Privacy-Preserving Obfuscation for Distributed Power Systems

1 Pri v ac y-Preserving Obfuscation for Distrib uted Po wer Systems T errence W .K. Mak ˚ , Ferdinando Fioretto ˚ : , and Pascal V an Hentenryck ˚ ˚ Georgia Institute of T echnology , Atlanta, GA, USA : Syracuse Univ ersity , New Y ork, NY , USA wmak@gatech.edu, fﬁorett@syr .edu, pvh@isye.gatech.edu Abstract —This paper considers the problem of releasing privacy-pr eserving load data of a decentralized operated power system. The paper focuses on data used to solve Optimal P ower Flow (OPF) problems and proposes a distributed algorithm that complies with the notion of Diff erential Privacy , a strong privacy framework used to bound the risk of re-identiﬁcation. The problem is challenging since the application of traditional differential privacy mechanisms to the load data fundamentally changes the nature of the underlying optimization problem and often leads to sever e feasibility issues. The proposed differentially private distrib uted algorithm is based on the Alternating Direction Method of Multipliers (ADMM) and guarantees that the released privacy-pr eserving data retains high ﬁdelity and satisﬁes the A C power ﬂow constraints. Experimental results on a variety of OPF benchmarks demonstrate the effectiveness of the approach. Index T erms —Differential Privacy , Optimal Po wer Flow , ADMM, Distributed computing I . I N T RO D U C T I O N The av ailability of test cases representing high-ﬁdelity power system netw orks is essential to foster research in se veral important power optimization problems, including optimal power ﬂow (OPF), unit commitment, and transmission plan- ning. Howev er, the release of such datasets poses signiﬁcant priv acy risks. For instance, rev ealing the electrical load of a customer may disclose sensitive b usiness activities and manufacturing processes, causing signiﬁcant economic loss. Indirectly , it may also rev eal how transmission operators operate their networks, raising security issues [ 1 ]. Differ ential Privacy (DP) [ 2 ] is a pri vacy frame work that has been sho wn effecti ve in protecting sensitiv e information during a data release process. It prevents the disclosure of sensitiv e information by introducing carefully calibrated noise to the result of a computation. While DP algorithms could be used dir ectly to generate pri vacy-preserving po wer system data, the y face signiﬁcant challenges when the released data is required to preserve domain speciﬁc properties, such as preserving the optimal cost and the feasibility of an A C Optimal Power Flow (A C-OPF) problem. Naive noise addition can drastically degrade the ﬁdelity to the original problem of interest and introduce severe feasibility issues, as sho wn in [ 1 ], [ 3 ], [ 4 ]. Fig. 1 , reported from [ 3 ], emphasizes these results. It shows the average load distance (as L 1 ) between the original and the priv acy-preserving loads for a set of 29 networks, at v arying obfuscation parameter α . The percentages of instances with a feasible AC-OPF solution are shown above the bars. 0.1 1 10 0 2 4 6 8 10 Indistinguishability ( ⍺) Error (p.u.) 43.2% 14.4% 3.04% Fig. 1. A verage L 1 error and percentages of feasible A C- OPF instances. Interestingly , a recent body of work has sho wn that it is possi- ble to release A C-feasible obfus- cated load data that also satisﬁes the notion of differential priv acy [ 3 ]–[ 5 ]. Despite the soundness and effecti veness of such data release techniques, these methods rely on the presence of a trusted data cura- tor that can collect sensitiv e loads from all the system participants. Howe ver , this is impractical in very large systems with distributed loads and generators (e.g., multiple microgrids). Even if the power system is operated centrally , it is typically owned and controlled by various parties, e.g., load customers, transmission system operators (TSO), distrib ution system oper - ators (DSO), and generation companies. These parties operate with speciﬁc customer and legal agreements, which render the transmission of proprietary data to a centralized server infeasible. T o ov ercome these limitations, this paper introduces the Privacy-pr eserving Decentralized OPF (PD-OPF) , a novel decentralized and priv acy-preserving framework that allows multiple power system parties to release their data pri v ately without relying on a trusted data curator . Crucially , the frame- work guarantees that the released data produces a feasible A C power ﬂo w , and that its OPF cost is close to that of the original OPF . The heart of the mechanism is a distributed optimization procedure that relies on the Alternating Dir ection Method of Multipliers (ADMM) to redistribute the noise introduce by traditional DP algorithms to satisfy the desired properties. While the paper focuses on preserving the priv acy of individual loads, the framew ork is general and can be used to protect other sensitive quantities (e.g., generator capabilities). Contributions The key contributions of this work are as follows: (1) It introduces DP-OPF , a nov el, distributed, differ- entially priv ate mechanism that relies on ADMM to obfuscate the individual loads while ensuring AC-OPF feasibility on the obfuscated data. (2) DP-OPF satisﬁes the notion of  -local differential priv acy , providing a strong priv acy guarantees. (3) Experimental results on a large collection of OPF benchmarks illustrate that the proposed approach ﬁnds high-quality AC- feasible solutions, and that the results are comparable to those obtained with a centralized version with a data curator . 