CONCLUSIONS AND FUTURE WORK
Security and resilience issues that arise in the smart grid constitute a pivotal con- cern in modern critical infrastructures. In this dissertation, we have discussed a six-layer security architecture for complex hierarchical systems, which allows us to adopt a divide-and-conquer approach for addressing problems residing on in- dividual layers as well as a holistic approach for cross-layer system-level analysis and design. In this dissertation, we have addressed the robust and resilient con- trol problem across the cyber and physical layers of the system, secure routing problem at the networking layer, and the management of information security and smart grid energy systems.
With increasing integration of information technologies into the power grid, robust and resilient control system design is essential for assuring the robust per- formance of power systems in face of adversarial attacks. We have presented a hybrid game-theoretic framework in which the occurrence of unanticipated events is modeled by a stochastic switching, and deterministic uncertainties are repre- sented by the known range of disturbances. The design of robust controller at the physical layer takes into account risks of failures due to cyber system, while the design of the security policies is based on its potential impact on the control system. The cross-layer coupled design results in solving a zero-sum differential game for robust control nested in and coupled with a zero-sum stochastic game for security policy. The joint design results in a robust and resilient controller switching between different modes for guaranteeing performance in face of unex- pected events. Interesting future work on this topic includes studying the impact of information structures, imperfect observations and delayed measurements on the control system design. This framework can also be extended to investigate multi-agent systems, where the interconnections of the cyber networks can lead to
distributed strategy and payoff learning algorithms. The protocol has the inherent feature of scalability and resilience, which equips the network with a self-recovery mechanism. It has been applied to cognitive radio networks as well as smart grid data networks arising from the adoption of IP-based network technologies due to the wide use of PMUs and smart meters. We have also illustrated the hybrid structure of the routing protocol to incorporate the desirable features of the cen- tralized and decentralized architectures. Future work on this topic includes an ex- tension of security protocol by including lightweight and scalable cryptographic techniques for providing security against eavesdropping and man-in-the-middle attacks. In addition, the hybrid learning algorithm can be further extended to gen- eral stochastic games, for which the estimation of the payoff function will become the learning of the Q function as in Markov decision problems. The learning al- gorithms proposed in the dissertation can also be used to compute Stackelberg equilibria in which the leader can observe the decisions made by the follower at each step.
At the higher level of the information security management layer, we have dis- cussed a series of policy-making decisions on vulnerability discovery, disclosure, patch development and patching. We have adopted a system approach to under- stand the interdependencies of these decision processes. In particular, we have studied an optimal patching decision problem for industrial control systems. This framework can be further extended to a dynamic noncooperative game by includ- ing decision model for the adversaries who decide on the attack rates. In addition, in order to provide a holistic picture of information security management, we need to extend the theoretical framework here to study the decisions involving vulner- ability disclosure and the R&D development of control patches.
The emerging smart grid applications impose many challenges for efficient management of distributed energy resources and demand response participated by active utility users. In the dissertation, we have established game-theoretic frameworks and addressed fundamental questions such as “What is the system equilibrium under distributed power generation?” and “What is the value that de- mand response management can bring to the smart grid?”. We have presented a set of new analytical methods and computational tools for analyzing games with coupled constraints based on game linearization. In addition, we have proposed the mirror Stackelberg equilibrium to characterize the equilibrium solution for Stackelberg games in which the decision of the follower relies on the dual vari- able of leader’s problem. We have shown that a consistency principle can be used
to find the equilibrium solution. We have observed that the value of demand re- sponse heavily depends on the behaviors of the users. The implementation of communication systems for demand response may not result in cost reductions at supply and demand sides. This framework provides fundamental understanding of pricing, energy policies and infrastructural investment. As future work, we will investigate the impact of communication security on demand response manage- ment based on this equilibrium concept. In addition, games-in-games structure can be applied to study the impact of demand response on physical layer control of power systems.
APPENDIX A
PRICE OF ANARCHY
Motivated by the flow control problem in Section 5.2.3, in this appendix, we study a class of flow control games and characterize its efficiency loss in a generalized differential game framework. Consider the infinite-horizon scalar N−person LQ DGs, with quadratic cost function
Li(u) = Z ∞ 0 qix2(t) + riu2i(t) dt, i ∈N , (A.1) ˙ x(t) = ax(t) + N
∑
i=1 biui(t), x(0) = x0, (A.2)where qi> 0, ri> 0, x06= 0, bi6= 0 are all scalar quantities. Let b := [b1, . . . , bN].
We are interested in strongly time-consistent state-feedback (SF) Nash equilib- rium (NE), where further the NE policies are required to be stationary (that is time invariant). We will refer to such equilibria in short as Feedback NE. The following theorem provides their characterization.
