advertisement

The Power Flow Problem Scott Norr For EE 4501 April, 2015 Electric Concepts • Ohm’s Law: V = IR (V = IZ) • Kirchoff: KCL: ∑ i = 0 (at any node) ∑ V = 0 (on closed path) • Power: P = VI (S = VI*) = V2/R = I2R • NOTE: i means electric current, j2 = -1 Previously: DC Circuits (RI=V) Resistance Matrix AC Circuits: Phasor Analysis ZI = V (Thanks to Euler, Steinmetz) Impedance Matrix DC Power: All electrical systems naturally seek an equilibrium point of lowest entropy Important to recognize that P ᴕ V2 (try to find the proportional symbol in powerpoint sometime…..) AC Power • The complex power, S = VI* = P + jQ • P is the average power (“real” power) in watts, attributable to resistive loads • Q is the reactive power (“imaginary” power) in VAr, attributable to capacitive and inductive loads The Power Problem: • On AC power systems, we don’t predetermine the phase angles on the sources, they are determined by the system (additional unknowns to solve for!) • Power is injected into nodes in the system via sources and is removed at nodes via loads (consumption points) • Additionally, power is lost in the network Consider an Example: 3 Node System Unknowns: • At each bus (node) there are 4 parameters: P, Q, V and Ө • There are three types of buses: – Load Buses: P, Q are known, V, Ө are unknown – Generator Buses: P, V are known, Q, Ө are unknown – Slack Bus: (unique) V, Ө are known, P, Q unknown (this special generator node is allowed to accumulate errors in the iterative solution of the system of equations) So, for N nodes, 2N unknown node parameters Balancing Power at Each Node: • ∑Si = o • SG-SL = Vi∑Ip* • SG-SL = Vi∑Vp*Yp* • • • Can separate the real (P) from the imaginary (Q)to form two equations at each Bus A system of 2N equations Sparse, largely diagonalized matrices Solve for Node Voltages and Angles: • Vi new = (1/Yii)(Si/Viold - ∑Vp*Yp*) • An iterative process, involving an initial starting estimate and convergence to a predetermined tolerance. • This is called the Gauss-Seidel Solution Method A better Method: • For analytic, complex differentiable systems, can compute the low order terms of the Taylor series and solve using Netwon’s method. • In two variables, an iterative approach: f1(x,y) = K = f(xo + Δxo, yo + Δyo) g1(x,y) = L = g(xo + Δxo, yo + Δyo) Computing the Taylor Series, and truncating it yields an equation exploiting a Jacobian matrix Newton – Raphson Solution: Conclusions: • Powerflow Software is used by every electric utility in the world. Many models contain 10,000 nodes or more. • There are quite a few solution techniques that are more efficient than the G-S and N-R methods outlined here: Fast-Decoupled N-R – decouples P,O from Q,V and solves the two, smaller systems Interior Point Newton - calculates a Hessian Mtx! PowerWorld Simulator References: • Stevenson, William D., Elements of Power System Analysis, McGraw-Hill, 1982 • Tylavsky, Daniel, Lecture Notes #19, EEE 574, Arizona State University, 1999 • PowerWorld Simulator, www.powerworld.com,2014