A New Method for Estimating Maximum Power Transfer and Voltage Stability Margins to Mitigate the Risk of Voltage Collapse

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1 A New Method for Estimating Maximum Power Transfer and Voltage Stability Margins to Mitigate the Risk of Voltage Collapse Bernie Lesieutre Dan Molzahn University of Wisconsin-Madison PSERC Webinar, October 15, 2013

2 Voltage Stability Static: - Loading, Power Flows - Local Reactive Power Support Dynamic: - Component Controls - Network Controls 2

3 OVERVIEW: Static Voltage Stability Margin Voltage, V Margin Power A common voltage stability margin measures the distance from a post-contingency operating point to the nose point on a power-voltage curve. 3

4 OVERVIEW: Static Voltage Stability Margin Voltage, V Margin (negative) Power It s even possible that a normal power flow solution won t exist, post contingency. 4

5 ISSUES Voltage, V Margin Power 1. We don t know the values on the curve. 2. We don t know whether a post-contingency operating point exists! 5

6 Typical Approach Run post-contingency power flow. This may or may not converge. If successful, determine nose point: use a sequence of power flows with increasing load, or a continuation power flow. 6

7 Our Approach Cast as an optimization problem: Minimize the controlled voltages while a solution exists. (Claim: a solution exists.) Exploit the quadratic nature of the power flow equations to directly obtain Traditional Voltage Stability Margin even when the margin is negative. 7

8 Advantages Eliminates need for repeated solutions (multiple power flows, continuation power flows) Often offers provably globally optimal results Works when the margin is negative, i.e. when there isn t a solution. 8

9 OVERVIEW: Static Voltage Stability Margin Voltage, V Margin V 0 V opt P 0 Optimal Solution P max Power 9

10 Power Flow Equations: Review (I D1 +ji Q1 ) (V D1 +jv Q1 ) Electrical Network I = YV Y = G+jB (V DN +jv QN ) (I DN +ji QN ) Current Injection equations in Rectangular Coordinates: 10

11 Power Flow Equations: Review (P 1 +jq 1 ) (V D1 +jv Q1 ) Electrical Network I = YV Y = G+jB (V DN +jv QN ) (P N +jq N ) Power Injection equations in Rectangular Coordinates: 11

12 Power Flow Equations The power flow equations are quadratic in voltage variables: For reference, power engineers almost always express these equations in voltage polar coordinates: 12

13 Classical Power Flow Constraints Bus Type Specified Calculated PQ (load) P, Q V, δ PV (generator) P, V Q, δ Slack V, δ = 0º P, Q The controlled voltages are the problem-specified slack and generator voltage magnitudes. Locking the controlled voltages in constant proportion, and allowing them to scale by α, we claim a power solution exists for any power flow injection profile. (subject to the unimportant small print.) 13

14 Optimization Problem Modify the power flow formulation to Slack bus voltage magnitude unconstrained PV bus voltage magnitudes scale with slack bus voltage Minimize slack bus voltage 14

15 Optimization Problem This optimization problem can be solved many different ways We ve been using the convex relaxation formulation for the power flow equations (Lavaei, Low) because we really want (provably) the minimum solution. The problem has a feasible solution The optimization using the convex relaxation can be solved for a global minimum (and hopefully a feasible power flow solution). 15

16 Relaxed Problem Formulation In Rectangular coordinates, define Then, power flow equations can be written in the form where Which is a rank one matrix by construction. 16

17 Relaxed Problem Formulation The convex relaxation is introduced by relaxing the rank of W. With that, we pose the following convex optimization problem: 17

18 Semidefinite Relaxation Example MISDP 18

19 Semidefinite Relaxation Example MISDP 19

20 Bonus result concerning solution to the power flow equations The existence of a power flow solution requires Necessary, but not sufficient, condition for existence Conversely, no solution exists if Sufficient, but not necessary, condition for non-existence 20

21 Controlled Voltage Margin A controlled voltage margin to the solvability boundary Upper bound (non-conservative) No power flow solution exists for Increasing the slack bus voltage (with proportional increases in PV bus voltages) by at least is required for solution. 21

22 Power Injection Margin Uniformly scaling all power injections scales Uniformly scale power injections until Corresponding gives a power flow voltage stability margin in the direction of uniformly increasing power injections at constant power factor. indicates that no solution exists for the original power flow problem 22

23 Power Injection Margin The power injection margin answers the question For a given voltage profile, by what factor can we change our power injections (uniformly at all buses) while still potentially having a solution? Answer: 23

24 Examples IEEE 14-Bus System IEEE 118-Bus System Tested many other systems and loadings 24

25 Controlled Voltage Margin 1.1 IEEE 14-Bus Continuation Trace: Bus V Injection Multiplier 25

26 Power Injection Margin 1.1 IEEE 14-Bus Continuation Trace: Bus V Injection Multiplier 26

27 IEEE 14 Bus System Injection Multiplier Newton- Raphson Converged? dim(null(a) Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No No No No No

28 Voltage Margin 28

29 Voltage and Power Injection Margins 1.2 IEEE 14-Bus Continuation Trace: Bus V Injection Multiplier 29

30 Controlled Voltage Margin IEEE 118-Bus Continuation Trace: Bus V Injection Multiplier 30

31 Power Injection Margin IEEE 118-Bus Continuation Trace: Bus V Injection Multiplier 31

32 IEEE 118 Bus System Injection Multiplier Newton- Raphson Converged? (lower bound) dim(null(a) 1.00 Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No No No No No No No No No No