Theorem 14 [Feedback NE, [13]] Let {ki, i ∈N } solve the set of coupled alge-
braic Riccati equations
2 a− N
∑
j=1 sjkj ! ki+ qi+ siki2= 0, i ∈N (A.3)satisfying the stability condition a− ∑Ni=1siki< 0 , where si:= b2i/ri. Then, the N-
tuple of policies γi∗(x) = −bi
rikix, i ∈N , constitutes a feedback NE, with the cor-
responding cost for Player i being Ji∗= kix20. Furthermore, the positively weighed total cost is Jµ∗ = ¯kx20, where ¯k= ∑Ni=1µiki.
If the set of coupled algebraic Riccati equations do not admit a solution which is also stabilizing, then the DG does not have a feedback NE. The main challenge in computing the feedback NE solution for this DG is that equation (A.3) is a nonlinear coupled system of equations. The fact that we have
a scalar problem alleviates the difficulty somewhat, since it is possible to turn it into a linear problem through a change of variables, as outlined in [160], [161]. Let σi= siqi, σmax= maxiσi, pi= siki, i = 1, . . . , N, and
λ =
N
∑
i=1
pi− a. (A.4)
Multiplying (A.3) by si, we rewrite it as
p2i − 2λ pi+ σi= 0, i = 1, . . . , N. (A.5)
Let Ω ⊂N be an index set, Ω−i= Ω\{i}, and nΩ = |Ω|. For every Ω 6= /0, we
have (after some manipulations)
∏
j∈Ω pjλ = 1 2nΩ− 1 (∑
i∈Ω σi∏
j∈Ω−i pj−∑
i/∈Ω∏
j∈Ω pjpi+ a∏
j∈Ω pj ) . (A.6) When Ω = /0, we define∏
j∈Ω pjλ := λ = N∑
j=1 pj− a. (A.7)Hence, for every Ω, we have an equation in the form of either (A.6) or (A.7). Let p = [1, p1, p2, . . . , pN, p1p2, . . . , p1pN, p2p3, . . . , pN−1pN, . . . , ∏Ni=1pi]T. We
can write (A.6) and (A.7) into
e
Mp = λ p. (A.8)
Let k := [1, k1, k2, . . . , kN, k1k2, . . . , k1kN, k2k3, . . . , kN−1kN, . . . , ∏Ni=1ki]T and D =
diag{1, s1, s2, . . ., sN, s1s2, . . ., s1sN,s2s3, . . . , sN−1sN, . . . , ∏Ni=1si} . Hence, we
can rewrite p = Dk and (A.8) into
Mk = λ k, where M := D−1MD .e (A.9) Equation (A.9) is an eigenvalue problem with each index set Ω corresponding to a row enumerated starting from the empty set. It has maximum 2N distinct eigen- values and 2Neigenvectors. The vector formed by the second entry to the N + 1-st entry of the eigenvectors yields the solution to (A.3) when the first entry of the
Theorem 15 [Feedback NE Computation, [162]] Suppose M is a nondefective matrix with distinct eigenvalues. Let (λ , k) be an eigenvalue-eigenvector pair such that λ ∈ R+ and λ > σmax. Then, a feedback NE γi∗(x) = −briikix, i ∈N ,
is yielded by k∗= 1Tk provided that the resulting solution is stabilizing, where 1 = [0, 1, . . . , 1, 0, . . . , 0]T is a vector whose2nd to N + 1-st entries are 1’s.
Theorem 16 [Uniqueness of Feedback NE] Let ¯p:= ∑j∈N pj, p−i:= ∑j∈N , j6=ipj.
There exists a unique feedback NE for the LQ DG described by (A.1) and (A.2) under either one of the following two conditions:
(i) N is sufficiently large such that p−i> a, ∀i, or (ii) a = 0.
Furthermore, the solutions to the coupled algebraic Riccati equations that char- acterize the feedback NE are of the following forms under the corresponding con- ditions above: (s-i) pi= ( ¯p− a) −p( ¯p− a)2− σ i; (s-ii) pi= ¯p−pp¯2− σ i, where ¯ p− a = 1 N− 1 N
∑
i=1 q ( ¯p− a)2− σ i+ a ! . (A.10)Moreover, the stability condition a− ∑N
i=1siki< 0 is satisfied, and hence the FB
NE is stabilizing.
