On Privacy in Smart Metering Systems with Periodically Time-Varying Input Distribution Yu Liu a , Ashish Khisti a , Aditya Mahajan b GlobalSIP Symposium on Privacy and Security 14 Nov, 2017 a University of Toronto b McGill University
Smart-meter privacy–(Liu, Khisti, and Mahajan) 1 Smart Meters empower smart grids Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .
Smart-meter privacy–(Liu, Khisti, and Mahajan) 1 Smart Meters empower smart grids Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .
Smart-meter privacy–(Liu, Khisti, and Mahajan) 1 Smart Meters empower smart grids Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .
Smart-meter privacy–(Liu, Khisti, and Mahajan) 1 Smart Meters empower smart grids Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .
What is the minimum information leakage rate if consumers obfuscate consumption using a rechargeable battery? What are privacy-optimal battery charging strategies?
Smart-meter privacy–(Liu, Khisti, and Mahajan) 2 Home Applicances Power Grid Smart Meter Controller Demand: X t Consumption: Y t
Smart-meter privacy–(Liu, Khisti, and Mahajan) 2 Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Demand: X t Consumption: Y t
Smart-meter privacy–(Liu, Khisti, and Mahajan) 2 Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State S t ) Demand: X t Consumption: Y t Y t − X t
Smart-meter privacy–(Liu, Khisti, and Mahajan) 2 Energy conservation Consumption: Y t Demand: X t ( State S t ) Battery Adversory Evesdropper/ Controller Smart Meter Grid Power Applicances Home Y t − X t S t+1 = S t + Y t − X t , S t ∈ 𝒯 (Size of battery)
Smart-meter privacy–(Liu, Khisti, and Mahajan) Adversory Randomized charging strategy Energy conservation Consumption: Y t Demand: X t ( State S t ) 2 Battery Evesdropper/ Controller Smart Meter Grid Power Applicances Home Y t − X t S t+1 = S t + Y t − X t , S t ∈ 𝒯 (Size of battery) q t (y t | x t , s t , y t−1 ) : Choose consumption given history . . .
Smart-meter privacy–(Liu, Khisti, and Mahajan) Demand: X t 𝐫 (X T ; Y T ) 1 T→∞ min lim Objective Randomized charging strategy Energy conservation 2 Consumption: Y t ( State S t ) Smart Meter Home Applicances Power Grid (mutual information rate) Controller Evesdropper/ Adversory Battery Y t − X t S t+1 = S t + Y t − X t , S t ∈ 𝒯 (Size of battery) q t (y t | x t , s t , y t−1 ) : Choose consumption given history . . . Choose battery charging strategy 𝐫 = {q t } t≥1 to T I
Smart-meter privacy–(Liu, Khisti, and Mahajan) 3 Why is the problem non-trivial? (Binary model) 𝒴 = 𝒵 = 𝒯 = {0, 1} , P X = [0.5, 0.5] Consv: S t + Y t − X t ∈ 𝒯
Smart-meter privacy–(Liu, Khisti, and Mahajan) 3 Why is the problem non-trivial? (Binary model) 𝒴 = 𝒵 = 𝒯 = {0, 1} , P X = [0.5, 0.5] Consv: S t + Y t − X t ∈ 𝒯 Empty state S t = 0 Full state S t = 1 X t = 0 ⟹ Y t ∈ {0, 1} X t = 0 ⟹ Y t = 0 X t = 1 ⟹ Y t = 1 X t = 1 ⟹ Y t ∈ {0, 1}
Smart-meter privacy–(Liu, Khisti, and Mahajan) 3 Why is the problem non-trivial? (Binary model) Consider performance of memoryless policies 𝒴 = 𝒵 = 𝒯 = {0, 1} , P X = [0.5, 0.5] Consv: S t + Y t − X t ∈ 𝒯 Empty state S t = 0 Full state S t = 1 X t = 0 ⟹ Y t ∈ {0, 1} X t = 0 ⟹ Y t = 0 X t = 1 ⟹ Y t = 1 X t = 1 ⟹ Y t ∈ {0, 1}
Smart-meter privacy–(Liu, Khisti, and Mahajan) 3 Why is the problem non-trivial? (Binary model) Consider performance of memoryless policies Deterministic Memoryless Policy 𝒴 = 𝒵 = 𝒯 = {0, 1} , P X = [0.5, 0.5] Consv: S t + Y t − X t ∈ 𝒯 Empty state S t = 0 Full state S t = 1 X t = 0 ⟹ Y t ∈ {0, 1} X t = 0 ⟹ Y t = 0 X t = 1 ⟹ Y t = 1 X t = 1 ⟹ Y t ∈ {0, 1} P(Y|X = 0, S = 0) = [1 0] ; P(Y|X = 1, S = 1) = [0 1] : Leakage = 1 ( ∵ Y t = X t ). P(Y|X = 0, S = 0) = [0 1] ; P(Y|X = 1, S = 1) = [1 0] : Leakage ≈ 1 ( ∵ Y t = 1 − S t ).
