Distributed Stability Analysis and Control of Dynamic Networks

Transcription

1 Distributed Stability Analysis and Control of Dynamic Networks M. Anghel and S. Kundu Los Alamos National Laboratory, USA August 4, 2015 Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

2 Research Goals Analysis and control of complex systems Paradigmatic Example Power grids comprised of thousands nonlinear components linked across thousands of kilometers. Problems The grid aging, centralized, and inflexible control infrastructure does not provide the flexibility which power systems need to guarantee its robust stability. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

3 Research Goals Paradigm Shift The new demands on the electricity delivery system are requiring that it function in ways for which it was never originally designed. Challenges (Opportunities) Distributed & renewable generation, emerging demand response, real time measurements (PMU). Stochastic and epistemic uncertainty. The system s dynamic components become increasingly independent and autonomous agents. Future Transformations Modern distributed measurements, control, and management concepts are necessary to operate the grid. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

4 Research Goals Project Goals Construct an algorithmic synthesis of nonlinear and distributed control techniques for large networked (power) systems. Build a decentralized approach which exploits separability and decomposition of the corresponding control problem into nonlinear sub-problems which can be solved locally and efficiently. Design scalable analysis and control algorithms. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

5 Background & Significance The problem we study Transient Stability Basic Stability Problem We assume a dynamical system described by an autonomous set of nonliner equations ẋ = f (x), x R m. (1) Assume that x = 0 is a stable equilibrium point (SEP), i.e. f (0) = 0. Fundamental Stability Question Assume that the system reaches the state x c when the disturbance is finally cleared. Does the trajectory x(t) with x(0) = x c converge to the SEP as t? Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

6 Background & Significance Power System Example I Power Grid Examples e δ v 0 Consider this model: G x d x L x Th ẋ 1 = x 2 ẋ 2 = 10λ 20 sin(x 1 ) x 2 The equilibrium points can be found from the steady-state (power flow) equations: 0 = x 2 0 = 10λ 20 sin(x 10 ) x 20 Milano, F., Power System Modelling and Scripting, Springer, Heidelberg. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

7 Equilibria Background & Significance Power Grid Examples The solutions are: [ x10 x 20 ] [ sin 1 ] (λ/2) = 0 (2) With two equilibrium points (and their periodic images): x 1s = sin 1 (λ/2) x 1u = π sin 1 (λ/2) Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

8 Background & Significance Power Grid Examples Stability and Region of Attraction λ = 0.2 λ = 0.5 λ = x 2 0 x 2 x x x x 1 λ = 1.5 λ = 1.8 λ = x 2 x 2 x x x x 1 Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

9 Background & Significance Power System Example II Power Grid Examples 5 ẋ 1 = x 2 ẋ 2 = sin(x 1 ) 0.5 sin(x 1 x 3 ) 0.4x 2 ẋ 3 = x 4 ẋ 4 = 0.5 sin(x 3 ) 0.5 sin(x 3 x 1 ) 0.5x where x 1 = δ 1, x 2 = ω 1, x 3 = δ 2, x 4 = ω 2. Remark No transfer conductances! Jump to Energy Function Jump to Classical Model δ δ 1 The boundary of the region of attraction for the SEP x s located at the origin ( ). This boundary contains 12 hyperbolic equilibrium points ( ). Four more equlibrium points are also shown ( ). H. D. Chiang, Direct Methods for Stability Analysis of Electric Power Systems,Wiley, Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

10 Methods Lyapunov Function Methods Lyapunov s Theorem If D R m containing x s = 0 and V : D R such that V (0) = 0, andv (x) > 0 x D {0}, V (x) = V f (x) 0 x D, x then the origin is a stable equilibrium. Moreover, if then the origin is asymptotically stable. V (x) < 0 x D {0}, ROA estimates Any level set Ω c = {x R m V (x) c}, such that Ω c D, describes a positively invariant region contained in the domain of attraction of the equilibrium point. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

