power systems engineering and economics - · pdf file3.10 types of fault ... connects...

Power Systems Engineering andEconomics

Will McLennanBased on lectures by Prof. R. Wallace

Short Description

The aims of the course are to provide a hands-on experience of the problems created bytransmission and distribution of energy from power stations to consumers and to cover arange of topics related to the privatisation and restructuring of electricity supply industryworldwide. The first part of the course will be simulation-based utilising PowerWorldload-flow simulation program. After some introductory lectures, the students will beinvestigating the problems of voltage drops, thermal transmission constraints, steady-state stability constraints, transmission losses. Each simulation session will finish withan assignment which will be marked. In the second part of the course the students willbe introduced to the principles of power system economics. Main regulatory regimeswill be discussed together with the pricing principles. Then PowerWorld program willbe used to evaluate the effect of networks on energy prices, i.e. locational marginalpricing. At the end of the course the students will write two essays on topics related tothe liberalisation of electricity supply industry.

Summary of Intended Learning Outcomes

Ability to use load flow packages. Understanding and modelling of AC network effectsof transmission and distribution. Application of iterative methods to solve non-linearnodal network problems. Principles of power system economics and how market-basedsolutions can be applied to a previously centrally controlled industry. Understanding ofhow network affects marginal prices at different locations. Understanding how humanreactions have to be taken into account when designing engineering solutions.

Contents

1 Revision of fundamentals 11.1 Per-unit system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Per-unit equivalent circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Short line (< 50 km)/Cable . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Medium line (80-250 km) . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Synchronous generator and motor . . . . . . . . . . . . . . . . . . 41.2.5 Induction machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.6 System loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Real and imaginary power flow . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Complex power and notation . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Complex power transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Power flow 102.1 Power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Node parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Bus admittance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Node voltage analysis (linear) . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Power flow analysis (non-linear) . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6.1 Gauss-Siedel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6.2 Newton-Raphson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 Busbar types and effect on solution . . . . . . . . . . . . . . . . . . . . . . 162.7.1 Slack busbar (reference) . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 Line flows and losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 DC power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10 Geographical and economic effects . . . . . . . . . . . . . . . . . . . . . . 182.11 Generation cost functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.12 Economic dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.13 Constrained dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.14 Loss minimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.15 System wide sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.16 Optimal power flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.16.1 Economic interpretation of Lagrange multipliers . . . . . . . . . . 222.16.2 Security constrained optimal power flow (SCOPF) . . . . . . . . . 23

3 Network integration 243.1 Drivers and timeframes for change . . . . . . . . . . . . . . . . . . . . . . 243.2 Effect of connecting distributed generation . . . . . . . . . . . . . . . . . . 243.3 Requirements and standards . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Power-flow, thermal ratings, losses . . . . . . . . . . . . . . . . . . . . . . 253.5 Voltage regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

i

3.6 Harmonic content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 Transient stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.8 Protection co-ordination and operation . . . . . . . . . . . . . . . . . . . . 263.9 Synchronous or induction generators? . . . . . . . . . . . . . . . . . . . . 263.10 Types of fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.11 Short circuit close to a generator . . . . . . . . . . . . . . . . . . . . . . . 303.12 Fault level studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.13 Per-unit equivalent impedance of a network . . . . . . . . . . . . . . . . . 323.14 Fault level changes due to new DG . . . . . . . . . . . . . . . . . . . . . . 33

4 The UK electricity market 344.1 UK & National Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Privatisation and unbundling . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Electricity markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 The electricity pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 New Electricity Trading Arrangements (NETA) . . . . . . . . . . . . . . . 364.6 British Electricity Transmission and Trading Arrangements (BETTA) . . 37

5 Ancillary services 385.1 System control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Voltage control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Cost of producing reactive power . . . . . . . . . . . . . . . . . . . . . . . 385.4 Reactive power markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5 Frequency regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Epochs of response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.7 Market for ancillary services . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Transmission networks and locational marginal pricing 416.1 Transmission Network Congestion . . . . . . . . . . . . . . . . . . . . . . 416.2 Interconnected Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.3 Constrained Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.4 Effects of Network Congestion . . . . . . . . . . . . . . . . . . . . . . . . . 41

7 UK renewables & support mechanisms 437.1 UK climate & renewable energy targets . . . . . . . . . . . . . . . . . . . 437.2 Recent history of UK renewables . . . . . . . . . . . . . . . . . . . . . . . 437.3 Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

ii

Power Systems Engineering and Economics 1

1 Revision of fundamentals

The distribution network (the grid) connects generation with demand. It is designedto share electricity around the grid and respond when faults occur. Changes to thisinfrastructure are expensive and politically sensitive. However with the geographicaldistribution of renewable sources we wish to harness, changes to the grid will have to bemade.

1.1 Per-unit system

The power system is an interconnected system of busbars (nodes) at different voltagelevels linked by transformers, lines and cables delivering electricity from generators toloads. We want to be able to calculate power flow from one section to another, acrossdifferent voltage levels, without worrying about the fact that the absolute voltages aredifferent. The per-unit system enables this. We need to choose a base value for powerflow as an apparent power in VA. The base can be any round number of kVA or MVA:1000 kVA; 10 MVA, 100 MVA This becomes 1.0 pu apparent power...everywhere. Toa base power of 1000 VA, for a base voltage at one part of the system of 250 V, basecurrent in that part would be 1000/250 = 4 A.

