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

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Simple Models of the Transmission System

Judy Cardell February 19, 2002 ESD.126

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Overview

• Load Flow Equations

– ‘dc’ load flow

• Kirchoff’s Laws
• Power Transfer Distribution Factors
• Uses of Transmission Models

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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

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Impedance and Ohm’s Law

V IR V IZ I R jX Y Z G jB I V Z Z V YV

= = = + = = + = = =

−

( ) 1

1

• Terms: Impedance, Admittance

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ac Voltage and Current

V V t V I I t I

i V V i I I

= + = ∠ = + = ∠

| |cos( ) | | | |cos( ) | | ϖ δ δ ϖ δ δ

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Load flow equations

• Complex power flow, Si, at node I

– Where ‘*’ indicates complex conjugate

S V I V y V P jQ

i i i i ik k k i i

= = = +

∑

* ( )*

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Real and Reactive Power Flow

• Power injected at node i

( ) ( )

[ ]

( ) ( )

[ ]

P V V g b Q V V g b

i i k ik i k ik i k k i i k ik i k ik i k k

= − + − = − − −

∑ ∑

cos sin sin cos

δ δ δ δ δ δ δ δ

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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

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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

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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)

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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 *

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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

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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

( ) ( )

P g V g V V b V V

ik ik i ik i k i k ik i k i k

= − − − −

2

cos sin

δ δ δ δ

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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

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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

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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

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The ‘dc’ Load Flow Problem

• In addition, use the following mathematical

simplifications

– The fact that x>>r – The property that ) ( ) sin(

k i k i

δ δ δ δ − ≈ −

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The ‘dc’ Load Flow Equation

• This is referred to as the ‘dc’ load flow

because it resembles Ohm’s law, not because it models dc power flow

• The power injected at node i, and the power

flow from i to k

) ( ) (

k i ik ik k k i ik i

b P b P δ δ δ δ − = − =∑

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Reactive Power Equation?

• Ignore Q, reactive power equation

altogether

– As long as the system is in normal

• perating mode this is a valid assumption

– If the system is not in normal mode then the assumptions for ‘dc’ equations do not hold

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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

2 2

) (

ik ik k i ik ik

P r g L = − = δ δ

• Losses have a significant economic impact

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Other Popular Simple Models

• Kirchoff’s circuit laws
• Power transfer distribution factors

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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

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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

• n 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

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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

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Example: APS to PJM Transfer

DQE MECS AEP OES OH NYPP PJM APS VAP

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Example: PTDFs for Transfer

Flow Gate Factor APS - PJM 0.70 APS - AEP 0.21 APS - VAP 0.08 AEP - VAP 0.05 VAP - PJM 0.16 AEP - MECS 0.08 MECS - OH 0.07 OH - NYPP 0.07 NYPP - PJM 0.12 In general, for a given transfer:

• Factors out of APS sum to 1.0
• Factors into PJM sum to 1.0

* Note that not all factors are shown in this table *

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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

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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

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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

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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

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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