11 Methods The trouble with Lyapunov Lyapunov Function Methods They are not constructive conditions; they do not tell us how to find a Lyapunov function for a particular system. Testing the positivity conditions required in the theorem is notoriously difficult. Even when both the vector field f and the Lyapunov function candidate V are polynomial, the Lyapunov conditions are essentially polynomial non-negativity conditions which are known to be N P-hard to test. Fortunately, if we relax the polynomial non-negativity conditions to appropriate sum of squares (SOS) conditions, testing SOS conditions can be done efficiently using SDP. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

12 Positive Polynomials Methods Sums of Squares Methods Gramm Matrix Representation Any p R m,2d can be represented as p(x) = z m,d (x) T Qz m,d (x), where z n,d (x) is a vector of monomials z m,d (x) := [1, x 1, x 2,..., x m, x 2 1, x 1x 2,..., x 2 m,..., x d m] T (3) Remark: z m,d (x) is a ( ) m+d d -vector. SOS polynomials p is a sum of squares (SOS) if there exist polynomials {p i } N i=1 such that p = N i=1 p2 i. If p is SOS then p 0. The reverse implications is not true except in some specific cases. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

13 Methods Sums of Squares Methods Theorem Fix p R m,2d. Then p(x) Σ m,2d iff there exists Q 0. Remark All solutions to p(x) = z m,d (x) T Qz m,d (x) can be expressed as Q = Q 0 + n i=1 λ iq i where p = z T m,d Q 0z m,d and each Q i satisfies z T m,d Q iz m,d = 0. Jump to SOS example Determining if a SOS decomposition exists for a given polynomial is equivalent to a linear matrix inequality feasibility problem. find λ 1,..., λ n n s.t. Q 0 + λ i Q i 0 i=1 (4) Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

14 SOS programming Methods Sums of Squares Methods Given w R s and polynomials {p k } s k=0 solve: min α R m w T α subject to: p 0 + s α k p k Σ[x] k=1 This SOS programming problem is an SDP: The cost is a linear function of α. The SOS constraint can be replaced with a LMI constraint. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

15 SOSTOOLS Methods Sums of Squares Methods SOSP SOSTOOLS SDP Converts the SOS program to a SDP. SOSP Solution SOSTOOLS SDP Solution SeDuMi/ SDPT3 Calls the SDP solver. Converts the SDP solution back to the solution of the SOS program. S.Prajna, A.Papachristodoulou, and P.A.Parrilo. SOSTOOLS A Sum of Squares Optimization Toolbox, Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

16 Methods Sums of Squares Methods SOS search for Lyapunov functions SOS technique is used to find polynomial functions V R m, with V (0) = 0 such that V (x) φ 1 (x) Σ m x D (5) V (x) φ 2 (x) Σ m x D (6) for some domain D around x = 0 and positive definite functions φ 1,2 (x). For example φ 1,2 (x) = ɛ 1,2 m i=1 x 2 i and ɛ 1,2 > 0. Notes It is convenient to define the domain D = {x R m p(x) β} for some positive polynomial p(x) = m i=1 x i 2 and β > 0. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

17 Methods Representation Theorems Sums of Squares Methods Putinar s Positivestellensatz Theorem Let K = {x R m g 1 (x) 0,..., g n (x) 0} be a compact set. If p(x) is positive on K, then p(x) = σ 0 + j σ jg j (x) σ 0, σ j Σ m, j. SOS problem Given p(x) find σ j Σ m, j = 1, 2,..., n such that p(x) j σ jg j (x) Σ m. We can also search for unknown coefficients of p so that p is positive on K. M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana University Mathematics Journal, vol. 42, no. 3, pp , J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

18 Power System Example Methods Sums of Squares Methods SOS search for Lyapunov functions If there exists a constant β > 0 and polynomial functions V, p 1,2 R m, and σ 1,2 Σ m such that V (0) = 0 and V σ 1 (β p) p 1 G φ 1 Σ m (7) V σ 2 (β p) p 2 G φ 2 Σ m (8) then x = 0 is a stable equilibrium point. Remark The vectors of polynomials G enforce algebraic constraints produced by a recasting procedure that transforms the non-polynomial ODEs into a set of polynomials DAEs. Jump to Analysis of Non-polynomial systems Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