2 I I . R E L A T E D W O R K There is a rich literature on theoretical results of DP (see for instance [ 6 ] and [ 7 ]). The literature on DP applied to power systems includes considerably fewer ef forts. ´ Acs and Castelluccia [ 8 ] exploit a direct application of the Laplace mechanism to hide user participation in smart meter data sets, achieving  -DP . Zhao et al. [ 9 ] study a DP schema that exploits the ability of households to charge and discharge a battery to hide the real energy consumption of their appliances. Liao et al. [ 10 ] introduce Di-PriD A, a pri v acy-preserving mech- anism for appliance-lev el peak-time load balancing control in the smart grid, aimed at masking the consumption of top-k appliances of a household. Finally , Zhou et al. [ 11 ] introduce the notion of monotonicity of DC-OPF operator , which requires that monotonic changes in the network loads induce monotonic changes in the DC-OPF objectiv e cost. This enables a characterization of the network, which is useful to preserve the priv acy of monotonic networks . There are also related work on priv acy-preserving imple- mentations of the ADMM algorithm. Zhang et al. [ 12 ] pro- posed a version of the ADMM algorithm for pri vac y preserv- ing empirical risk minimization problems, a class of con ve x problems used for regression and classiﬁcation tasks. Huang et al. [ 13 ] proposed an approach that combines an approximate augmented Lagrangian function with time-varying Gaussian noise for general objecti ve functions. Finally , Ding et al. [ 14 ] proposed P-ADMM, to provide guarantees within a relaxed model of dif ferential priv acy (called zero-concentrated DP). The priv acy-preserving distributed learning literature fo- cuses almost entirely on problems whose objectiv e func- tions are smooth and strongly con vex. Additionally , most approaches suffer one shortcoming: The priv acy loss being provided as a guarantee is a function of the iteration counts of the algorithm, which can be huge if a large number of iterations is required to con verge to a feasible solution. In contrast, this work provides bounded priv acy loss irrespectiv e of the number of iterations. It also ensures that the priv acy- preserving data is A C-OPF feasible and that the solution cost stays close to the original ones. I I I . P R E L I M I N A R IE S A. Optimal P ower Flow Optimal P ower Flow (OPF) is the problem of determin- ing the most economic generator dispatch to meet the load demands in a power network. A power network N can be viewed as a graph p N , E q where the set of buses N “ r n s represents the nodes and the set of lines and transformers E Ď tp i, j q P N ˆ N u represents the directed arcs. The paper denotes with G and L as for the set of generators and loads in the network, and uses E R to indicate the set of arcs, but in the rev erse direction. The AC-OPF problem ( P OPF ) is speciﬁed in Model 1 , where I , V , Y , and S denote the complex quantities for current, voltage, admittance, and power , respecti vely . The model takes as input the power network N and returns the optimal generator dispatch costs (with ties broken arbitrarily). The objecti ve function O p S g q captures the cost of the generator dispatch, with S g “ x S g 1 , . . . , S g n y denoting Model 1 AC Optimal Power Flow: P OPF variables: S g i , @ i P G ; V i , @ i P N ; S ij , @p i, j q P E Y E R minimize: O p S g q “ ÿ i P N c 2 i p < p S g i qq 2 ` c 1 i < p S g i q ` c 0 i (1) subject to: = V s “ 0 , D s P N (2) v l i ď | V i | ď v u i @ i P N (3) ´ θ ∆ ij ď = p V i V ˚ j q ď θ ∆ ij @p i, j q P E (4) S gl i ď S g i ď S gu i @ i P G Ď N (5) | S ij | ď s u ij @p i, j q P E Y E R (6) S g i ´ S d i “ ř p i,j qP E Y E R S ij @ i P N (7) S ij “ Y ˚ ij | V i | 2 ´ Y ˚ ij V i V ˚ j @p i, j q P E Y E R (8) the vector of generator dispatch values. Constraint ( 2 ) sets the reference angle to zero for the slack bus s P N to eliminate numerical symmetries. Constraints ( 3 ) and ( 4 ) capture the voltage and phase angle dif ference bounds. Constraints ( 5 ) and ( 6 ) enforce the generator output and line ﬂow limits. Finally , Constraints ( 7 ) capture Kirchhoff ’ s Current Law and Constraints ( 8 ) capture Ohm’ s Law . The solution set satisfying Constraints ( 2 ) to ( 8 ) for a giv en set of load demands S d “ x S d 1 , . . . , S d n y is denoted by C P F p S d q . T able I summarizes the common notations used throughout the paper . B. Alternating Dir ection of Multipliers Method (ADMM) ADMM is a widely used distributed procedure solving optimization problems with coupling constraints. Consider an optimization problem of the follo wing form: min x P X ,z P Z f p x q ` g p z q s.t. A x ` B z “ c , (9) where X Ď R n and Z Ď R m are two disjoint sets, x P R n and z P R m denote v ariable v ectors o wned by two distinct groups of agents, and A x ` B z “ c describes the set of coupling constraints between the two groups of agents with A P R ` ˆ n , B P R ` ˆ m , and c P R ` . The functions f and g denote the objectives over x and z , respecti vely , and are commonly assumed to be con ve x. The augmented Lagrange function L p x , z , λ q of ( 9 ) is: f p x q ` g p z q ` λ p A x ` B z ´ c q ` ρ 2 | A x ` B z ´ c | 2 (10) where λ P R ` is a vector of Lagrangian multipliers and ρ ą 0 is a penalty term. The vector of Lagrangian multipliers are the dual v ariables associated with the coupling constraints A x ` B z “ c . Giv en a solution tuple p x i , z i , λ i q at iteration i , ADMM [ 15 ] proceeds to the next iteration i ` 1 computing p x i ` 1 , z i ` 1 , λ i ` 1 q as follo ws, in three sequential steps: x i ` 1 “ argmin x P X L p x , z i , λ i q (11) z i ` 1 “ argmin z P Z L p x i ` 1 , z , λ i q (12) λ i ` 1 “ λ i ` ρ p A x i ` 1 ` B z i ` 1 ´ c q . (13) The algorithm terminates when a desired termination condition is reached (e.g., an iteration limit or a con ver gence factor). The 3 T ABLE I C O MM O N N OTA T I ON U S E D I N T H E PA PE R . N Power network  Priv acy budget S g V ector of power generator dispatch α Indistinguishability value S d V ector of load demands β Faithfulness value/parameter P OPF Function solving AC-OPF , with input S d and output S g O ˚ The optimal costs of the original problem C PF The set of feasible AC power ﬂow for P OPF x A vector of variables/values M x A mechanism of x x l , x u Upper and lower bounds of quantity x P x The optimization problem for the accuracy phase of M x < p¨q , = p¨q Real / imaginary component of a complex number Y ˚ , I ˚ , V ˚ Conjugate of admittance matrix Y , current I , and voltage V c 2 , c 1 , c 0 Cost function coefﬁcients λ d , λ g , λ V , λ S Lagrange multiplier for load, generation, voltage, and power ﬂow ρ ADMM penalty parameter quality of the solution at iteration i can be measured by the primal infeasibility (residue) vector [ 16 ] r i p “ A x i ` B z i ´ c , (14) indicating the distance to a primal feasible solution, and the dual infeasibility (residue) vector [ 16 ] r i d “ ρA T B p z i ´ z i ´ 1 q , (15) indicating the distance from the previous local minima. When both infeasibility vectors are zero, ADMM conv erges to a (local) optimal and feasible solution. C. Dif fer ential Privacy Notions The need for data priv acy emerges in two main contexts: the global context, as in when institutions release datasets containing information of se veral users or answer queries on such datasets (e.g., US Census queries [ 17 ], [ 18 ]), and the local context, as in when individuals disclose their personal data to some data curator (e.g., Google Chrome data collection process [ 19 ]). In both contexts, priv acy is achieved through a randomizer M adding noise to the data before releasing. Differ ential privacy [ 2 ] (DP) is an algorithmic property that characterizes and bounds the priv acy loss of an individual when its data participates into a computation. It has originally been proposed in the global priv acy context and, informally , ensures that an adversary would not be able to reliably infer whether or not a particular individual participates in the dataset, even with unbounded computational power and access to every other entry of the dataset. The setting adopted in this work studies the local priv acy context (LDP) [ 20 ], in which each load customer i holds a datum, S d i P C , describing the complex load consumption of the b us i P N . While the standard local dif ferential priv acy frame work is concerned with protecting the participation of an individual into a dataset, in a power system, the individual identity is not a sensitiv e information: It is a public knowledge that each bus may con- nect to a demand. The sensiti ve information is represented by the load magnitude. T o accommodate such notion of priv acy risk, the paper uses the deﬁnition of generalized differ ential privacy for metric spaces [ 21 ] and adapts it to the local differential pri v acy context. W ithout loss of generality , we focus on Lebesgue spaces L 1 , and in particular, consider the complex space C equipped with norm 1. For a given value α ą 0 , a randomized mechanism M is  -LDP for α distances (a.k.a. local α -indistinguishable), if for all x and x 1 P C s.t. } x ´ x 1 } 1 ď α , and for any output response o P C : Pr r M p x q “ o s ď e  Pr r M p x 1 q “ o s . (16) Informally , the LDP deﬁnition adopted ensures that an attacker obtaining access to a priv acy-preserving load value cannot detect, with high probability , the distance between the priv acy-preserving value and its original val ue. The lev el of privacy is controlled by the priv acy loss parameter  ě 0 , with small values denoting strong priv acy . The le vel of indistin- guishability is controlled by the parameter α ą 0 . The abov e deﬁnition allows us to obfuscate load values that are close to one another while retaining the distinction between those that are far apart. Local Differential Priv acy (LDP), including its extension for generic metric spaces, satisﬁes sev eral important properties. In particular , it is immune to post-processing as deﬁned in the following theorem. Theorem 1 (P ost-Pr ocessing Immunity): [ 6 ] Let M be an  -(local) differentially pri v ate mechanism and g be an arbitrary mapping from the set of possible output sequences to an arbitrary set. Then g ˝ M is  -(local) differentially priv ate. I V . D E C E N T R A L I Z E D L O A D O B F U S C A T I O N The decentralized load obfuscation problem is the problem of coordinating the release priv acy-preserving load data in a power system owned and controlled by multiple parties. W e consider a set of agents, each coordinating