33 Voltage Margin IEEE 118 Bus System 33

34 Voltage and Power Injection Margins 1.2 IEEE 118-Bus Continuation Trace: Bus V Injection Multiplier 34

35 Alternate Power Injection Profiles Power injection margin is in the direction of a uniform, constant-power-factor injection profile We can alternatively specify any profile that is a linear function of powers and squared voltages However, insolvability condition is not necessarily valid 35

36 Reactive Power Limits Previous work models generators as ideal voltage sources Detailed models limit reactive outputs Limit-induced bifurcations Two approaches to modeling these limits: Reactive Power versus Voltage Magnitude Characteristic Q max Reactive Power Q min Mixed-integer semidefinite programming Infeasibility certificates using sum of squares programming 0 V Voltage Magnitude 36

37 MISDP Formulation Model reactive power limits using binary variables Reactive Power versus Voltage Magnitude Characteristic Q max Reactive Power Q min 0 V Voltage Magnitude Insolvability 37

38 MISDP Formulation Model reactive power limits using binary variables Reactive Power versus Voltage Magnitude Characteristic Q max Reactive Power Q min 0 V Voltage Magnitude Insolvability 38

39 MISDP Formulation Model reactive power limits using binary variables Reactive Power versus Voltage Magnitude Characteristic Q max Reactive Power Q min 0 V Voltage Magnitude Insolvability 39

40 MISDP Formulation Model reactive power limits using binary variables Reactive Power versus Voltage Magnitude Characteristic Q max Reactive Power Q min 0 V Voltage Magnitude Insolvability 40

41 Reactive Power Limits Results Bus System Power vs. Voltage V P-V Curve 0.99 η max Infeasibility Certificate Found PQ Bus Injection Multiplier 41

42 Conclusions Cast the problem of computing voltage stability margins as an optimization problem to minimize the slack bus voltage. Calculated voltage stability margins power injection/flows, and controlled voltages. Tested with numerical examples Advantages: Eliminates repeated solution (multiple power flows, continuation power flows) Often offers provably globally optimal results Works when the margin is negative, i.e. when there isn t a solution. 42

43 Related Publications [1] B.C. Lesieutre, D.K. Molzahn, A.R. Borden, and C.L. DeMarco, Examining the Limits of the Application of Semidefinite Programming to Power Flow Problems, 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2011, pp , Sept [2] D.K. Molzahn, J.T. Holzer, and B.C. Lesieutre, and C.L. DeMarco, Implementation of a Large- Scale Optimal Power Flow Solver Based on Semidefinite Programming, To appear in IEEE Transactions on Power Systems. [3] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, An Approximate Method for Modeling ZIP Loads in a Semidefinite Relaxation of the OPF Problem, submitted to IEEE Transactions on Power Systems, Letters. [4] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, "A Sufficient Condition for Global Optimality of Solutions to the Optimal Power Flow Problem, to appear in IEEE Transactions on Power Systems, Letters. [5] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, A Sufficient Condition for Power Flow Insolvability with Applications to Voltage Stability Margins, IEEE Transactions on Power Systems, Vol 28, No. 3, pp [6] D.K. Molzahn, V. Dawar, B.C. Lesieutre, and C.L. DeMarco, Sufficient Conditions for Power Flow Insolvability Considering Reactive Power Limited Generators with Applications to Voltage Stability Margins, presented at Bulk Power System Dynamics and Control - IX. Optimization, Security and Control of the Emerging Power Grid, 2013 IREP Symposium, Aug [7] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, Investigation of Non-Zero Duality Gap Solutions to a Semidefinite Relaxation of the Power Flow Equations, To be presented at the Hawaii International Conference on System Sciences, January Conclusion 43

44 Questions? 44

45 Extra Slides: Feasibility 45

46 Feasibility For lossless systems: 1. Show a solution exists for zero power injections at PQ buses and zero active power injection at PV buses, for α = Use implicit function theorem to argue that perturbations to zero power injections solutions also exist. Specifically choose one in the direction of desired power injection profile. 3. Exploit the quadratic nature of power flow equations to scale voltages and power to match injection profile. 46

47 Feasibility: Zero Power Injection Solution Easy: Construct a solution. Open Circuit PQ buses for zero power, zero current injection. Use Ward-type reduction to eliminate PQ buses. (not really necessary, but clean) small print Choose uniform angle solution for all buses. Directly use reactive power flow equation to calculated the reactive power injections at generator buses

48 Note: Zero Power Injection Solutions for Lossy Systems Not all systems have a zero-power injection solution Ability to choose θ such that P PV = 0 depends on Ratio of V PV to V slack Ratio of g to b Systems with small resistances and small voltage magnitude differences are expected to have a zero power injection solution 48

49 Nearby Solutions A nearby non-zero solution exists provided the Jacobian is nonsingular. For a connected lossless system at the zero-power injection solution, the appropriate Jacobian is nonsingular, generically. small print 49

50 Feasible Solution Exploit the quadratic nature of power flow equations to scale voltages to match desired power profile: 50

51 Extra Slides: Infeasibility Certificates 51

52 Infeasibility Certificates Guarantee that a system of polynomial is infeasible Positivstellensatz Theorem If such that then the system of polynomials has no solution 52

53 Power Flow in Polynomial Form Power Injection and Voltage Magnitude Polynomials Reactive Power Limit Polynomials Reactive Power versus Voltage Magnitude Characteristic Q max Reactive Power Q min 0 V Voltage Magnitude 53

54 Power Flow Infeasibility Certificates Find a sum of squares polynomial of the form such that by finding polynomials and sum of squares polynomials Then the power flow equations have no solution. 54

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