outputΓ
i consists of
P
∀i∈V λi(t) and P∀i∈Vyi(t) for t = {0, 1, ..}. This
security definition guarantees in the worst case that any adversary only gains knowledge of outputΓ
i. This is not the strongest security notion, where any
adversarial node i would only gain knowledge of the optimal power output of node i. However, even with the knowledge of outputΓ
i or
P
∀i∈V λi(t) and
P
∀i∈V yi(t) for t = {0, 1, ..}, the adversarial node i can not gain any information
about (aj, bj, cj, xj, xj) or the optimal xjfor j 6= i. Note that even if the attacker
node i knows Dj for some other node j, the attacker can only gain knowledge
that yj(0) might be zero (if xj ≤ Dj ≤ xj then xj(0) = Dj ) and that λj(1)
might be P
k∈Vλk(0) but everything else remains private as individual λj(t)
and yj(t) values are unknown.
Theorem 3. (PPED protocol is private against any non-colluding semi-honest adversary i if n ≥ 4.)
Proof. The view of node j consists of (λ(j)i (t), λiS4(t), yi(j)(t), ySi4(t)), for i ∈ V, i 6= j, t ∈ {0, 1, . . .}. For node i, all of these are uniformly distributed with the constraints, X i∈V λi(t) = X i∈V,i6=j λ(j)i (t) + X i∈V,i6=j λiS4(t) X i∈V yi(t) = X i∈V,i6=j yi(j)(t) + X i∈V,i6=j ySi4(t) This shows that the view of node i only depends on outputΓ
i, not on input
matrices Z or W , unless |V | = n ≤ 3 (in which case node j can recover values of λi(t) for i 6= j).
8.5.1
Communication Cost
Now, let us see how much communication overhead is produced by our PPED protocol compared to a non-private economic dispatch protocol. A non-private protocol finds the solution by sending 2tn(n − 1) messages, whereas the PPED protocol takes 4tn(n − 1). In PPED, for step 3 there are n(n − 1) messages sent forP
∀i∈V λi(t) and n(n − 1) messages sent forP∀i∈V y(t). In step 5, another
2n(n−1) messages are sent. Hence, the total number of communication messages is 4tn(n − 1) for a t round PPED protocol. In terms of order of complexity, both protocols have O(tn2).
8.6
Communication
Cost
Improvement
for
PPED
In Yang et al. [8], at every round t, the IC λi(t) and the mismatch yi(t)
are broadcasted to other nodes. Every node needs to know Pn
i=1λi(t) and
Pn
84 CHAPTER 8. PRIVACY-PRESERVING ED PROTOCOL I
still calculatePn
i=1λi(t) and P n
i=1yi(t) with a reduced communication round
than in Yang et al. [8] for a fully connected network. The communication rounds can be reduced by a factor of two for Yang et al. as follows:
1. Every node broadcasts only λi(0) at t = 0.
2. Every node broadcasts yi(t) at every round.
We show how every node can calculate Pn
i=1λi(t) and yi(t) at every round
t from the aforementioned steps:
– At first at t = 0, each node finds: Pn
i=1λi(0) and P n
i=1yi(0).
– At t = 1, every node i finds:
λi(1) = 1 n n X i=1 λi(0) + yi(0)
Node i calculates power output:
xi(1) = βiλi(1) + αi
And the power mismatch at node i:
yi(1) = 1 n n X i=1 yi(t) − (xi(1) − xi(0)) Notice that: n X i=1 λi(1) = n X i=1 λi(0) + n X i=1 yi(0)
Hence, node i finds Pn
i=1λi(1) as i knows P n i=1λi(0), P n i=1yi(0), and the parameter .
– Therefore by mathematical induction, node i can calculatePn
i=1λi(t) and
yi(t) at round t.
Similarity, the communication complexity of PPED can be reduced by a factor of two with this approach instead of the original Yang et al. protocol. Hence, the total number of communication messages will be 2tn(n − 1) for t rounds.
8.7. SUMMARY 85
8.7
Summary
We believe the attack scenario and our privacy solution will be applicable to other existing ED solutions. This work assumes a fully connected bidirectional topology. It would be interesting to extend this work to a relaxed topological constraint. Also, analyzing a stronger security notion with malicious adversaries is a future direction. Our assumption for the cost function here is quadratic as per the standard used by power engineers. However, the cost function could behave differently with different requirements in the smart grid. In the next chapter, we will analyze the security of ED solutions when the cost function changes to non-convex.
Chapter 9
Privacy-preserving ED
Protocol II
9.1
Introduction
In this chapter, we provide our second privacy solution for distributed ED. The content of this chapter is previously published in [17] and [13]. We have shown a privacy-preserving distributed ED protocol for quadratic cost function in the previous chapter 8. In real-world smart grid scenarios, more practical consider- ations include valve point loading effects, multiple fuel options and prohibited operating zones. For example, to consider a valve-point loading effect, a sinu- soidal term is added to the cost function with some non-differentiable points which makes the cost function non-convex. More details about the non-convex cost function for ED can be found in chapter 7, section 7.3. A realistic dis- tributed ED solution is provided by Binetti et al. [9] for a non-smooth and non-convex cost function. In this chapter, we aim to solve the privacy problem in distributed ED with non-convex cost function. We use the same notations for ED problem defined in the Chapter 7, section 7.3. The main contributions of this work are as follows:
– We show that the ED protocol by Binetti et al. [9] leaks confidential information of the generator nodes.
– We transform the Binetti protocol into a privacy-preserving distributed protocol for ED calculation for smart grid systems.
– We analyze the security of our proposed protocol and give results of a prototype implementation of our protocol.
88 CHAPTER 9. PRIVACY-PRESERVING ED PROTOCOL II