Forecasting Regime Shifts in Natural and Man-made Infrastructure - PowerPoint PPT Presentation

Forecasting Regime Shifts in Natural and Man-made Infrastructure M.D. Lemmon Department of Electrical Engineering University of Notre Dame, USA UCONN - August, 2014 M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 1 / 28

Motivation - why do this? Human society relies on many natural and man-made processes to deliver a consistent level of essential goods and services. Ecosystem services - clean air, water, food, and fiber Man-made systems - power, water, and traffic networks These processes are infrastructure systems since they form the basis of a civil society and disruptions to infrastructure service delivery has a strong impact on public health and welfare. 2014 National Climate Assessment warns of climate change’s potential to disrupt those services, with negative impact on public health and welfare. One kind of infrastructure disruption is a regime shift . M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 2 / 28

What is a Regime Shift? A regime shift occurs when a system shifts its state trajectory from a nominal operating point to an alternative operating point for an extended period of time. Example is cultural eutrophication of aquatic ecosystem Nutrient enrichment from human sources triggers a toxic algae bloom. Negative impacts include reduced water quality, fish kills, toxic algae blooms, reduced biodiversity M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 3 / 28

Regime Shift Mechanisms in Aquatic Ecosystems Regime shifts occur in response to the disturbance of a system’s internal parameters and can be categorized with regard to disturbance time-scale Fast disturbances that are impulsive in nature give rise to shock-induced shifts. Slow disturbances give rise to bifurcation-induced shifts P = phosphorus in water column “low-nutrient” “eutrophic” u = rate of external nutrient loading dP/dt bifurcation-induced shift α = rate at which P sorbs to sediment generated by a step change k +u(t) ρ = rate at which sediment P desorbs external loading term, k. 0 k 0 sink term ρP 3 dP { dt = k ( t ) − αP + θ 3 + P 3 k + δ (t) { 0 external { internal shock-induced shift generated by an impulsive change load load external loading term, k. source terms M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 4 / 28

Regime Shift Mechanisms in Power Systems The same mechanisms V 2 trigger voltage collapse in z=.02+.06j G two bus system with two V = 1 ∠ 0° equilibria Y =.25j S = 1+0.5j pu c load Voltage collapse occurs if 0.2 increase in demand 0.15 causes nominal voltage 0.1 nominal-voltage Imag V2 (pu) 1 0.05 equilibrium to vanish equilbrium 0 shock-induced voltage collapse increasing Sload may Voltage collapse occurs if − 0.05 cause nominal low-voltage − 0.1 equilibrium to vanish fault causes voltage to equilbrium - bifurcation-induced − 0.15 voltage collapse jump into low voltage − 0.2 0 0.5 Real V2 (pu) 1 1.5 RoA M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 5 / 28

Shock-induced Shifts as First Passage Time Problem We want to bound the likelihood of a shock-induced shift for dx ( t ) = f ( x ( t )) dt + dJ ( t ) where J ( t ) is a shot noise process. x 2 1 alternative equilibrium 0.9 If the initial state is at nominal 0.8 X T equilibrium, x 0 , and X T is the ROA 0.7 jump event 0.6 passage to of the alternative equilibrium, then s t alternative state a 0.5 t e t separatrix r X 0 a 0.4 j the first passage time is the first time e c t o 0.3 r y nominal equilibrium the system state x ( t ; x 0 ) enters X T . 0.2 0.1 This time is a random variable. 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 1 The distribution of this first passage time (FPT) characterizes the likelihood of a shock-induced regime shift M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 6 / 28

Distance to Bifurcation-induced Shifts (D2B) We want to determine the range of parameters about a nominal parameter k 0 for which system ˙ x = f ( x ; k ) does not have a bifurcation. A parameter, k , is topologically parameter space equivalent to the nominal bifurcation manifold Ω (k 0 ) parameter if the flows of their k* minimum D2B equilibria can be smoothly mapped into each other. k 0 k=k + ε η 0 Bifurcation occurs for a system Ω (k 0 ) = set of parameters whose that is not topologically equilibria are topologically equivalent to k 0 equivalent to the nominal system. The minimum distance-to-bifurcation (D2B) is the distance to the bifurcating system that is closest to k 0 . This distance characterizes sensitivity to bifurcation-induced shifts. M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 7 / 28