19 Methods Sums of Squares Methods Remark An algorithm to maximize the size of the invariant subset is needed in order to improve the estimated ROA. Wloszek, Lyapunov based analysis and controller synthesis for polynomial systems using SOS optimization, Ph.D. dissertation, UC Berkeley, Anghel et al, Algorithmic Construction of Lyapunov Functions for Power System Stability Analysis. TCS 60, , Go to Bretas model Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

20 The Trouble with SOS Methods Sums of Squares Methods There are serious difficulties before these algebraic methods can be applied to large power systems. The difficulties are not conceptual but numerical. It is currently very hard to construct Lyapunov functions of systems with large state dimension, for cubic vector fields and quartic Lyapunov functions. This limitation renders the proposed algorithm impractical in its current formulation. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

21 Large Scale Systems Analysis Decomposition/Aggregation Approach System decomposition and analysis of its components (subsystems) and their interconnections. J. Anderson and A. Papachristodoulou, A decomposition technique for nonlinear dynamical system analysis, IEEE TAC 57, , June Fundamental Questions How to guarantee the global stability of a nonlinear interconnected system form the local stability of its subsystems? How to design local controllers that respond to local disturbances? Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

22 Large Scale Systems Analysis Generic Analysis Tools 1 vector Lyapunov functions and linear comparison principles 2 composite Lyapunov functions (do not scale well!) 3 small-gain theorems for networked systems (do not scale well!) Proposed Approach 1 We employ a system decomposition technique in order to derive a collection of low order, weakly interacting subsystems. 2 We perform a Lyapunov stability analysis for each isolated subsystem. 3 We analyze the stability of the full system using the subsystem Lyapunov functions and disturbance analysis techniques. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

23 Large Scale Systems Analysis Network of Dynamical systems System Decomposition We seek a decomposition ẋ 1 = f 1 (x 1 )+g 1 (x 1, x N1 ) ẋ 2 = f 2 (x 2 )+g 2 (x 2, x N2 ). ẋ n = f n (x n )+g n (x n, x Nn ) where f i (0) = 0, g i (x i, 0) = 0, i, and N i : {neighbors of i }. Fundamental Questions Is it asymptotically stable, i.e. lim t x i (t) = 0? If not, is there a distributed stabilizing control u i (x i )? Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

24 Large Scale Systems Analysis Isolated Subsystem Analysis Lyapunov Functions and Interactions Analyze the isolated dynamics ẋ i = f i (x i ) If D i R m i and V i : D i R such that V i (0) = 0, V i (x i ) > 0 x i D i {0}, V i (x i ) = V T i f i (x i ) < 0 x i D i, Finaly, R 0 i = {x i R m i V i (x i ) 1} D i, Let s point the obvious With no interactions R = R 0 1 R R0 n approximates the global ROA. Under interactions from neighbors V i (x i ) = Vi T (f i (x i ) + g i ) < 0 only if g i 2 is sufficiently small. This is not quite so as x i 0. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

25 Large Scale Systems Analysis Generic Network Stability Stability of Two Connected Systems Assume two interacting subsystems ẋ 1 = f 1 (x 1 ) + g 1 (x 1, x 2 ), ẋ 2 = f 2 (x 2 ) + g 2 (x 2, x 1 ). We assume that f i (x i ) = 0 and g i (x 1, x 2 ) = 0, for i = 1, 2. There Lyapunov functions V i (x i ) for each isolated subsystem ẋ i = f i (x i ), and ROA estimates given by Ω i 1 = {x i R n i V i (x i ) 1} for i = 1, 2. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