1.0 pu power (everywhere) = 1000 VA1.0 pu voltage (locally) = 250 V1.0 pu current (locally) = 4 A

With per-unit system, the electricity network can be represented by a set of PU impedancesoperating with 1.0 pu voltage at each end. If we operate in pu, without differing absolutevoltage levels we can deal with transformers as single impedances. PU impedances ex-hibit all the same relationships as absolute values, obey all the same circuit laws, such asOhm’s law and Kirchhoff’s Law, but the arithmetic involved in the per-unit calculationsis much simpler.

The essence of per-unit calculations is

(a) Choose a base for all circuit parameters. NB - one chosen value of base may defineothers

(b) Convert all absolute values into per-unit

(c) Carry out the very much simpler calculations

(d) Convert all per-unit values back into absolute terms using base in (a)

PU analysis depends on a consistent choice of a common base, and an accurate conver-sion of circuit parameters into and out of that base. Four base quantities required tocompletely define a per-unit system, voltage (V), current (I), complex apparent power(S) and impedance (Z).


If a current Iphase flows in a circuit under the action of voltage Vphase it seems as if thereis an impedance somewhere in the circuit limiting the current to that value. In powersystems analysis Iphase is determined by power transfer and actual circuit impedanceplays no part in determination of Zbase.

Base impedance (Ω) Zbase =Base voltage (V )

Base current (A)=VbaseIbase

=kV 2

b

MVAb(1.1)

For three phase use Vline and total three phase MVAbase

Zbase(Ω) =kV 2

line

MVAbase(total)(1.2)

PU impedances are always initially quoted to Zbase of the device. Generators Z(Ω) isstator reactance, MVAbase = MWout/ cosφ In transformers Z(Ω) is summed into singleleakage reactance. In power systems analysis need to assert a common MVAbase andconvert all ohmic values into per unit based on operating voltage and this MVAbase.If reactances or impedances are already in per-unit to the base of the component theyneed to be convert to same MVAbase. Thus,

Xpu(new) = Xpu(old)

(MVAbase(new)

MVAbase(old)

)= Xpu(old)

(KV 2

line(old)

kV 2line(new)

)(1.3)

In power systems analysis, pu values of devices are quoted after calculation fromMVAbaseand Vline, where Vline is usually 1.0pu. Occasionally transformers/motors/generators areoperated at a voltage off-nominal and per-unit calculations must allow for this.

1.2 Per-unit equivalent circuits

Where systems are balanced - i.e., all phase voltages, impedances, loads, and currentsare exactly equal and at each phase may be represented identically at 120 to each othercan use single phase equivalent circuits to describe behaviour of devices within system.Normally a combination of R, L & C resistive, inductive and capacitive components torepresent real and imaginary power flow over range of operating conditions. Equivalentcircuits do not represent components that could be dismantled out of device or measureddirectly.

1.2.1 Transformer

Where Rp, Rs are the ohmic resistances giving losses in prim & sec windings, Rc repre-sents heating loss in iron core due to eddy currents, Xm represents reactance associatedwith magnetising current drawn and Xl represents series inductance of winding notlinked by magnetic flux.


Xl puV1

I1 Xl

I 2

nV1

nI2

V2

Rp Rs

Xm

Rc

Figure 1: PU equivalent transformer

For constant rate of change of core flux

E2 = N2dφ

dt, E1 = N1

dφ

dt(1.4)

Transformers are usually > 99% efficient, so ohmic losses are close to negligible, seriesRp, Rs → 0 and parallel Rc → ∞. If primary and secondary are at 1.0 pu voltage,components for transformation and Xm disappear. Per-unit model becomes the PUseries reactance Xl which represents voltage drop across device - usually 5%.

1.2.2 Short line (< 50 km)/Cable

Rs XlRc Xc

Rs Xl

Figure 2: PU equivalent short line/transformer

Rc is the insulation resistance to neutral or ground - usually approaching infinite.Xc is the stray capacitance across insulation media - which is neglected for short lines.Rs is the series ohmic resistance that represents heating losses I2Rs in each phase.Xl is the series inductive reactance due to self inductance and proximity of other twophase conductors.

1.2.3 Medium line (80-250 km)

Rc is the insulation resistance to neutral or ground - usually approaching infinite.Yc is the shunt admittance (1/stray cap reactance) across insulation media - which is


Rs XlRs XlRc Xc Yc/2 Yc/2

Figure 3: PU equivalent medium line

included in π configuration.Rs is the series ohmic resistance that represents heating losses I2Rs in each phase.Xl is the series inductive reactance due to self inductance and proximity of other twophase conductors.