Model-based Method for Regime Shift Prediction We are using a common analysis framework to solve both the FPT and D2B problems The benefits of the proposed method are that it obtains bounds on parameters for which no regime shifts occur. obtains tighter bounds than other approaches analysis can be done using existing computational tools In exchange for these benefits, we need to realize the process as a polynomial system For such polynomial realizations, one can uses barrier certificates and concepts from algebraic geometry to recast the FPT/D2B problem as a polynomial optimization problem Approximations of this problem’s solution are then computed using a sequence (hierarchy) of semidefinite programming (SDP) or linear programming (LP) relaxations. M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 8 / 28

Polynomial System Realizations A polynomial realizations does not mean the system is polynomial. Consider a system whose original state ˜ x satisfies ˙ x = F (˜ ˜ x, k ) where F is a smooth vector field with system states ˜ x and parameters k . n → R n and define the new state Introduce a diffeomorphsim T : R ˜ x = T ˜ x . The original system has a polynomial realization if one can find a diffeomorphism T such that x = T ˙ x = TF ( T − 1 ( x ); k ) = f ( x ; k ) ˙ ˜ in which f is an element of the polynomial ring R ( k )[ x ] with variables x and real coefficients k M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure UCONN - August, 2014 9 / 28

How Restrictive are Polynomial Realizations? Polynomial realizations were studied back in the 1980’s which established necessary and sufficient conditions for the existence of such realizations. Primary result in [Bartosiewicz, Minimal polynomial realizations, Mathematics of Control, Signals, and Systems , 1988] made use of algebraic characterizations of observability and provided a constructive approach. So in many cases, even a system that is not polynomial may have a polynomial realization with a clever variable substitution. Even if this cannot be done, one can still approximate any smooth vector field on a compact domain using polynomials. UCONN - August, 2014 10 / M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure 28

Polynomial Realization of Two bus System 1 Introduce the variable R+jX transformation x = cos δ and GEN. Load y = sin δ and add ODE’s for ˙ x and ˙ y 1 1 ! ˙ = M [ P m − P e 1 ( δ, V ) − D G ! ] ! ˙ = M [ P m − ( G − V ( Gx − By )) − D G ! ] 1 1 x = cos δ ˙ ˙ = ! − [ P e 2 ( δ, V ) − P d ] δ = ! − � − V 2 G + V ( Gx + By ) − P d � δ D L D L 1 1 ˙ � − V 2 B − V ( Gy − Bx ) − Q d � V = ˙ V = τ [ Q e ( δ, V ) − Q d ] τ � 1 �� � − V 2 G + V ( Gx + By ) − P d x ˙ = − y ! − = G − V ( G cos δ − B sin δ ) y = sin δ P e 1 D L − V 2 G + V ( G cos δ + B sin δ ) P e 2 = � 1 �� � − V 2 G + V ( Gx + By ) − P d y ˙ = x ! − D L − V 2 B − V ( G sin δ − B cos δ ) = Q e UCONN - August, 2014 11 / M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure 28

Polynomial Realization of Foodweb Secondary consumer (black crappie) - x 3 Three level aquatic food web consisting of producers ( x 1 ), primary consumers Primary consumer (Daphnia) - x 2 ( x 2 ) and secondary consumers ( x 3 ). 1 Introduce new variable x 4 = k 2 + x 1 and add ODE for the new state. Producer (Algae) - x 1 ˙ = x 1 (1 − x 1 ) − k 1 x 1 x 2 x 4 x 1 (1 − x 1 ) − k 1 x 1 x 2 x 1 x 1 ˙ = k 2 + x 1 x 2 ˙ = k 3 x 1 x 2 x 4 − k 4 x 2 x 3 − k 5 x 2 k 3 x 1 x 2 x 2 ˙ = − k 4 x 2 x 3 − k 5 x 2 ˙ = k 6 x 2 x 3 − k 7 x 3 x 3 k 2 + x 1 1 x 4 = − x 2 k 2 + x 1 x 4 ˙ = 4 [ x 1 (1 − x 1 ) − k 1 x 1 x 2 x 4 ] x 3 ˙ = k 6 x 2 x 3 − k 7 x 3 UCONN - August, 2014 12 / M.D. Lemmon (ND) Forecasting Regime Shifts in Natural and Man-made Infrastructure 28