

– Step 1: Initialization of every node at t = 0:

xi(0) =      xi, if xi< Di Di, if xi≤ Di≤ xi xi, if Di< xi ∀i ∈ V λi(0) = xi(0) − αi βi yi(0) = Di− xi(0)

– Step 2: In step 2 to step 6 we map λi(t) and yi(t) values to integer values

and map it back in step 7. We consider ZM as the additive group of inte-

gers from 0 to M − 1 (M is a large integer such that M >Pn

i=1λi(t) and

M >Pn

i=1yi(t)). Every node i, chooses (n − 1) numbers independently

with uniform distribution from ZM, such that their sum (in modulo M ) is

equal to λi(t). The same is done for breaking yi(t) into n − 1 parts where

8.4. PRIVACY-PRESERVING ED (PPED) PROTOCOL 81

segment y(j)_i (t) for the node j. All calculations are done in modulo M . λi(t) = X ∀j∈V ,j6=i λ(j)_i (t) yi(t) = X ∀j∈V ,j6=i y(j)_i (t)

– Step 3: Every node i sends segments λ(j)_i (t) and y_i(j)(t) to the respective j in the network. We can use some token to distinguish between λi(t) and

yi(t) segment values.

– Step 4: Each node i will receive a total of n−1 data segments for λ(t) and y(t) respectively from the other n − 1 nodes in the graph. All the nodes add all their received segments for λ(t) and y(t) separately. At node i, the total received sums λi_S4(t) and yi_S4(t) is calculated as follows:

λi_S4(t) = X ∀j∈V ,j6=i λ(i)_j (t) yi_S4(t) = X ∀j∈V ,j6=i y_j(i)(t) – Step 5: Every node i sends λi

S4(t) and yS4i (t) to the remaining n − 1

nodes in the network.

– Step 6: Every node i adds all received λj_S4(t)’s from the other n − 1 nodes and its own λi

S4(t). Hence, every node gets sum value without knowing

the individual inputs:

X ∀i∈V λi(t) = X ∀j∈V ,j6=i λj_S4(t) + λiS4(t)

Do the same for y(t).

– Step 7: Node i finds:

λi(t + 1) = 1 n X ∀i∈V λi(t) + yi(t)

– Step 8: For all nodes i, the power output is calculated as:

xi(t + 1) = βiλi(t + 1) + αi

– Step 9: For all nodes i, the power difference is compared as:

yi(t + 1) = 1 n X ∀i∈V yi(t) − (xi(t + 1) − xi(t))

82 CHAPTER 8. PRIVACY-PRESERVING ED PROTOCOL I

– Step 10: Check: if

∀i ∈ V , yi(t + 1) ≈ 0

then ED solution found break

else t = t+1, Repeat ∀i ∈ V Step 2 to Step 10.

Remark. (Correctness of PPED) If Yang et al.’s ED protocol [8] is correct, the PPED protocol is also correct.

Proof. Correctness of Yang et al.’s ED protocol is shown in [8, Thm 2 and 3]. Hence the correctness of PPED protocol follows from, for any round for t = {0, 1, . . . , T }, P ∀i∈Vλi(t) =P∀j∈V ,j6=iλ j S4(t) + λ i S4(t) and P ∀i∈V yi(t) = P ∀j∈V ,j6=iy j S4(t) + y i S4(t).