some power system component, e.g., loads, generators, buses, or power lines. The goal of the problem is to release load data, which is controlled by the load agents. The problem has three desiderata. (1) It requires obfuscation of the loads up to some amount α ą 0 . (2) It requires that the A C-OPF objective induced by the obfuscated loads is close to that attained using the original data. (3) It requires its agents to coordinate the data release process using a decentralized and conﬁned communication process. Formally , the decentralized load obfuscation problem ﬁnds the activ e and reactive, pri v acy-preserving load values ˆ S d i for each load agent i P N that satisfy the following criteria: 1) Privacy : The original load S d i and its priv acy-preserving counterpart ˆ S d i are local α -indistinguishable, for ev ery load i P N . 2) F idelity : For ev ery generator i , the optimal AC-OPF dispatch cost O p ˆ S g i q obtained by using the obfuscated 4 loads ˆ S d i is required to be close to the original A C-OPF dispatch cost O p S g i q up to a user -deﬁned factor β ą 0 : | O p ˆ S g i q ´ O p S g i q| ď β O p S g i q @ i P N . Finally , it requires the computation mechanism to be per- formed in a decentralized fashion. In the following, we denote with O ˚ i as for the original optimal generation costs O p S g i q , which are assumed to be publicly known [ 5 ] (e.g., from the market information). V . T H E P D - O P F M E C H A N I S M This section introduces the Privacy-pr eserving Distributed OPF (PD-OPF) mechanism to solve the decentralized load obfuscation problem. PD-OPF agents operate in two phases: 1) Priv acy Phase During the ﬁrst phase, each load agent i P N applies a LDP protocol to obtain an α -local obfuscated version ˜ S d i of its original load S d i . This process is executed independently and autonomously by each load agent in the system. 2) Fidelity Phase In the second phase, the agents coordinate a distributed process to adjust the priv ate load values ˜ S d i , to new values ˆ S d that achiev e the ﬁdelity goal, while deviating as little as possible from the local α -obfuscated loads ˜ S d i . The next sections describe in details the PD-OPF phases. A. Privacy Phase In the pri vac y phase, each (load) agent i perturbs its load data S d i , independently from other agents, so to generate an α - local indistinguishable load ˜ S d i . T o do so, the agents use a ver- sion of the Laplace Mechanism , a method used to guarantee an  -LDP priv ate responses to numeric functions [ 6 ]. The Laplace distribution with 0 mean and scale b , denoted by Lap p ξ q , has a probability density function Lap p x | ξ q “ 1 2 ξ e ´ | x | ξ . Let Lap p ξ q to be the Laplace distribution with parameter ξ , f a numeric function that maps datasets to R , and z to be a random variable drawn from Lap ` α  ˘ . The Laplace mechanism for local differential priv acy for α distances is deﬁned as follows: Theorem 2 (Laplace Mechanism): The Laplace mechanism that outputs f p x q ` z achieves α -local indistinguishability . Since the load data is represented in the comple x form, agents use the Polar Laplacian mechanism [ 3 ], [ 22 ], which is a generalization of the Laplace mechanism to Euclidean spaces. The mechanism satisﬁes α -local obfuscation [ 3 ], [ 21 ]. For simplicity , the paper refers to the the Laplace mechanism as for the Polar Laplace mechanism. B. F idelity Phase While simply adding Laplace noise to each load satisﬁes local α -indistinguishability , the resulting po wer system data may no longer be A C feasible, nor it may induce a similar optimal dispatch costs. T o ﬁnd a set of loads ˆ S d that satisfy the ﬁdelity criteria, a post-processing step that uses a bi-lev el program P B L can be formulated as follo ws [ 3 ]: P B L “ min } ˆ S d ´ ˜ S d } 2 (17) s.t.: ˇ ˇ O p S g q ´ O ˚ ˇ ˇ ď β O ˚ (18) S g “ P OPF p ˆ S d q . (19) The upper lev el objecti ve Eq. ( 17 ) minimizes the L2 distance between the noisy loads ˜ S d and the (post-processed) load variables ˆ S d . Constraint ( 19 ) captures the A C-OPF require- ment. It computes an AC optimal generator dispatch S g for the post-processed loads ˆ S d . Finally , Constraint ( 18 ) requires the generator dispatch to satisfy the ﬁdelity goal. Solving bilev el programs is challenging computationally , being strongly NP-Hard [ 23 ]. T o address the underlying com- putational challenge, an ef ﬁcient relaxation of problem P B L can be provided as in [ 3 ]: P RBL “ min } ˆ S d ´ ˜ S d } 2 (20) s.t.: ˇ ˇ O p S g q ´ O ˚ ˇ ˇ ď β O ˚ (21) A C Power Flo w: ( 2 ) ´ ( 8 ) . (22) It relax es the optimality requirement Eq. ( 19 ) and only requires A C feasibility (Eq. ( 22 )). The mechanism restores feasibility of the loads and ensures the existence of a dispatch whose cost is close to the optimal one. C. Decentralized F idelity Phase T o coordinate the resolution of problem P RBL in a decen- tralized fashion, the problem is expressed into the format of Eq. ( 9 ) and solved using an ADMM protocol. The ADMM mechanism used follows the component-based dual decompo- sition framework [ 16 ], [ 24 ] and models each power system component as an individual agent. The framew ork considers four types of agents: load demand agents D , generator agents G , line agents L , and bus agents B . Figure 2 illustrates the ADMM communication scheme of adopted by each agent ( i P N , if it is a bus, load, or generator agent), or ( p i, j q P E , if it is a line agent). It is summarized in the following three steps. At each iteration: 1) Load, generator, and line agents compute their individ- ual consensus variables , respectiv ely , S d p D q i , for load agent i , S g p G q i , for generator agent i , and S p L q ij , V p L q ij (and S p L q j i , V p L q j i for the rev erse direction) for line agent p ij q . Collectively , they form a consensus vector x “ x S d p D q i , S g p G q i , S p L q ij , V p L q ij , S p L q j i , V p L q j i y (see Eq. ( 11 )), which is sent to their connecting bus agents. 