Smart-meter privacy–(Liu, Khisti, and Mahajan) 3 Is this the optimal policy? Is this the best memoryless policy? P(Y|X = 0, S = 0) = [0.5 0.5] ; P(Y|X = 1, S = 1) = [0.5 0.5] : Leakage = 0.5 . Randomized Memoryless Policy Deterministic Memoryless Policy Consider performance of memoryless policies How do we evaluate the performance of an arbitrary policy? Need ℙ (X T , Y T ) ? Why is the problem non-trivial? (Binary model) 𝒴 = 𝒵 = 𝒯 = {0, 1} , P X = [0.5, 0.5] Consv: S t + Y t − X t ∈ 𝒯 Empty state S t = 0 Full state S t = 1 X t = 0 ⟹ Y t ∈ {0, 1} X t = 0 ⟹ Y t = 0 X t = 1 ⟹ Y t = 1 X t = 1 ⟹ Y t ∈ {0, 1} P(Y|X = 0, S = 0) = [1 0] ; P(Y|X = 1, S = 1) = [0 1] : Leakage = 1 ( ∵ Y t = X t ). P(Y|X = 0, S = 0) = [0 1] ; P(Y|X = 1, S = 1) = [1 0] : Leakage ≈ 1 ( ∵ Y t = 1 − S t ).
Smart-meter privacy–(Liu, Khisti, and Mahajan) 4 Literature overview Evaluate privacy of specific battery management policies [Kalogridis et al., 2010] Monte-Carlo evaluation of best-efgort policy [Varodayan Khisti, 2011] Computing performance of battery conditioned [Tan Gündüz Poor, 2012] Generalized results of [Varodayan Khisti] to include models with energy harvesting. [Giulio Gündüz Poor, 2015] Bounds on performance of best-efgort and hide-and-store policies for infjnite battery size. stochastic charging policies using BCJR algorithm.
Smart-meter privacy–(Liu, Khisti, and Mahajan) 4 Literature overview Evaluate privacy of specific battery management policies [Kalogridis et al., 2010] Monte-Carlo evaluation of best-efgort policy [Varodayan Khisti, 2011] Computing performance of battery conditioned [Tan Gündüz Poor, 2012] Generalized results of [Varodayan Khisti] to include models with energy harvesting. [Giulio Gündüz Poor, 2015] Bounds on performance of best-efgort and hide-and-store policies for infjnite battery size. Dynamic programming decomposition to identify optimal policies [Yao Venkitasubramanian, 2013] Dynamic program, computable inner and upper bounds. Li Kshiti Mahajan, 2016 Dynamic program, closed form optimal strategy for i.i.d. case. stochastic charging policies using BCJR algorithm.
Smart-meter privacy–(Liu, Khisti, and Mahajan) 5 [LKM] Main results: Markovian demand Structure of optimal strategies Charging strategies of the form q t (y t |x t , s t , π t ) are optimal. Defjne belief state π t (x, s) = ℙ (X t = x, S t = s|Y t−1 )
Smart-meter privacy–(Liu, Khisti, and Mahajan) Then, [LKM] Main results: Markovian demand Structure of optimal strategies Charging strategies of the form q t (y t |x t , s t , π t ) are optimal. Dynamic programming decomposition Let denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯) . 5 q ∗ (y|x t , s t , π t ) = a ∗ (y|x t , s t ) x,s,y π(x, s)a(y|x, s)v(φ(π, y, a))} Defjne belief state π t (x, s) = ℙ (X t = x, S t = s|Y t−1 ) Suppose there exists a J ∈ ℝ and v∶ 𝒬 X,S → ℝ that satisfjes the following: J ∗ + v(π) = inf a∈ {I(a; π) + ∑ J ∗ is the minimum leakage rate Let f ∗ (π) denote the arg min of the RHS and a ∗ = f ∗ (π) . Then, J ∗ is achieved by the charging policy (note a ∗ depends on π t )
Smart-meter privacy–(Liu, Khisti, and Mahajan) x,s,y is concave rather than linear. The DP is similar to the DP for POMDPs but the per-step cost Permuter et al 2008) with feedback (Goldsmith Varaiya 1996, Tatikonda Mitter 2009, Inspired by the approach used for capacity of Markov channels q ∗ (y|x t , s t , π t ) = a ∗ (y|x t , s t ) Then, 5 π(x, s)a(y|x, s)v(φ(π, y, a))} Dynamic programming decomposition Let denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯) . Charging strategies of the form q t (y t |x t , s t , π t ) are optimal. Structure of optimal strategies [LKM] Main results: Markovian demand v(π) is concave. So, computational approaches for POMDPs work. Defjne belief state π t (x, s) = ℙ (X t = x, S t = s|Y t−1 ) Suppose there exists a J ∈ ℝ and v∶ 𝒬 X,S → ℝ that satisfjes the following: J ∗ + v(π) = inf a∈ {I(a; π) + ∑ J ∗ is the minimum leakage rate Let f ∗ (π) denote the arg min of the RHS and a ∗ = f ∗ (π) . Then, J ∗ is achieved by the charging policy (note a ∗ depends on π t )
Smart-meter privacy–(Liu, Khisti, and Mahajan) 6 [LKM] Main results: i.i.d. demand Solution of the dynamic program J ∗ ∶= min θ∈𝒬 S I(S − X; X) where X ∼ P X and S ∼ θ . Let θ ∗ denote the arg min of the RHS. Then, J ∗ is the minimum leakage rate
Smart-meter privacy–(Liu, Khisti, and Mahajan) ⎩ t (y|x t , s t , π t ) = b ∗ (y|x t , s t ) q ∗ . otherwise 0, if y ∈ 𝒴 and y is feasible Normalize P X (y)θ ∗ (y + x − s) ⎨ 6 ⎧ Defjne b ∗ (y|x, s) = Optimal strategies Solution of the dynamic program [LKM] Main results: i.i.d. demand J ∗ ∶= min θ∈𝒬 S I(S − X; X) where X ∼ P X and S ∼ θ . Let θ ∗ denote the arg min of the RHS. Then, J ∗ is the minimum leakage rate Then, J ∗ is achieved by time-invariant action (note b ∗ does not depend on π t )
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