1.2.4 Synchronous generator and motor

EtVp

Xs

Ip

EtVp

Xs

Ip

Figure 4: PU synchronous generator and motor

Treated as induced phase emf Et between line and neutral conductors. Et is normallygreater than phase voltage Vp when machine is loaded and producing or taking currentIp. The series (stator) reactance Xs is introduced to resolve phasor voltage triangle

Et = Vp ± Ip.jXs (1.5)

1.2.5 Induction machines

R1, R2 - series resistance of stator and rotor windings to represent I2R losses in eachphase.Rc - represents heating loss in iron stator due to eddy currents.Xm - represents reactance that would draw magnetic current.X2 - represents standstill reactance of rotor winding.


Vp Rc Xm

X1+ X2 R1+R2

R2(1-s)/s

Ip

Figure 5: PU induction machine

X1 - represents leakage reactance of stator winding.R2(1− s)/s - represents power developed as ohmic I2R losses.

1.2.6 System loads

RloadXl load

RloadXc load

Figure 6: PU system loads

Loads are series/parallel combination of RC or RL to draw from the supply a current atthe correct phase angle to supply the active and reactive power demanded by the load.Loads add in parallel, never series.

1.3 Real and imaginary power flow

P flows A→B depending on MW load connected to busbar B and network impedance,Q flows A↔B depending on MVAr load connected to busbar B and network impedance.S flows from A→B. Since P & Q are in quadrature they can flow in the same conductorquite independently of one another. Consider a single series (inductive) impedance, with


phase voltage V applied across it.

I =V 6 0

Z 6 φ=

(V

Z

)6 0− φ = I cosφ− jI sinφ (1.6)

Multiply through by V to get complex power

S = V I = V I 6 − φ = V I cosφ− jV I sinφ = P − jQ (1.7)

In this series circuit the current is determined by V and Z. Phase angle is determinedby tan−1(X/R). Since I cosφ is in phase with V , P is in phase with V and since I sinφis in quadrature, Q is orthogonal to V . Also I is in phase with S, displaced by φ fromP and V . Alas this is not generally true in a multi-bus power system.

S =√P 2 +Q2 6 − φ (1.8)

1.4 Complex power and notation

In a multi-source, multi node system power and voltage are only asserted to be in phaseat one busbar in power systems analysis. At all other busbars the apparent powerconveyed by current I passing a node at voltage V ,

S = V I∗ V A (1.9)

α βP

Q

φ

V

I

Figure 7: Complex power flow

The real power conveyed by the cosine component of current I in phase with V is:

P = V.I cos(α− β) W (1.10)

The imaginary power conveyed by sine component of current I in phase with V

Q = V.I sin(α− β) V Ar (1.11)


Take the voltage phasor as reference for just now

S = P + jQ (1.12)

= V.I[cos(α− β) + j sin(α− β)] (1.13)

= V.I[(cosα. cosβ + sinα. sinβ) + j(sinα cosβ − cosα sinβ)] (1.14)

= V (cosα+ j sinα)xI.(cosβ − j. sinβ) V A (1.15)

Polar forms may also be used to express voltage and current. If V = V.ejα and I = Iejβ

then,

S = V I∗ (1.16)

= V.I.ej(α−β) (1.17)

= V.I.ejφ (1.18)

= V.I(cosφ+ j sinφ) (1.19)

While V remains ahead of I, or I lagging V , (α−β) is positive, Q is positive and the loadon the busbar absorbs imaginary or reactive power to operate at a lagging power factor.If I is ahead of V , or V lags I, (α − β) is negative, Q is negative and the load on thenode produces imaginary or reactive power to operate at a leading power factor. Reactivepower is produced by, over-excited sync machines, capacitors, cables and lightly loadedoverhead lines. Reactive power absorbed by under-excited sync machines, inductors,induction motors, heavily loaded overhead lines and transformers.

1.5 Complex power transfer

The power system is represented as a set of busbars or nodes, between adjacent nodes Aand B the system impedance is represented as ZAB = R+ jX. There may be more thanone system route connected in parallel contributing to the overall system impedanceZ. Current flows because the magnitude and angles of the voltages on the busbars willdiffer. (VA − VB) is asserted by the system across impedance Z.

Current I leaves sending end under action of VS and flows through Z = R + jX intoreceiving end which is at VR. Kirchoff’s law applies around the single phase equivalentcircuit.

VS − VR = IR+ IjX (1.20)

V S = VR + IZ (1.21)

= VR 6 0 + IZ 6 (γ − φ) (1.22)

Resolve in-phase and quadrature voltage drops

a = RI cosφ+XI sinφ b = −RI sinφ+XI cosφ (1.23)

a = RPRVR

+XQRVR

b = RPRVR−XQR

VR(1.24)


= VS ∠δ°

VR

VS

IR

jXI

Ia

b

φδ

VR

VS

I R

daoLecruoS

jX

IR IjX

γ

Z ∠+γ°

I ∠-φ°

∠δ°

Figure 8: Complex power transfer

Consider right angled triangle with Vs as hypotenuse

sin δ =b

Vs= X

(PRVSVR

)−R

(QRVSVR

)=XPR −RQR

VSVR(1.25)

cos δ =a+ VRVs

= R

(PRVSVR

)+X

(QRVSVR

)+VRVS

(1.26)