2) Upon recei ving its neighboring load, generator , and line consensus variables, b us agent i computes the response value z “ x S d p B q i , S g p B q i , S p B q ij , V p B q i y (see Eq. ( 12 )) and send v alue S d p B q i to load agent i , v alue S g p B q i to generator agent i , and values S p B q ij , V p B q i to line agent p ij q , for each line ( i, j ) connected to bus i . 3) Finally , each agent updates its corresponding dual vari- ables: λ d i , for load agent i , λ g i , for generator agent i , and λ V ij , λ S ij , for line agent p i, j q . Collectiv ely , they are identiﬁed with λ “ x λ d i , λ g i , λ V ij , λ S ij y , using the notation in Eq. ( 13 ). 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The ADMM-based LDP post-processing step of PD-OPF . Model 2 ADMM: Load agent D i p P load q inputs: x ρ, λ d i , ˜ S d i , S d p B q i y variables: S d p D q i minimize: } S d p D q i ´ ˜ S d i } 2 ` λ d i ¨ S d p D q i ` ρ 2 } S d p D q i ´ S d p B q i } 2 (23) Model 3 ADMM: Generator agent G i p P g en q inputs: x ρ, λ g i , O ˚ i , S g p B q i y variables: S g p G q i minimize: λ g i ¨ S g p G q i ` ρ 2 } S g p G q i ´ S g p B q i } 2 (24) local constraints: S gl i ď S g p G q i ď S gu i (25) O ˚ i p 1 ´ β q ď O p S g p G q i q ď O ˚ i p 1 ` β q (26) the bus agents to those of their connected components x . The ﬁdelity constraint (Eq. ( 21 )) and the AC Power Flow constraints (Eq. ( 22 )) are enforced as local constraints by each agent. Finally , the load agents control the minimization term Objectiv e ( 20 ) of problem P RBL , to control the deviation of the new , post-processed load w .r .t. the Laplace obfuscated coun- terpart. A detailed description of the individual optimization problems computing the local Lagrange functions (Eq. 11 ) for the load, generator , and line agents, and (Eq. 12 ) for the bus agents is gi ven as follows. Load agent The optimization step performed by each load agent i ( i P N ), at each iteration, produces a load value S d p D q i and is sho wn in Model 2 . Eq. ( 23 ) captures the load augmented Lagrange function (see Eq. ( 20 )) and the agent coupling constraints described as penalty terms. The ﬁrst term of the objecti ve is the L2 distance between the load v alue S d p D q i and the Laplace load value ˜ S d i . The remaining terms correspond to the load coupling constraint, matching the load values S d p D q i to the feedback signal S d p B q i from the connecting bus. Generator agent The objective of the generator agent i ( i P N ), at each iteration, is that of producing a dispatch value S g p G q i that matches the feedback signal S g p B q i from the connecting bus. The problem is reported in Model 3 . Therein, Model 4 ADMM: Line agent L i,j p P line q inputs: x ρ, λ S ij , λ V ij , S p B q ij , V p B q ij , λ S j i , λ V j i , S p B q j i , V p B q j i y variables: S p L q ij , S p L q j i , V p L q ij , V p L q j i minimize: ÿ p e,f qPtp i,j q , p j,i qu r λ S ef ¨ S p L q ef ` λ V ef ¨ V p L q ef ` ρ 2 p} S p L q ef ´ S p B q ef } 2 ` } V p L q ef ´ V p B q ef } 2 qs (27) local constr . = V p L q ij “ 0 , if i “ s ; = V p L q j i “ 0 , if j “ s ; (28) v l e ď | V p L q ef | ď v u e , @p e, f q P tp i, j q , p j, i qu (29) ´ θ ∆ ef ď = p V p L q ef V p L q˚ f e q ď θ ∆ ef (30) | S p L q ef | ď s u ef , @p e, f q P tp i, j q , p j, i qu (31) S p L q ef “ Y ˚ ef | V p L q ef | 2 ´ Y ˚ ef V p L q ef V p L q˚ ef @p e, f q P tp i, j q , p j, i qu (32) Model 5 ADMM: Bus agent B i p P bus q inputs: x ρ, λ d i , S d p D q i , λ g i , S g p G q i y , x λ S ij , S p L q ij , λ V ij , V p L q ij |@p i, j q P E Y E R y variables: S d p B q i , S g p B q i , V p B q i , S p B q ij @p i, j q P E Y E R minimize: λ d i ¨ S d p B q i ` ρ 2 } S d p B q i ´ S d p D q i } 2 ` λ g i ¨ S g p B q i ` ρ 2 } S g p B q i ´ S g p G q i } 2 ` ÿ p i,j qP E Y E R r λ S ij ¨ S p B q ij ` ρ 2 } S p B q ij ´ S p L q ij } 2 ` λ V ij ¨ V p B q i ` ρ 2 } V p B q i ´ V p L q ij } 2 s (33) local constraint: S g p B q i ´ S d p B q i “ ÿ p i,j qP E Y E R S p B q ij (34) Eq. ( 24 ) describes the generator agent coupling constraints as penalty terms. The optimization model ensures that the dispatch values satisfy the feasible bounds (Eq. ( 25 )), and that the dispatch cost stays within the ﬁdelity requirement (Eq. ( 26 )). Line agent The objective of the line agent ( ij ) ( p i, j q P E ), is that of ﬁnding ﬂow values S p L q ij and S p L q j i and v oltage v alues V p L q ij and V p L q j i that match the corresponding feedback signals S p B q ij , V p B q i and S p B q j i , V p B q j , computed by the buses i and j , respectiv ely . The optimization is illustrated in Model 4 . It describes four coupling constraints: two associated to the voltage v alues and two associated to the ﬂo w values (Eq. ( 33 )). The model also ensures the voltages and power ﬂo ws are within the feasible bounds (Eqs. ( 29 ) to ( 31 )), and that the A C power ﬂo w constraints are satisﬁed (Eq. ( 32 )). The voltage angle = V p L q ij { = V p L q j i is zero if it connects to a slack bus (Eq. ( 28 )). Bus agent At each iteration, bus agent i performs the opti- mization described in Model 5 . Its objective is that of ﬁnding load value S d p B q i , generator value S g p B q i , voltage value V p B q i , and ﬂo w values S p B q ij , for each connecting line p i, j q P E Y E R , that match the state variables sent from the load, generator , and line agents, respecti vely . The model also ensures the satisfaction of the ﬂow balance constraint (Eq. ( 34 )). The ADMM coordination process coordinating all agents is 6 illustrated in the Appendix (Algorithm 2 ). Even though the ADMM agent structure comes from [ 16 ], the ADMM scheme used by PD-OPF serves as a distributed resolution of P RBL , rather than a traditional scheme for solving OPF . It redistributes the noise introduced by the Laplace mechanism optimally to satisfy the ﬁdelity criteria. Theorem 3: PD-OPF satisﬁes local α -indistinguishability . Pr oof . By Theorem 2 , the load values obtained by the applica- tion of the Laplace mechanism satisfy α -local indistinguisha- bility . The ADMM mechanism makes use of exclusiv ely the priv acy-preserved load v alues ˜ S d (computed by the application of the Laplace mechanism), as well as additional public information (e.g. the local cost values O ˚ i ). Therefore, by post- processing immunity of differential priv acy , PD-OPF satisﬁes local α -indistinguishability . l V I . E X P E R I M E N TA L R E S U LT S This section reports on the obfuscation quality and ability to conv erge of PD-OPF . Additionally , the proposed method is compared with a centralized version that solves problem P RBL , thus admitting the presence of a centralized data curator . The experiments are performed on a variety of NEST A [ 25 ] bench- marks. Parameter  is ﬁxed to 1.0, the indistinguishability level α varies from 0.01 to 0.1 in p.u. (i.e. 1 MV A to 10 MV A), and the ﬁdelity level β varies from 10 ´ 2 to 10 ´ 1 (i.e. from 1% to 10% of the optimal cost difference). PD-OPF is limited to use 5000 iterations. All the models are implemented using PowerModels.jl [ 26 ] in Julia with nonlinear solver IPOPT [ 27 ]. Choosing a ﬁxed penalty factor ρ to driv e con vergence is challenging [ 16 ]. Thus, the experimental routine adjusts ρ dynamically , using the maximum primal and dual infeasibility values,  p “ max r p and  d “ max r d , respecti vely (in spirit of [ 16 ]). Higher values of ρ encourage the satisfaction of the primal constraints, while lower values shift weights to the objecti ves and reduce the dual infeasibilities [ 16 ]. The heuristic adopted changes ρ when the distance between  p and  d becomes too large: ρ “ min tp 1 ` c q ρ, ρ u , if  p ą c t  d , ρ “ max t ρ 1 ` c , ρ u , if  d ą c t  p . The scaling factor c is set to 2% , the threshold parameter c t to 7 . 0 , and upper ρ and lower ρ bounds to 10 6 and 5 , respecti vely . T o allow PD-POPF to restore primal feasibility , a feasi- bility boosting procedure is implemented as follows. When the iteration counter 4500 iterations, if the maximum primal infeasibility is larger than 10 ´ 3 , ρ will be increased by: min tp 1 ` c q ρ, ρ u . W e call this phase feasibility boosting . A. Quality of Load Demand Obfuscation Figure 3 depicts the original load values (Orig.) associated to the IEEE-57 bus systems, and compares them with those generated by the Laplace mechanism (Lap.) and by PD-OPF . The ﬁgure illustrates that the post-processing step used in PD- OPF modiﬁes the original loads. Since the Laplace mechanism does not conv erge to an AC feasible solution, PD-OPF further modiﬁes the Laplace-generated loads. The ﬁgure does not 0 5 Active Load (p.u.) 5 0 Reactive load (p.u.) = 0 . 0 0 5 0 5 Active Load (p.u.) Reactive load (p.u.) = 0 . 0 1 Fig. 3. Original loads distance from the Laplace and the PD-OPF Mechanisms on the IEEE-57 bus system, at varying of the indistinguishability value α “ 0 . 005 (left) and α “ 0 . 01 (right). report the A C-feasible loads due to large overlaps with PD- OPF values. B. Quality of Privacy Loss Minimization Figure 4 illustrates the difference between the loads pro- duces by PD-OPF and those produces by a centralized imple- mentation of problem P RBL [ 5 ]. The difference is measured in terms of distance from the Laplace obfuscated loads (av eraged ov er 50 instances). The dif ferences in the IEEE-39 test case are due to the feasibility boosting phase, acti v ated to improve the primal feasibility . In the other test cases the dif ferences between the two approaches are negligible, thus validating the use of a decentralized solution for releasing loads when a centralized trusted data curator is unav ailable. C. Quality of Fidelity Restoration Figure 5 illustrates the average percentage difference on the dispatch cost dif ferences ˆ O between the original and obfus- cated loads produced by PD-OPF: 100 ˆ O p ˆ O ´ O p P OPF p S d qq O p P OPF p S d qq . Since a PD-OPF implements a relaxation of Constraint ( 19 ), the Figure also reports a comparison using a centralized procedure that solves an A C-OPF with the PD-OPF loads as input. The experimental results indicate that PD-OPF is able to restore the problem ﬁdelity well, even when the ﬁdelity requirement β are as small as 0 . 