26 Example I Large Scale Systems Analysis Generic Network Stability Coupled Van der Pol oscillators Dynamics: ẋ 1 = x 2 + ax 2 x 3 ẋ 2 = x 1 + (x1 2 1)x 2 + bx 1 x 4 ẋ 3 = x 4 + ax 4 x 1 ẋ 4 = x 3 + (x3 2 1)x 4 + bx 3 x 2, There quadratic Lyapunov functions : V 1 = 0.77x x 1x x2 2, V 2 = 0.77x x 3x x4 2. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

27 Large Scale Systems Analysis Generic Network Stability Problem Formulation Assume that a disturbance moves the coupled system to the state (x1 0, x 2 0 ) after which the disturbance is removed. Will the system evolve back to its stable equilibrium (0, 0)? x 1 x 2 x 1 x 2 Compute Stability Curves Find the γ 2 (γ 1 ) such that Ω 1 γ 1, γ 1 1, remains invariant under bounded disturbances Ω 2 γ 2, γ 2 1, and the action of the system s dynamics. GoAnghel to Disturbance & Kundu Analysis (LANL) Distributed Analysis & Control of Networks August 4, / 55

28 Example I Large Scale Systems Analysis Generic Network Stability γ a= 0.10 b= 0.10 a= 0.01 b= 0.01 a= 0.05 b= 0.05 γ 2 (γ 1 ) γ 1 Stability Condition For two connected systems the answer is very simple: If for γ 1 = V 1 (x1 0 ) the state x2 0 {x 2 R n V 2 (x 2 ) γ 2 (γ 1 )}, If for γ 2 = V 2 (x2 0 ) the state x1 0 {x 1 R n V 1 (x 1 ) γ 1 (γ 2 )}, then the composite system is stable. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

29 Large Scale Systems Analysis Generic Network Stability Network Stability Conditions Let s assume a system composed of n subsystems: ẋ i = f i (x i ) + g ij (x i, x j ), for k = 1,..., M. (9) j N (i) Assume that the system is in the state (x 0 1,..., x 0 n ). Compute γ i = V i (xi 0 ). For each i solve the SOS programs s 2 Vi s 1j (γ j V j (x j )) q(v i γ i ) Σ n+m. (10) j N (i) If a solution exists for all i the system is stable. The subsystems for which a solution is not found might be unstable. Apply local controllers to restore stability. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

30 Example II Large Scale Systems Analysis Generic Network Stability Coupled 1D Systems ẋ 1 = 0.1x x x 2, V 1 = 0.09x1 2, R 0 1 = [ 3.3, 3.3]. ẋ 2 = 0.6x x x 1, V 2 = 0.11x2 2, R 0 1 = [ 3.0, 3.0]. 3 1 x 1 V 1 2 x V 2 states Lyapunov level sets time time Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

31 Large Scale Systems Analysis Generic Network Stability Iterative Stability Algorithm: Step # 1 V l = ɛ 0 l V j = ɛ 0 j V i = ɛ 0 i V i = ɛ 1 i + Communicate ɛ 0 i = V i (x i (0)) Then, for each i, find the minimum ɛ 1 i [0, ɛ 0 i ), such that Vi T (f i + g i ) < 0 on {x V i [ɛ 1 i, ɛ0 i ], V j [0, ɛ 0 j ] j N i } Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

32 Large Scale Systems Analysis Generic Network Stability Iterative Stability Algorithm: Step # k V l = ɛ k l V j = ɛ k j V i = ɛ k i If ɛ k i =0 : STOP. Else, Communicate ɛ k i (from step k) + V i = ɛ k+1 i Then, for each i, find the minimum ɛ k+1 i [0, ɛ k i ), such that Vi T (f i + g i ) < 0 on {x V i [ɛ k+1 i, ɛ k i ], V j [0, ɛ k j ] j N i } Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

33 Large Scale Systems Analysis Generic Network Stability Example I: Network of Van Der Pol Oscillators ẋ i1 = x i2 ẋ i2 = µ i x i2 (1 x 2 i1 ) x i1+ j N i \{i} ζ ij x i1 x j Polynomial Lyapunov functions, using Expanding interior algorithm, see Wloszek 2003, Anghel et al 2013, and Estimates of isolated ROAs for varying polynomial orders. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