(VS cos δ − VR) =RPR −XQR

VR(1.27)

If X >> R

PR =

(VRX

)VS sin δQR =

(VRX

)(VS cos δ − VR) (1.28)

This shows that P is influenced by voltage angle δ and Q is influenced by VS − VR. Ifthe sending end power flow tries to push VS further ahead than 90, the receiving endwill start to absorb less power i.e., PR will decrease again. Generators at sending endwill quickly move further than 90 ahead and will rise out of synchronism. Part of thenetwork will become islanded and asynchronous which is seriously dangerous. Sendingend generators would be tripped on loss of synchronism to avoid islanded operation, andpower flow from sending end would reduce to zero. Receiving end generators will nowattempt to supply sending end load and current flow in line will reverse. Directionalover-current protection may trip line and the system will collapse.

Due to the steady state stability limit, δ must be kept fairly small (less than 30−40). Forsmall δ, cos δ is nearly 1 and slowly changing. For very approximate analysis, assume


cos δ = 1. Reactive power flow is always down voltage gradient and proportional tovoltage difference. Plot QR as a function of VR, assuming Vs = constant.

capacitive inductiveQ

Operating part

Vr

VsCriticalpoint

Figure 9: Reactive power - limits of stability

Operation is in the upper part of the characteristic as VS and VR have to be quite close(large voltage deviations are unacceptable). Voltage may serve as the reactive powerbalance indicator as falling voltages indicate reactive power deficit. Increasing voltagesindicate reactive power surplus.

To control system voltages, Static VAr compensation is used, thyristor control of parallel-connected capacitors and inductors allows flexible operation from leading to laggingpower factors.

transmission line

Figure 10: Static VAr compensation


The most popular means of voltage control is tap changing transformers

Figure 11: Tap changing transformers, one-line diagram, off-load tap changer and on-load tap changer respectively

System heating losses may be expressed in terms of the complex power flow in thenetwork.

Losses = I.I∗R =(P + jQ)(P − jQ)

V V ∗=P 2 +Q2

V 2(1.29)

This shows that losses are proportional to the sum of the squares of P and Q, and in-versely proportional to the square of voltage. Reducing Q reduces losses and CO2.

2 Power flow

2.1 Power flow

Power flow (or load flow) calculations predict and verify the most effective means ofproviding bulk electricity supplies between one or more sources of power and one ormore loads, over a number of circuits which may be configured in a number of ways.The most important objectives that must be met in normal operation of a generation,transmission and distribution network are

1. Maintenance of real power balance: Pgen = Pdemand + Plosses at all times to ensurecontrol of frequency.

2. Maintenance of reactive power balance: Qgen = Qstored in load + Qstored in system atall times to ensure control of the voltage profile.

Power flow studies are carried out to

• Predict the flow of real (MW) and (MVAr) imaginary power

• Estimate the system voltages at all points in the network


• Assess the effects of system reconfiguration

• Study voltage compensation

• Assess the effects of system outages

• Project for the addition of new plant

Power flow analysis may therefore be defined as establishing the solution of the networkequations that define the steady state condition of the power system. It prescribes thatthe three phase system is balanced and may be represented and analysed as a singlephase equivalent circuit using per-unit values.

2.2 Node parameters

At every busbar there is a bus voltage, V , a bus voltage angle, δ, an inflow/outflow ofP = Pgen−Pload and an inflow/outflow of Q=Qgen−Qload. Busbars are Kirchhoff Nodes,they cannot store P or Q, thus

∑P = 0 and

∑Q = 0.

VB

VA

δAB

VAB

PgenB , QgenBBA

ZAB

±P ±jQ

Generation and loads

Generation and loads

VA

δA δB

PgenA, QgenA

PloadA, QloadA

PloadB, QloadB

Figure 12: Node parameters

The voltage angle δAB between buses A and B across ZAB is largely determined by thenet flow of active power PAB. The voltage VAB between buses A and B across ZAB islargely determined by the net flow of reactive power QAB. The flow of PAB & QAB toand from bus A to bus B is determined by the voltage angle, δAB = δA−δB, and voltagedifferences between A and B, VAB = VA − VB.

While busbars are Kirchhoff nodes, interconnecting equipment like lines, cables andtransformers have resistance (and P losses) and reactance (and Q stored). Nonetheless,real power must be conserved across the connection. If current IAB flows through ZAB =RAB + jXAB,

losses = I2ABRAB (2.1)

storage = I2ABXAB (2.2)


2.3 Bus admittance matrix

Transmission lines are usually represented by their π-equivalents but in this exampleinitially only the series reactances X will be used. Resistances and capacitances areneglected.

transformer

Transmission line

generatormotor

Node (bus)

Single-line diagram Single-phase equivalent circuit

internal reactancesof generator/motor

transformer reactances

Series reactancesof lines

emf

Figure 13: Transmission lines are usually represented by their π-equivalents, in this caseonly series reactances are used.

It is convenient to replace all impedances by admittances (Y = 1/Z) and emfs by currentsources (I = E/Z). Busbar 0 is at earth potential.