01 % of the original costs. D. Con verg ence Quality & Runtime Finally , table II presents the maximum and dual infeasibil- ities (in p.u.), before and after (marked with ˚ ) activ ating the feasibility boosting procedure. The table clearly illustrates the beneﬁts of the boosting procedure. It is able to reduce the primal infeasibility of up to two order of magnitude, albeit at a cost of a larger dual infeasibility . Figure 6 illustrates the details of one run on the IEEE-39 benchmark. After a few iterations, both the primal and the dual infeasibilities stabilize in the range r 10 1 , 10 ´ 1 s (top-left), and the generator costs stabilize after 2000 iterations (bottom-left). When the feasibility boosting is acti vated, the coordination agent increases the parameter ρ (bottom right), inducing all agents to re-optimize with a higher penalty for violating the coupling constraints. This is obtained at a cost of a larger dual feasibility (top-right). 7 0.1 0.01 0 10 20 30 40 L2 distance from Laplace sol. (p.u.) = 0 . 0 5 0.1 0.01 0 20 40 60 80 L2 distance from Laplace sol. (p.u.) = 0 . 1 PF-OPF Centralized 0.1 0.01 0 10 20 30 40 50 60 70 L2 distance from Laplace sol. (p.u.) = 0 . 0 5 0.1 0.01 0 20 40 60 80 100 120 140 L2 distance from Laplace sol. (p.u.) = 0 . 1 PF-OPF Centralized 0.1 0.01 0 20 40 60 80 L2 distance from Laplace sol. (p.u.) = 0 . 0 5 0.1 0.01 0 25 50 75 100 125 150 175 L2 distance from Laplace sol. (p.u.) = 0 . 1 PF-OPF Centralized Fig. 4. L2 distance between the Laplace obfuscated data and the PD-OPF and the Centralized obfuscated data. IEEE-39 (left), IEEE-57 (center), IEEE-189 (right). α “ r 0 . 05 , 0 . 1 s , β “ r 0 . 01 , 0 . 1 s . 0.1 0.01 20 15 10 5 0 5 10 percentage (%) Mechanism PD-OPF AC-OPF 0.1 0.01 20 15 10 5 0 5 10 percentage (%) Mechanism PD-OPF AC-OPF 0.1 0.01 20 15 10 5 0 5 10 percentage (%) Mechanism PD-OPF AC-OPF 0.1 0.01 20 15 10 5 0 5 10 percentage (%) Mechanism PD-OPF AC-OPF Fig. 5. Dispatch costs differences between the optimal and the PD-OPF solution (PD-OPF) and its centralized AC-OPF counterpart. IEEE 39 (top) & IEEE 57 bus (bottom), α “ 0 . 01 (left), 0 . 1 (right), β “ r 0 . 1 , 0 . 01 s . T ABLE II P R I M A L & D UA L I N F E A S I B I L I T Y , A N D S I M . RU N T I M E . α “ 0 . 1 , β “ 0 . 1 . Primal Primal ˚ Dual Dual ˚ Time (min.) nesta case3 lmbd 0.036 0.001 0.173 0.079 1.147 nesta case4 gs 0.023 0.001 0.092 13.953 2.110 nesta case5 pjm 1.580 0.015 3.094 380.243 3.501 nesta case6 c 0.203 0.001 0.835 7.088 2.607 nesta case6 ww 0.094 0.001 0.419 7.919 3.215 nesta case9 wscc 0.197 0.001 1.224 5.908 2.776 nesta case14 ieee 0.579 0.001 2.228 19.762 5.141 nesta case24 ieee rts 0.293 0.006 1.276 540.403 11.157 nesta case29 edin 0.216 0.128 2.393 3027.724 38.686 nesta case30 as 0.386 0.001 1.685 15.223 12.592 nesta case30 fsr 0.416 0.001 2.215 14.001 10.738 nesta case30 ieee 0.621 0.001 2.831 37.137 11.167 nesta case39 epri 0.291 0.026 1.597 1849.358 15.273 nesta case57 ieee 0.584 0.001 2.951 62.776 19.614 nesta case73 ieee rts 0.402 0.008 2.762 691.576 45.131 nesta case118 ieee 0.968 0.004 4.427 394.885 82.160 nesta case189 edin 3.214 0.017 14.780 871.245 86.908 V I I . C O N C L U S I O N This paper presents a distributed framework based on ADMM and Local Differential Priv acy (LDP) to preserve the priv acy of customer loads while maintaining po wer ﬂows close to the optimal solution. W e formally present the distributed priv acy-preserving problem, and a tw o phase distrib uted mech- anism Pri v acy-preserving Distributed OPF (PD-OPF) to guar- antee pri vacy and ﬁdelity . The mechanism satisﬁes pri v acy properties. Experimental ev aluations on the NEST A bench- 0 2000 4000 Iterations 0 5000 10000 15000 Infeasibilities (p.u.) Primal Infeas. Dual Infeas. 4400 4600 4800 5000 Iterations 0 200 400 600 Infeasibilities (p.u.) Primal Infeas. Dual Infeas. 0 2000 4000 Iterations 87500 90000 92500 95000 97500 Cost Dispatch Cost 0 2000 4000 Iterations 0 20000 40000 60000 80000 100000 p e n a l t y Fig. 6. IEEE-39 bus: Primal  p and dual  d infeasibilities (Full-scale: top left, ą 4400 iterations: top right), generator dispatch costs (bottom left), penalty ρ (bottom right); α “ 0 . 1 , β “ 0 . 1 . The vertical dotted line marks the activ ation of the boosting procedure. marks sho w that the mechanism provides high obfuscation quality , satisﬁes the ﬁdelity requirements, and achieves com- parable results when compared to a centralized approach. A C K N O W L E D G E M E N T The authors would like to thank Kory Hedman for extensiv e discussions on various obfuscation techniques. This research is partly funded by the ARP A-E Grid Data Program under Grant 1357-1530. 8 R E F E R E N C E S [1] F . Fioretto, T . W . K. Mak, and P . V . Hentenryck, “Pri vac y-preserving obfuscation of critical infrastructure networks, ” in Pr oceedings of the International Joint Conference on Artiﬁcial Intelligence (IJCAI) , 2019, pp. 1086–1092. [2] C. Dwork, F . McSherry , K. Nissim, and A. 