Proof: From (A.5), we obtain
p2i + 2(p−i− a)pi− σi= 0, (A.11)
which admits the solutions:
pi= (a − p−i) ±
q
(a − p−i)2+ σi. (A.12)
Since we need pi > 0, we retain the one with “ + ” sign. By rearranging the
positive solution of (A.12), we arrive at
( ¯p− a)2= (p−i− a)2+ σi, (A.13)
and, therefore, in terms of ¯p, we have pi= ( ¯p− a) ±
q
( ¯p− a)2− σ
Under condition (i), we have pi− ¯p + a < 0, hence we obtain the unique solution
(s-i). Under scenario (ii), (A.14) reduces to pi= ¯p±
p ¯ p2− σ
i. Since, pi< ¯p, we
again obtain the unique solution (s-ii).
By summing over (A.14), we have a fixed point equation (A.10). Let
¯ P( ¯p) := 1 N− 1 N
∑
i=1 q ( ¯p− a)2− σ i+ a ! − ( ¯p − a) .Its derivative is given by d ¯P dp¯ = −1 + ¯ p− a N− 1 N
∑
i=1 1 p (a − ¯p)2− σ i ! .Since σi> 0 and ¯p − a > 0, it follows that
d ¯P dp¯ > −1 + ¯ p− a N− 1 N ( ¯p− a) (A.15) = 1 N− 1 > 0, for N > 2. (A.16) This says that ¯Pis a monotonically increasing function, and hence the solution to
¯
P= 0 is unique. Hence, under (i) or (ii), there exists a unique feedback NE. The fact that the solution is stabilizing follows directly from (A.3), where the first term has to be negative because the second and third terms are positive.
A.1 Team Model
When players form a team to achieve an optimal social objective, a specific total cost is minimized. Let ¯qµ= ∑Ni=1µiqi, Rµ = diag{µ1r1, . . . , µNrN}, and consider
(FOC) min u(t) Z ∞ 0 ¯ qµx2(t) + uT(t)Rµu(t) dt s.t. x(t) = ax(t) +˙ N
∑
i=1 biui(t) , x(0) = x06= 0 .are γi◦(x) = − bi µiri ˆkµx, ˆkµ := a+ p a2+ ¯q¯b ¯b , (A.17)
with ¯b:= ∑Ni=1(b2i/µiri), and minimum cost is Jµ◦ = ˆkµx20.
The optimal control can also be expressed in open-loop form, as: u◦i = − bi
µiri
ˆkµΦ(t, 0)x0,
where Φ(t, 0) is the unique solution to
˙ Φ(t, 0) = a− N
∑
i=1 b2i µiri ˆkµ ! Φ(t, 0), Φ(0, 0) = 1.A. 2 Price of Anarchy (PoA)
Here, we provide a closed-form expression for the PoA in the feedback LQ DG, where we make the natural assumption that x06= 0, as otherwise the costs are all
zero.
Theorem 18 The PoA of the LQ feedback DG described by (A.1) and (A.2) is characterized by the following:
(i) Given a weight vector µ, the PoA ρµ is equal to
ρµFB= max
k∈K [ µ
Tk ] /ˆk , (A.18)
where µ = [0, µT, 0, . . . , 0]T andK is the set of all eigenvectors of the matrix M.
(ii) Suppose µi= ¯µi:= si/∑Nj=1sj, i ∈N . Then,
ρµFB¯ 6 [ ρ(M) + a ] /
N
∑
i=1
siˆk ,
where ρ(M) is the spectral radius of M.
(iii) Let µmaxs = maxi∈N µi/si. Given a weight vector µ that satisfies ∑Ni=1µi= 1,
the PoA is bounded by
Proof: The proof is a direct application of the results in Theorem 14 and The- orem 17. PoA is the worst-case ratio of the game cost under feedback NE to the optimum social cost. Under the feedback IS, an LQ DG has
ρµFB= max k∗ ∑Ni=1µik∗i(x0)2 ˆk(x0)2 = maxk∈K µTk ˆk . This leads to statement (i). The price of anarchy under ¯µ is
ρµFB¯ = max k ∑Ni=1µ¯iki ˆk = maxk siki ∑Ni=1siˆk = max λ λ + a ∑Ni=1siˆk . (A.20)
The last equality is due to (A.4). Hence, by taking the largest eigenvalue, we obtain (ii). The equality is achieved when ρ(M) is an eigenvalue in the eigenvalue- eigenvector pair that yields the equilibrium from Theorem 15. For an arbitrarily picked µ, (A.18) yields
ρµFB¯ = max k ∑Ni=1 µi sisiki ˆk 6 maxk usmax∑Ni=1siki ˆk = max λ usmax(λ + a) ˆk 6 usmax(ρ(M) + a) ˆk . (A.21)
Using (A.4) and taking the worst case, we obtain statement (iii). Since max i∈N ¯ µi si = 1 ∑Nj=1sj ,
the last inequality is achieved when µ = ¯µ .