Kirchhoff’s Current Law at each node gives∑I = 0, (entering + , leaving -)

Bus 1: 0− I12 − I13 − I14 = 0 (2.3)

Bus 2: 0− I21 − I23 − I24 = 0 (2.4)

Bus 3: I3 − I30 − I31 − I32 = 0 (2.5)

Bus 4: I4 − I40 − I41 − I42 = 0 (2.6)

Rearrange to separate current injections onto left

Bus 1: 0 = I12 + I13 + I14 = 0 (2.7)

Bus 2: 0 = I21 + I23 + I24 = 0 (2.8)

Bus 3: I3 = I30 + I31 + I32 = 0 (2.9)

Bus 4: I4 = I40 + I41 + I42 = 0 (2.10)


Express branch currents as vector products of admittance x voltage difference

Bus 1: 0 = y12(V 1 − V 2) + y13(V 1 − V 3) + y14(V 1 − V 4) = 0 (2.11)

Bus 2: 0 = y21(V 2 − V 1) + y23(V 2 − V 3) + y24(V 2 − V 4) = 0 (2.12)

Bus 3: I3 = y30V 3 + y31(V 3 − V 1) + y32(V 3 − V 2) = 0 (2.13)

Bus 4: I4 = y40V 4 + y41(V 4 − V 1) + y42(V 4 − V 2) = 0 (2.14)

Rearrange to separate and group individual products by voltage

Bus 1: 0 = (y12 + y13 + y14)V 1 − y12V 2 − y13V 3 − y14V 4 (2.15)

Bus 2: 0 = −y21V 1 + (y21 + y23 + y24)V 2 − y23V 3 − y24V 4 (2.16)

Bus 3: I3 = −y31V 1 − y32V 2 + (y30 + y31 + y32)V 3 (2.17)

Bus 4: I4 = −y41V 1 − y42V 2 + (y40 + y41 + y42)V 4 (2.18)

Introduce new admittances as follows

Y 11 = (y12 + y13 + y14) (2.19)

Y 22 = (y21 + y23 + y24) (2.20)

Y 33 = (y30 + y31 + y32) (2.21)

Y 44 = (y40 + y41 + y42) (2.22)

Y 12 = Y 21 = −y12 = −y21 (2.23)

Y 13 = Y 31 = −y13 = −y31 (2.24)

Y 14 = Y 41 = −y14 = −y41 (2.25)

Y 23 = Y 32 = −y23 = −y32 (2.26)

Y 24 = Y 42 = −y24 = −y42 (2.27)

Bus current equations reduce to depict connections 1-2, 1-3, 1-4, 2-3, 2-4. Buses 3 and4 are not connected by any single admittance. Introduce I1 = I2 = Y34 = Y43 = 0 torepresent no current injections on buses 1 and 2 and no connections between 3 and 4.Bus current equations reduce to

Bus 1: I1 = Y 11V 1 + Y 12V 2 + Y 13V 3 + Y 14V 4 = 0 (2.28)

Bus 2: I2 = Y 21V 1 + Y 22V 2 + Y 23V 3 + Y 24V 4 = 0 (2.29)

Bus 3: I3 = Y 31V 1 + Y 32V 2 + Y 33V 3 + Y 34V 4 (2.30)

Bus 4: I4 = Y 41V 1 + Y 42V 2 + Y 43V 3 + Y 44V 4 (2.31)

Writing similar nodal equations for all the nodes gives the following nodal equationI = YV with a symmetrical bus admittance matrix.

I1 = 0

I2 = 0

I3I4

=

Y 11 Y 12 Y 13 Y 14

Y 21 Y 22 Y 23 Y 24

Y 31 Y 32 Y 33 Y 34

Y 41 Y 42 Y 43 Y 44

V 1

V 2

V 3

V 4

(2.32)


Diagonal elements Y (i, i) are the sum of admittances connected to node i, termed self-admittance, e.g.,

Y 33 = Y 30 + Y 31 + Y 32 = −j0.8− j4− j4 = −j8.8 (2.33)

Off-diagonal elements Y (i, j) are negative admittance connecting nodes i and j, termedmutual admittance e.g.,

Y 12 = Y 21 = −(−j8) = j8 (2.34)

2.4 Node voltage analysis (linear)

The general formulation of the nodal equation for a system with N buses is I = YVwhereV i = |Vij |6 δij is the voltage at node i,Ii is the current injection at node i,Y ij = |Yij |6 θij is the mutual admittance between nodes i and j (equal to the negativeof the branch series admittance),Y ii =

∑Ni=1 yij is the self-admittance of node i and is equal to the sum of all the

admittances terminating on node i (including any shunt admittance)

The node voltage method of current and voltage analysis may be used to systematicallyformulate the network equations for power flow analysis. Forming the network equationsusing nodal admittance and node current injections results in a system of complex, linear,simultaneous equations that can be described in matrices using the bus admittancematrix. Once the node current injections or drains are specified, the linear equationscan be solved (by numerous means) to determine node voltages. The bus admittancematrix is symmetrical about the leading diagonal, and can be reduced to upper triangularfor solution. Additionally many elements are zero to represent busbars that are notconnected. This leads to a sparse, upper triangular matrix that can be decomposed andinverted.