34 Large Scale Systems Analysis Generic Network Stability Example I: Network of Van Der Pol Oscillators The red lines are the contour level sets defined by the initial conditions: ɛ 0 i = V i (x i (0)). Evolution of the stability analysis algorithm. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

35 Large Scale Systems Analysis Stabilizing Control Design Generic Network Stability Necessary when the stability algorithm fails Attributes: Decentralized: activated and designed locally Minimal: applied on certain "rings" in the state-space Design method (at each step k, for each i), Decide if control is needed, i.e. if V T i (f i + g i ) Vi =ɛ k i, V j ɛ k j j i 0. If needed, compute control u k i (x i ), so that V T i (f i + g i + u k i ) Vi =ɛ k i, V j ɛ k j j i < 0 Find the minimum ɛ k+1 i [0, ɛ k i ) with the controlled dynamics. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

36 Control Example Large Scale Systems Analysis Generic Network Stability V 1 V 2 V 3 V 4 V 5 V 6 V 7 x(t) V(t) t t Control applied at subsystems 1, 2 and 5. Conservative design, due to (1) its distributed nature, and (2) the inherent conservativeness of the Lyapunov functions Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

37 Control Example Large Scale Systems Analysis Generic Network Stability x(t) V(t) V 1 V 2 V 3 V 4 V 5 V 6 V t t Control applied at subsystems 1, 2 and 5. Conservative design, due to (1) its distributed nature, and (2) the inherent conservativeness of the Lyapunov functions Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

38 Large Scale Systems Analysis Generic Network Stability Example II: Power System Example Network preserving model: D i δi = P Di P Ei (δ), i = 1,..., L, δ L+i = ω i, i = 1,..., G, M i ω i + D L+i ω i = P Mi P EL+i (δ). Bergen & Hill, A structure preserving model for power 5 6 Load node system stability analysis, TPAS, 100, 25-35, Generator node Overlapping decomposition: δ in = δ i δ n, for i = 1,..., L + G 1, ω in = ω i ω n, for i = 1,..., G 1, ω G = ω n. Jump to Dynamic Load Model Subsystem node 4 1 Reference node Shadow reference Network of three generators and six load nodes. We perform an overlapping decomposition in which the speed dynamics of the reference node (generator node 3) is shared with all the subsystems. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

39 Large Scale Systems Analysis Generic Network Stability Example II: Power System Example Disturbance Analysis V x 2 (a) ROA of subsystem 2 (node 5) (b) ROA of subsystem 7 (node 2) Estimates of regions of attraction of isolated subsystems using expanding interior algorithm. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

40 Large Scale Systems Analysis Generic Network Stability Stability in the sense of Lyapunov, under control A disturbance was created by tripping the line between nodes 5-7 for the duration t [0, 3s] and also tripping the line between nodes 7-8 for t [1s, 3s], essentially disconnecting the nodes 7 and 2 from the rest of the network for t [1s, 2s]. k ɛ k 1 ɛ k 2 ɛ k 3 ɛ k ɛ k ɛ k ɛ k ɛ k State-Feedback Control State-feedback example U 0 7 = 5.1 cos δ 2,1 38.4ω 2,1 1.8ω sin δ 2,1 5.1 which is applied to the speed dynamics equation of the generator 2 of S 7. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

41 Conclusions Key Points Distributed, scalable, and parallel algorithm for large-scale nonlinear dynamical systems Robust to structural changes in the network Local communications and measurements Can be extended to cases when Lyapunov function level-sets show initial increase (Kundu & Anghel 2015) The method opens up the possibility to include more refined subsystem models in the analysis of large (power) systems. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

42 Acknowledgements Many thanks to: Pablo Parrilo Zachary W. Jarvis- Wloszek Antonis Papachristodoulou Stephen Prajna Weehong Tan Federico Milano Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