The next corollary further characterizes the bound on PoA. Corollary 3 The following follow from Theorem 18:
(i) Given a µ and a 6= 0, PoA is bounded above by
ρµFB6 1 + 1 2a(N + σmax− 1) s•, (A.22)
where σmax= maxi∈N σi, and s•:= N
∑
i=1 si minj∈N sj. The upper-bound is independent of µ.(ii) If a= 0, PoA is bounded above by
ρµFB6 µ s max √ ¯ qpµmins √ N(N + σmax− 1), (A.23)
where µmins = mini∈N µi/si.
Proof: The matrices M = [mi j] and eM = [ ˜mi j], i, j = 1, . . . , 2N, share the same
set of eigenvalues. From Gersgorin theorem, we can obtain
ρ ( eM)6 min ( max i 2N
∑
j=1 | ˜mi j|, max j 2N∑
i=1 | ˜mi j| ) 6 max i 2N∑
j=1 | ˜mi j|.From (A.6) and (A.7), the absolute row sum RSk, k = 1, . . . , 2N, can easily be evaluated by letting pi= 1:
RSk= [a +
∑
i∈Ω
σi+ (N − nΩ)] / [2nΩ− 1],
where k is the row index corresponding to the set Ω. When Ω = /0, we let RS1=
N+ a. From (A.19),
ρµFB6 [ ρ(M) + a ]/(ˆk/µmaxs ). The numerator is upper-bounded by (skipping some steps):
ρ (M) + a 6 max max 16nΩ6N (2a + σmax− 1)nΩ+ N 2nΩ− 1 , 2a + N − 1 6 max {2a + N + σmax− 1, 2a + N − 1}
6 2a + N + σmax− 1. (A.24)
The second inequality holds because the quantity (2a + σmax− 1)nΩ+ N
increases with nΩ. The denominator has a lower bound: 2a ¯bµs max > 2a ∑Ni=1 max i∈N µi/si µi b2 i ri > 2a ∑Ni=1minsi i∈N si = 2a s• . (A.25)
The last inequality is due to maxiµi/si6 maxiµimaxis1i. Combining (A.24) and
(A.25), we have, for a 6= 0,
ρµFB6 1 + 1 2a(N + σmax− 1) s• When a = 0, ˆk =qq/¯b =¯ s ¯ q ∑Ni=1µsi i > p ¯qµ s min √ N .
Using this together with (A.24), we arrive at the inequality (A.23). The upper bound on price of anarchy in the preceding corollary provides a worst case of efficiency loss.
The next result studies the large population game and its proof relies on the Taylor series expansion of the square-root term in (A.14).
Theorem 19 Suppose the number of players in the LQ DG is sufficiently large so that
(C-i) p−i> a, ∀i ∈N , (C-ii) a N , (C-iii) σmax ¯σ ,
where ¯σ = ∑Ni=1σi. Then, the following quantities can be approximated as given:
(i) pi∼ σi √ 2 ¯σ , (ii) ui∼ − σi bi√2 ¯σ x, (iii) J∗∼√q¯ 2 ¯σ (x0)2, (iv) J∗∼ ¯ q √ 2 ¯σ (x0)2, (v) ρµFB∼ q¯ ˆk√2 ¯σ , and for a = 0, ρ FB µ ∼ s ¯ q¯b 2 ¯σ.
Proof: By Taylor series expansion, (A.14) can be written as pi = ( ¯p− a) 1 − r 1 − σi ( ¯p− a)2 = σi 2( ¯p− a) 1 + O σi ( ¯p− a)2 , (A.26)
where O(·) is a function such that limx→0O(x) = 0. In a similar way, (A.10) can
be rewritten as (skipping some steps):
¯ p− a =p¯− a N− 1 N
∑
i=1 r 1 − σi ( ¯p− a)2+ a ! =p¯− a N− 1 N ¯σ 2( ¯p− a)2 1 + O σmax 2( ¯p− a)2 + a . (A.27)Hence, we obtain for large N
¯ p− a = r ¯σ 2 1 + O σmax 2( ¯p− a)2 . (A.28)
Note that ¯p− a > 0 due to the stability condition. Let ¯σ = ∑Ni=1σi, as before. Let
a solution of (A.28) be ¯p=p ¯σ /2 + a, i.e.,
¯ p− a =r ¯σ 2 h 1 + Oσmax ¯ σ i . (A.29)
(A.29) is consistent provided that σmax σ and a N. Since, by Theorem
16, the solution is unique under (C-i), ¯p can indeed be approximated by ¯p∼ a+p ¯σ /2, which leads to pi∼√σ2 ¯i
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