Vbus = Y−1busIbus (2.35)

Solving Ibus by inversion is very inefficient. It would be faster to set about direct solutionusing triangular factorisation. Matlab does this using matrix division. After solutionwe know all of the bus voltage vectors and we still have all of the admittances of theinterconnections between buses, and can calculate branch currents.

2.5 Power flow analysis (non-linear)

Node voltage analysis took current injections and admittance and multiplied them outto solve linear equations for bus voltage. It resolved bus voltage difference and branchadmittance into branch currents. Then it calculated branch power flow and losses.


This is ok for very small networks and (perhaps artificially) supposes we know currentinjections. The power system is not really like this.

There are 3 types of node in power system and power flow studies.

• The slack busbar - must be chosen as the one and only infinite busbar, whosevoltage, frequency and voltage angle are prescribed to be constant.

• Load or PQ busbars - are specified by the complex representation of apparentpower passing through them, such as S = P ± jQ

• Generation or PV busbars - are specified as a generation voltage and real powerflow. Limits are set on imaginary power defined by the limits of the generator.

Absolute values or correctly expressed per unit values may be used for system calcu-lations. Inputs are bus power injections PGen, QGen and load demands PLoad, QLoad.Outputs are branch power flows, bus voltages and losses.

From the complex power and the current we get the following equation:

V =(P − jQ)(R+ jX)

V ∗(2.36)

This has V and V ∗ on opposite sides of the equation (or V 2 on one side) it is thereforenon-linear and can only be solved by iterative numerical methods.

If we consider one busbar connecting to n others, the equation can be changed to:

Pi − jQiV ∗i

=

n∑j=0

yij

Vi − n∑j=1

yijVj (2.37)

There are four variables P,Q, δ and V for every one of N busbars in the network. Com-plex power flow and voltages are described by vectors or complex numbers. It requiresN simultaneous non-linear equations in complex number form to describe system. So-lution is iterative and employs numerical methods and matrix algebra to solve powerflow on computers. All the calculations are in per-unit. There are two (main) andalternative numerical methods used in the power flow solvers: GaussSeidel and Newton-Raphson.

2.6 Numerical methods

Newton Raphson converges faster than Gauss Siedel, and is more stable. It still needs agood first estimate of the solution, however, or it may diverge.


2.6.1 Gauss-Siedel

Consider a non-linear function of x: f(x) = 0Rearrange to make x the subject of the equation: x = g(x)Make an initial estimate of x, x(0) and calculate x(1): x(1) = g(x(0))Repeat the iteration: x(k+1) = g(x(k))Until a solution is obtained when successive values |x(k+1) − x(k)| ≤ εdiffer by the required accuracy.

2.6.2 Newton-Raphson

Newton-Raphson uses the gradient of the function compared to the error to calculatethe next iteration. This leads to the Newton Raphson algorithm:

x(k+1) = x(k) + ∆x(k) (2.38)

where

∆x(k) =∆c(k)(∂f∂x

)(k) (2.39)

2.7 Busbar types and effect on solution

PQ buses only have loads, where the real and imaginary power demands are known. PVbuses have generators, (and sometimes loads) and the busbar voltage (see AVR) and thereal power output (see Turbine Governor) are controlled. The preset values of P are setby optimal dispatch (to be discussed later). The preset values of voltage V are set bySystem Operator to maintain a required voltage profile. As the voltages at PV busesare set (i.e. known), there are fewer unknowns. The columns for voltages in PV buses,and rows for reactive power injections in PV buses, will be missing.

If there is both a generator and a load connected to the same busbar it is PV type.The value of Pbus input to the solution is held equal to the net injection of Pbus =(+PgenPdemand). The value of V is held equal to the AVR voltage setting of the generator.Bus voltage angle δ and net imaginary power injection are determined as output bysolution of the network power flow. There are limits on δ determined by Pmax andthe governor limits. Synchronous generators have Qmax as an imaginary power outputcapability (set by the excitation limit). A specified bus voltage can only be maintaineduntil the excitation limit is reached. Then either the bus voltage starts to fall, or thegenerator is switched from PV control to PQ but with Q equal to Qmax. The value ofQbus arising from the power flow solution is used to calculate Qgen the actual value ofgenerator’s output imaginary power: Qgen = (Qbus +Qdemand).


2.7.1 Slack busbar (reference)

At all times the power balance must be kept. The power flow solution iterates to findthe real and imaginary power injected from every busbar by adjustment of power angleson all buses and voltage levels on PQ buses. As the solution converges the final powerflows and losses are determined - we do not know them beforehand.

This creates a chicken-and-egg situation: to fully specify data for the power flow solver,we’d need to know the losses but they are only calculated once the network power flowhas been solved. One network busbar is selected as the slack bus in which the voltage isspecified (usually 1.0 pu) with zero voltage angle (reference bus).

It has zero upstream impedance and can absorb or produce any amount of P orQ withoutvoltage or voltage angle changing. Hence two rows and two columns corresponding tothat node are removed from the formulation - this is a mathematical fix that allowssolution.