43 Appendices Classical Power System Model Power System Model In a classical power system model consisting of n synchronous generators the dynamics of the generator phase angles are modeled by the swing equations: δ i = ω i, ω i = λ i ω i + 1 M i (P mi P ei (δ)), (11a) (11b) where P ei (δ) = E 2 i G ii + j,j i E i E j [B ij sin(δ i δ j ) + G ij cos(δ i δ j )]. Return to Power System Example Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

44 Energy Function Appendices Energy Function Methods When G ij = 0 the classical model has the following energy function: V (δ, ω) = 1 M i ωi 2 P i (δ i δi s 2 ) i i E i E j B ij {cos(δ i δ j ) cos(δi s δj s )} j,j i Estimating ROA The Closest UEP algorithm The Boundary of stability region-based Controlling Unstable equilibrium point (BCU) method. Return to Power System Example δ x u1 x u2 Return to Troubles with Lyapunov δ 1 Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

45 Appendices Energy Function Methods... and some Remarkable Weaknesses No analytical energy functions for power systems with transfer conductances exists. The task of computing the critical energy value is very difficult. counter-examples in which these methods produce wrong results. A. Llamas et al, Clarifications of the BCU method for transient stability analysis, PES 10, ,1995. The assumption of the method do not hold generically for power system models. There are no theoretical guarantees. F. Paganini et al, Generic Properties, One-Parameter Deformations, and the BCU Method, TCS 46, , The only methods with zero miss probability are those who survey all UEPs. R. Fischl et al, A comparison of dynamic security indices based on direct methods, IJEPES 10, ,1988. Return to Power System Example Return to Troubles with Lyapunov Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

46 Example Appendices SOS Example The polynomial p = 2x x 1 3x 2 x1 2x x 2 4 can be written as p = z2,2 T Qz 2,2 where x z 2,2 = x 1 x 2 Q 0 = Q 1 = x We can define an affine subspace of symmetric matrices related to p as S p = {Q z T n,d Qz n,d = p(x)} = { Q 0 + } h λ i Q i λ i R i=1 Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

47 Appendices SOS Example p = 2x x 3 1 x 2 x 2 1 x x 4 2 is SOS since Q 0 + λ 1 Q 1 0 for λ 1 = 5. An SOS decomposition can be constructed from a Cholesky factorization: Q + λ 1 Q 1 = L T L where: L = 1 [ ] Thus p = (Lz) T (Lz) == 1 2 (2x 2 1 3x x 1x 2 ) (x x 1x 2 ) 2 A Tutorial on Sum of Squares Techniques for Systems Analysis Antonis Papachristodoulou and Stephen Prajna, ACC Return to SOS Polynomials Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

48 Appendices Analysis of Non-polynomial Systems Analysis of Non-polynomial systems Theorem Any system with non-polynomial nonlinearities can be converted to a polynomial system with a larger state dimension, but with a series of equality constraints restricting the states to a manifold of the original state dimension. 1 Define the new state variables z 3i 2 = sin(δ i ), z 3i 1 = 1 cos(δ i ), z 3i = ω i for i = 1,..., (n 1). 2 Use the chain rule of differentiation to derive the dynamics of the new state variables. 3 Derive the equality constraints that arise from the recasting process: where i = 1,..., n 1. G i (z) = z 2 3i 2 + z2 3i 1 2z 3i 1 = 0, (12) Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

49 Appendices Analysis of Non-polynomial Systems Recasted dynamics: ż = f (z), where z = F (x), f : R M R M, with M = m 1 + m 2, and m 1 = n 1 and m 2 = 2(n 1). ż 1 = z 3 z 2 z 3 ż 2 = z 1 z 3 ż 3 = z 4 0.4z z z z 1 z z 1 z z 2 z z 2 z 5 ż 4 = z 6 z 5 z 6 ż 5 = z 4 z 6 ż 6 = z z z z 5 0.5z z 1 z z 1 z z 2 z z 2 z 5 The recasting process introduces the following constraints: G 1 (z) = z1 2 + z z 2 = 0 G 2 (z) = z4 2 + z z 5 = 0 Return to Power System Example Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

