Simple Models of the Transmission System Judy Cardell February 19, - PDF document

Simple Models of the Transmission System Judy Cardell February 19, 2002 ESD.126 1 Overview • Load Flow Equations – ‘dc’ load flow • Kirchoff’s Laws • Power Transfer Distribution Factors • Uses of Transmission Models 2

Objective of Discussion • Understand origin of load flow programs • Understand use of load flow programs • Become comfortable with these basic equations, since they are also important for the economics of operating power systems – Economic dispatch, security constrained dispatch – Unit commitment 3 Impedance and Ohm’s Law • Terms: Impedance, Admittance = V IR = = + V IZ I R ( jX ) 1 = = + Y G jB Z V Z − = = = 1 I Z V YV 4

ac Voltage and Current = ϖ + δ = ∠ δ V | |cos( V t ) | | V i V V = ϖ + δ = ∠ δ I | |cos( I t ) | | I i I I 5 Load flow equations • Complex power flow, S i , at node I – Where ‘*’ indicates complex conjugate ∑ = = S V I * V ( y V )* i i i i ik k k = + P jQ i i 6

Real and Reactive Power Flow • Power injected at node i [ ] ( ) ( ) ∑ = δ − δ + δ − δ P V V g cos b sin i i k ik i k ik i k k [ ] ( ( ) ) ∑ = δ − δ − δ − δ Q V V g sin b cos i i k ik i k ik i k k 7 The Load Flow Problem • Objective: To solve for complex S and V – S = P + jQ – V = |V| Ê * • There are two equations and four unknowns at each node, so we must make assumptions – Assume the load is known, P and Q – Assume generators can control P, |V| and that these are known 8

Using a Load Flow Program • The user specifies – The line parameters (impedance), g and b – P and Q at all loads – P and |V| at all generators – |V| and Ê * at the swing bus • The load flow program solves for the remaining variables 9 Running a Load Flow Program • Enter initial values for a ‘flat start’ – All |V| = 1 – Swing bus Ê * = 0 – P and Q, normalized, based on knowledge of the system – Line parameters, usually given as part of problem definition (data from company, filed with government, or standard test case) 10

Output from Load Flow Program • The program iterates to find a solution – If the algorithm does not converge, this indicates there is no solution • The solved parameters are – S = P + jQ – V = |V| Ê * 11 Using a Load Flow Program • Calculates line flow and loss for each line • Reveal constraints on power handling capability of transmission lines • Fundamental to power system planning 12

Using a Load Flow Program • Use input and output data from the load flow program to find the power flow on each line • The real power flow from node i to node k ( ) ( ) 2 = − δ − δ − δ − δ P g V g V V cos b V V sin ik ik i ik i k i k ik i k i k 13 Limitations of Load Flows • Models only a single snap-shot in time • Do not provide information on a better choice of specified generator outputs ( i.e., do not optimize system as economic dispatch and optimal power flows, discussed next week) • There is not always a solution (or a unique solution) for a given set of input parameters 14

Limitations of Load Flows • There are three modes of operation – Normal – Alert – Emergency • Normal operation is assumed when running a load flow program – Unity voltage amplitude, |V| = 1 all nodes – Real and reactive power are decoupled - they independent of each other • No help for alert or emergency situations 15 The ‘dc’ Load Flow Problem • Writing computer programs to solve (iterate) the full set of equations is difficult • Simplifying assumptions – ) P is independent of ) |V| – ) Q is independent of ) * – P and Q are decoupled – Normal operations have |V|=1 16

The ‘dc’ Load Flow Problem • In addition, use the following mathematical simplifications – The fact that x>>r δ − δ ≈ δ − δ – The property that sin( ) ( ) i k i k 17 The ‘dc’ Load Flow Equation • The power injected at node i, and the power flow from i to k = ∑ δ − δ P b ( ) i ik i k k = δ − δ P b ( ) ik ik i k • This is referred to as the ‘dc’ load flow because it resembles Ohm’s law, not because it models dc power flow 18

Reactive Power Equation? • Ignore Q, reactive power equation altogether – As long as the system is in normal operating mode this is a valid assumption – If the system is not in normal mode then the assumptions for ‘dc’ equations do not hold 19 Calculating Power Loss • The loss equation also resembles ‘Ohm’s’ law • To get the final form – Substitute in the dc load flow equation – Simplify the impedance coefficient = δ − δ 2 L g ( ) ik ik i k = 2 r P ik ik • Losses have a significant economic impact 20

Other Popular Simple Models • Kirchoff’s circuit laws • Power transfer distribution factors 21 Kirchoff’s Circuit Laws • The voltage law – The sum of voltages around any loop equals 0 • The current law – The sum of currents into any node equals 0 • These concepts are used in numerous 3-bus examples in the literature • They are redundant if the dc load flow equations are also used 22

Power Transfer Distribution Factors • Power transfer distribution factors are a linear simplification • A power transfer distribution factor (PTDF) is the percentage of power flow that flows on a given line, due to a power transfer between any two nodes on the system • The loading on each line is determined by summing all PTDFs for that line 23 Transmission Lines and PTDFs • Each line has a set of PTDFs associated with it – one for each pair of nodes that can exchange power across the line • Each node has a different set of PTDFs for every possible nodal power transaction, representing the distinct impact of each transaction on each line 24

Example: APS to PJM Transfer OH MECS NYPP OES DQE PJM APS AEP VAP 25 Example: PTDFs for Transfer Flow Gate Factor In general, for a given transfer: APS - PJM 0.70 • Factors out of APS sum to 1.0 APS - AEP 0.21 APS - VAP 0.08 • Factors into PJM sum to 1.0 AEP - VAP 0.05 VAP - PJM 0.16 * Note that not all factors are shown in this table * AEP - MECS 0.08 MECS - OH 0.07 OH - NYPP 0.07 NYPP - PJM 0.12 26

Why Have Simple Equations? • Generator output is easy to understand • Load and load shapes are easy to understand • Power flows are non-intuitive • Transmission constraints are hard to predict • Stability issues are non-intuitive 27 Transmission System Modeling • Analyze the scheduled commercial, or market, transactions • Analyze the physical power flows, generator output and load consumption • Predict transmission congestion • Predict (and hopefully prevent) system failure 28

Reactive Power • Typically ignored in policy making and in economic literature • Very important for system operation – Power system west of the Rocky Mountains is voltage limited, highlighting importance of reactive power – Markets for system services (ancillary services) – Possibility of voltage collapse 29 Availability of Power System Models • System operators have full ac models • Relatively recent user-friendly model from University of Illinois, Power World – We have early version for class projects • Common to write own dc models for papers • No widely available distribution system models • Loads typically modeled static, exogenous variables 30

Course Project • Will not be required to use the load flow program unless groups want to • It is important to be aware of basic requirements of the transmission system • El Salvador data can be used with the load flow • There will be homework problems for everyone 31