2.8 Line flows and losses

On solution, after convergence we know the complex voltages on each busbar i and j,and we know the actual or mutual admittances yij between them. From this we cancalculate the complex power flow on each line

Sji = VjI∗ji = Pji + jQji (2.40)

Complex losses are determined from the sum of opposite line complex power flows fromeach busbar

SLji = Sij + Sji (2.41)

Ploss + jQstored = (Pij + jQij) + (Pji + jQji) (2.42)

2.9 DC power flow

A pragmatic simplification is to concentrate on real power flows only. In HV transmissionlines rij << xij and Bc can be neglected. Then real power flow in line i-j is:

Pij =ViVjxij

sin δij =ViVjxij

sin(δi − δj) (2.43)

As we are concentrating on real power flows and neglecting Q, voltages can be assumedall equal to 1 per-unit. Angle δ has to be small due to steady-state stability constraints.For small δ, sin δ ≈ δ and (in per-units):

Pij ≈1

xij(δi − δj) (2.44)


The problem is now linear (power is proportional to δ) and therefore δ can be calculateddirectly (without using Newton-Raphson method). The method is referred to, confus-ingly, as DC power flow as the power flow is directly proportional to the difference inangles, similarly as in DC circuits a current is directly proportional to the difference involtages. However we are still dealing with AC circuits.

2.10 Geographical and economic effects

The simplest power flow aims to exactly balance real and imaginary power generationand neglects generation economics, geographical effect and transmission losses in theallocation of generation. In a real network: Pscheduled > Pgen to provide reserve margin.Generators will cost more or less to operate than one another and generators will bedifferent distances from load and incur greater or lesser system losses.

In a real network objective of generator dispatch is to:

• Schedule sufficient capacity to meet planned, second by second, and emergency(contingency) demand and losses;

• Do so with minimum fuel cost;

• Supply load demands with minimum transmission losses.

The connected capacity, amount and location of P&Q generated can be varied to meetdemand, meet and minimise losses and do so at minimum financial cost. Optimal PowerFlow (OPF) is used to optimise the power flow solution against one, more or all of theseobjectives.

Generation can be ascribed fuel and operating costs (£/MWh) to a formula, transmissionlosses can be ascribed a cost (£/MWh) and connected capacity beyond specified reservecan be penalised. Other strategic factors can be assigned a cost. An overall cost function(CF - sometimes objective function) may be determined that need not equate to real£/MWh because some components of the CF may be weighted. OPF minimises thecost function while finding an acceptable converged electrical power flow solution whosecapacity margin and line loadings are acceptable.

2.11 Generation cost functions

Economic merit order scheduling loads up the cheapest generator up to its limit, thenadds the next cheapest, etc until the total system demand + losses are met. Demand isusually assumed to be price-inelastic giving a vertical characteristic. Assuming no twoidentical units, there is always just one marginal unit supplying the last MW.

The operating costs (£/MWh) of thermal plants are not constant or independent ofoutput MW, nor are they linear with MW, due to the variation of thermal efficiency.


They are typically quadratic, of form:

Ci = αi + βiPi + γP 2i (2.45)

The incremental or marginal fuel cost curve (£/MWh) is obtained from derivative offuel cost curve. It is the cost of increasing generation by 1 MW (not the average).

λi =∂Ci∂Pi

= βi + 2γiPi (2.46)

2.12 Economic dispatch

Operating cost = Fuel + (O&M, depreciation, civic costs) and is usually expressed as a% of fuel costs. Additionally there are start-up and shut-down costs, and minimum andmaximum loading limits. When the costs were constant, there was always one unit atthe margin (supplying the last MW). For differing costs, there may be more than oneunit at the margin. Economic dispatch chooses the optimal combination of outputs forunits with a range of differing costs.

All units in an economic dispatch should operate at equal incremental (marginal) oper-ating cost. This is common sense since if any unit is operating at a higher incrementalcost than the others, then costs would be reduced by loading up cheaper units whilebacking off the more expensive one.

Determine combination of shares of P1, P2, . . . , Pn that minimises total production costsCtotal.

Ctotal =∑

Ci(Pi) (2.47)

subject to the equality constraint arising from real power balance

PD =∑

(Pi) (2.48)

The equality constraint can be turned into a mismatch cost, ie the penalty cost of failingto balance. The general solution to a constrained optimisation is found by adding (orsubtracting) the constraints, multiplied by an unknown Lagrange multiplier λ, to thecost function to form the Lagrangian function:

L(P, λ) =

n∑i=1

(αi + βiPi + γP 2

i

)︸︷︷︸

cost function

+λ

(PD −

n∑i=1

Pi

)︸︷︷︸mismatch penalty

(2.49)

This can be minimised with respect to Pi and λ separately and combining the solutionsgives:

n∑i=1

λ− βi2γi

= PD (2.50)


where λ is given by:

λ =PD +

∑ni=1

βi2γi∑n

i=112γi

(2.51)

2.13 Constrained dispatch

In reality generators operate within limits. The maximum output of the generator is itsthermal limit, Pi(max), determined by losses, cooling and ambient temperature. Theminimum output of the generator is its stability limit Pi(min), determined by the boileror electromechanical characteristics. The output is then within these limits:

Pi(min) ≤ Pi ≤ Pi(max) (2.52)

This means three new additional conditions (Kuhn-Tucker) apply in the Lagrangianfunction:

λi =dCidPi

for Pi(min) ≤ Pi ≤ Pi(max) (2.53)

λi ≤dCidPi

when Pi = Pi(max) (2.54)

λi ≥dCidPi

when Pi = Pi(min) (2.55)

The system marginal cost, or incremental fuel cost thus has three different values de-pending on whether Pi is within operating limits. This cannot be solved linearly - itneeds iteration.