50 Another Example Appendices A model with transfer conductances A model with transfer conductances: ẋ 1 = x 2 ẋ 2 = cos(x 1 x 3 ) cos(x 1 ) sin(x 1 x 3 ) sin(x 1 ) x 2 ẋ 3 = x 4 ẋ 4 = sin(x 1 x 3 ) cos(x 1 x 3 ) cos(x 3 ) x sin(x 3 ) where x 1 = δ 1, x 2 = ω 1, x 3 = δ 2, x 4 = ω 2. Return to Power System Example Bretas and Alberto, Lyapunov function for power systems with transfer conductances: Extension of the invariance principle, TPS 18, , Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

51 Disturbance Analysis Appendices Disturbance Analysis Problem Formulation We consider a polynomial dynamical system ẋ = f (x) + g(x, w), (13) where x(t) R n, w(t) R m, f R n n, f (0) = 0, and g R n m n. We define the peak of w to be bounded by w(t) γ 2. Define the invariant set as Ω γ1 = {x R n V (x) γ 1 }, where V (x), is the Lyapunov function of the isolated system (w = 0). Find the γ 2 such that Ω γ1 remains invariant under these bounded disturbances and the action of the system s dynamics. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

52 Appendices Disturbance Analysis SOS formulation If V (x, w) 0 on the boundary of Ω γ1 for all w(t) γ 2, then Ω γ1 is invariant. Solving the SOS program s 1 (γ 2 w T w) s 2 V q(v γ1 ) Σ n+m. (14) by searching over the polynomials q and the s i, i = 1, 2 to maximize γ 2 subject to (14). Compute stability curve γ 2 (γ 1 ). Remarks SOS techniques can be used to perform Nonlinear Reachable Set Analysis and to incorporate Parametric Uncertainty. Return to Stability of Two Connected Systems Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

53 Network Preserving Model Dynamic Load Modeling In classical power system model, the dynamics of the system is given by the swing equations of the generators alone. The loads in the network are simply modeled as constant impedance for the purpose of stability analysis. This is not necessarily a valid assumption. Especially considering that in a future grid there will be increasingly higher demand side participation which is likely to introduce faster dynamics at the load bus (or node). One usually adopted dynamic load model represents the frequency dynamics at the load nodes as a function of their real (or active) power. However, other dynamics can also be considered, for example, the voltage magnitude dynamics as a function of the reactive load. A. R. Bergen and D. J. Hill, A structure preserving model for power system stability analysis, IEEE PAS-100(1), 25-35, Jan Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

54 Network Preserving Model Frequency-dependent Load Power Each load is modeled as a frequency-dependent power load, while assuming that the load bus voltage magnitude remains constant (an assumption that can be relaxed if desired). δ i = 1 D i (P Li P ei (δ)), (15a) where P ei (δ) = E 2 i G ii + j,j i E i E j [B ij sin(δ i δ j ) + G ij cos(δ i δ j )]. and P Li is the nominal (or rated) active power load. At the limit D i 0+, it is constant power load, with P ei = P Li. Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55

55 Network Preserving Model Frequency-dependent Load Power (Cont.) The frequency-dependent load model has its advantages: In Lyapunov stability analysis, the transfer conductances in lines are often ignored, which becomes invalid if active loads are absorbed as equivalent resistances. Frequency-dependent load model resolves that issue. The structural aspect of the network is nicely preserved. Thus the Lyapunov function truly represents the spatial distribution of stored energy in the network (a topological Lyapunov function ). This also gives us the tool to consider other dynamics, such as a voltage magnitude dependent reactive power load model. There are demand side control algorithms that changes the active power consumption on a real-time basis based on the network condition). Preserving the network topology helps us explicity account for such load control techniques in analyzing stability of the network. Return to Network Preserving Model Anghel & Kundu (LANL) Distributed Analysis & Control of Networks August 4, / 55