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Instead of using Polar Laplace mechanism in the Pri v acy Phase, this section showcases another LDP mechanism: the Piecewise Mechanism [ 28 ]. The Piecewise Mechanism also satisﬁes the L 1  -LDP for α distances deﬁnition. It requires all input data x i to be normalized within r´ 1 , 1 s from r x i , x i s . Let: C “ e  { 2 α ` 1 e  { 2 α ´ 1 , L p x i q “ C ` 1 2 x i ´ C ´ 1 2 , and R p x i q “ L p x i q ` C ´ 1 . The mechanism perform obfuscation based as in Algorithm 1 . T o implement the Piecewise Mechanism, linear transforma- Algorithm 1: Piecewise Mechanism for LDP 1 Sample p „ Uniform pr 0 , 1 sq 2 if p ď e  { 2 α e  { 2 α ` 1 then 3 Sample ˜ x i „ Uniform pr L p x i q , R p x i qsq 4 else 5 Sample ˜ x i „ Uniform pr´ C, L p x i qs Y r R p x i q , C sq 6 Return ˜ x i tions are used by each of the load agents D i to normalize activ e and reacti ve parts of the load value S d i into [-1,1]. T o transform from a bounded domain x i P r x i , x i s to y i P r´ 1 , 1 s (and vice versa), the following equation is used: y i “ 2 x i ´ x i x i ´ x i ´ 1 . T able III sho ws the primal and dual conv ergence quality and simulation runtime similar as in previous section. Figure 7 shows the ﬁdelity can again be restored by the ADMM mech- anism. Figure 8 shows comparable obfuscation quality when comparing to the Laplace mechanism in Figure 3 . Finally , Figure 9 shows the ADMM algorithm can achiev e comparable priv acy loss minimization results to centralized optimization. T ABLE III P R I M A L A N D D UA L I N F E A S I B I L I T Y , A N D S I M U L ATI O N R U N T I M E . α “ 0 . 1 , β “ 0 . 1 . Primal Primal ˚ Dual Dual ˚ Time (min.) nesta case3 lmbd 0.001 0.001 0.015 0.015 0.089 nesta case4 gs 0.031 0.001 0.151 11.733 3.505 nesta case5 pjm 1.820 0.015 3.290 382.929 3.416 nesta case6 c 0.006 0.001 0.038 0.180 0.479 nesta case6 ww 0.217 0.072 1.064 20667.869 4.165 nesta case9 wscc 0.023 0.001 0.119 1.596 1.445 nesta case14 ieee 0.085 0.001 0.392 5.402 8.722 nesta case24 ieee rts 0.133 0.008 0.859 611.856 11.294 nesta case29 edin 0.197 0.098 2.676 3810.460 82.134 nesta case30 as 0.161 0.001 0.847 5.090 9.244 nesta case30 fsr 0.050 0.001 0.250 1.525 9.824 nesta case30 ieee 0.211 0.001 1.074 9.788 9.714 nesta case39 epri 0.920 0.020 4.371 1029.756 37.142 nesta case57 ieee 1.201 0.001 4.947 104.336 40.772 nesta case73 ieee rts 0.219 0.011 1.654 777.139 43.815 nesta case189 edin 1.432 0.016 6.904 799.463 83.319 0.1 0.01 20 15 10 5 0 5 10 percentage (%) Mechanism PD-OPF AC-OPF 0.1 0.01 20 15 10 5 0 5 10 percentage (%) Mechanism PD-OPF AC-OPF Fig. 7. IEEE 57 bus. Percentage Dif ference on the Dispatch Costs after ADMM mechanism and AC validation: α “ 0 . 01 p l ef t q , 0 . 1 p rig ht q , β “ r 0 . 1 , 0 . 01 s . A verage over 50 instances. 5 0 5 Active Load (p.u.) 2 0 2 Reactive load (p.u.) = 0 . 0 0 5 5 0 5 Active Load (p.u.) Reactive load (p.u.) = 0 . 0 1 Fig. 8. Loads from the original dataset and the Piecewise Linear & ADMM Mechanisms on the IEEE-57 bus system. B. ADMM Coor dination Pr ocess The ADMM coordination process ex ecuted by all PD-OPF agents is illustrated in Algorithm 2 . Lines 1 to 3 initialize all v ariables associated to load, generator , and line agents, respectiv ely . These agents, hence, perform their optimization step (lines 6 to 8), independently , and send their state (consen- sus) variables to the corresponding bus agents. Next, the bus agents perform the associated local optimization step and send the feedback values back to the corresponding load, generator, and line agents (line 10). Finally , the multipliers v ariables λ are updated by each individual agents (lines 12 to 14). At the end of each iteration, the parameter ρ can be updated bu all agents. 10 0.1 0.01 0 50 100 150 200 250 L2 dist. from Piecewise Linear (p.u.) = 0 . 0 5 0.1 0.01 0 100 200 300 400 500 L2 dist. from Piecewise Linear (p.u.) = 0 . 1 PD-OPF Centralized 0.1 0.01 0 20 40 60 80 100 120 L2 dist. from Piecewise Linear (p.u.) = 0 . 0 5 0.1 0.01 0 50 100 150 200 250 L2 dist. from Piecewise Linear (p.u.) = 0 . 1 PD-OPF Centralized Fig. 9. L2 distance between ADMM/Centralized Mechanisms and Piecewise Linear Obfuscated Dataset on the IEEE-39 (top) and IEEE-57 (bottom) bus systems, with α “ r 0 . 05 , 0 . 1 s , and β “ r 0 . 01 , 0 . 1 s . Algorithm 2: ADMM: Main routine Inputs : x N , ρ init , t max y , x ˜ S d i |@ D i y , x O ˚ i |@ G i y 7 ρ Ð ρ init 8 x λ d i , S d p B q i y Ð x 0 , 0 y @ D i ; x λ g i , S g p B q i y Ð x 0 , 0 y @ G i ; 9 x λ S ij , S p B q ij y Ð x 0 , 0 y ^ x λ V ij , V p B q ij y Ð x 0 , 0 y @p i, j q P L i,j 10 f or t “ 1 , 2 , . . . , t max do 11 Optimization of load, generator , and line agents 12 @ D i : S d p D q i Ð P load ( x ρ, λ d i , ˜ S d i , S d p B q i y ) 13 @ G i : S g p G q i Ð P gen ( x ρ, λ g i , O ˚ i , S g p B q i y ) 14 @ L i,j : S p L q ij , V p L q ij , S p L q j i , V p L q j i Ð P line ( x ρ, λ S ij , λ V ij , S p B q ij , V p B q ij , λ S j i , λ V j i , S p B q j i , V p B q j i y ) 15 Optimization of bus agents 16 @ B i : S d p B q i , S g p B q i , V p B q i , S p B q ef Ð P bus px ρ, ´ λ d i , S d p D q i , ´ λ g i , S g p G q i y , x´ λ S ef , S p L q ef , ´ λ V ef , V p L q ef yq 17 Lagrange multiplier update 18 @ D i and B i : λ d i Ð λ d i ` p S d p D q i ´ S d p B q i q 19 @ G i and B i : λ g i Ð λ g i ` p S g p G q i ´ S g p B q i q 20 @ L i,j and B i { B j : λ S ij Ð λ S ij ` p S p L q ij ´ S p B q ij q , λ V ij Ð λ V ij ` p V p L q ij ´ V p B q i q 21 Coordinating agent penalty ρ update (optional) 22 ρ Ð update p() Output : S d i

Privacy-Preserving Obfuscation for Distributed Power Systems

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