2.14 Loss minimisation

Now we can

• Optimise dispatch to minimise fuel costs,

• Penalise excess generation (mismatch) in the cost function by the equality con-straint

• Constrain generation within limits in the economic dispatch.

We need to reflect the reality that transmitting power over a long distance will result inohmic heating losses in the network that (for optimisation) must be minimised and willinfluence the optimal dispatch. The losses on a line are given by:

Losses = I.I∗R =(P + jQ)(P − jQ)

V V ∗=

(P 2 +Q2)

V 2(2.56)


Thus I2R transmission losses are approximately proportional to the complex power flowsquared. For the purposes of loss minimisation transmission losses can be approximatedin per unit. Any losses are a ‘shortage’ that must be accounted for in the optimisa-tion.

Generally need to minimise the sum of generation costs,

∇CT =

n∑i=1

Ci(Pi) (2.57)

subject to an equality constraint or a mismatch penalty:

φ = PD + Ploss(P1, P2, . . . , Pn)−n∑i=1

Pi (2.58)

and minimise the Lagrangian:L = CT + λφ (2.59)

∂L

∂Pi=dCi(Pi)

dPi+ λ

(∂Ploss∂Pi

− 1

)= 0 (2.60)

This can be rearranged to give λ:

λ =1

1− ∂Ploss∂Pi︸︷︷︸

penalty factor

dCi(Pi)

dPi(2.61)

The penalty factor for each bus increases with increasing sensitivity of losses to additionalgeneration. It is also called the marginal loss factor for a busbar and is geographically andconfiguration sensitive. The generalised equal lambda criterion arises as the marginalcosts of generation multiplied by penalty factors must be the same and equal to λ (systemincremental costs). Penalty factors in the transmission network are usually greater than1 as the denominator is usually less than one (since an increase in generation usuallytends to cause a small increase in losses).

When generation is less than local demand the penalty factor is less than 1 because themarginal loss factor for that bus is negative (increase in generation causes a reductionin losses). This happens generally when a generator creates a counterflow against thedominant flow.

2.15 System wide sensitivity

In a highly interconnected network it is very difficult to express power losses as a functionof individual generators. Penalty factors can be obtained directly from solution of theAC power flow. Any real power imbalance in a power flow solution will be identified inthe slack busbar. The marginal loss factors ∂Ploss/∂Pi can be explored by disturbing


generation at each busbar in turn by ∆Pi and observing the effect it has on the slackbusbar (reference) generation ∆Pref . If there are no losses, an increase in generation∆Pi by 1 MW results in an equal reduction in generation at the slack busbar (reference)and ∆Pref = - 1 MW.

However if an increase in generation ∆Pi causes an increase in transmission losses, thenthe reduction in generation at the slack busbar is smaller.

2.16 Optimal power flow

To ensure power system security of supply, the system must not only be operated in themost economical manner but also in an acceptably secure way. In a liberalised marketenvironment there can be a conflict between the two drivers.

Example of security requirements / network constraints are:

• Generators must operate within their MW and MVAr limits

• The lines and transformers must not be overloaded

• Bus voltages must be maintained within limits

Optimal power flow is an economic dispatch problem subject to security constraints.Apart from adjusting generation to supply demand in the most economic manner, othercontrol variables may be manipulated:

• Generator voltage set points

• Transformer tap positions

• Switched capacitor settings

• Reactive injection for static VAR compensators

• Load shedding

The overall method of solution is the same, i.e. constraints are multiplied by Lagrangemultipliers and added to the objective function (total generation costs). To minimisethe cost function, subject to the constraints, the gradient of the Lagrange function (sumof partial derivatives) must be equal to zero. Solution of the resulting equations is noteasy and the subject of on-going research worldwide.

2.16.1 Economic interpretation of Lagrange multipliers

If a given Lagrange multiplier is zero, the corresponding constraint is not active orapplied. A non-zero Lagrange multiplier gives the shadow marginal cost of a givenconstraint, i.e. the reduction in the total cost if a given constraint is marginally relaxed.For example if there is a constraint on the transmission line flow, the corresponding


Lagrange multiplier shows by how much the generation cost would be reduced if theconstraint was relaxed (i.e. the line flow limit increased) by 1 MW.

2.16.2 Security constrained optimal power flow (SCOPF)

Secure operation must be maintained not only for a given operating state, but also im-mediately following ‘credible’ contingencies (tripping of lines, generators, transformers).The almost universally adopted reliability criterion is (N-1), i.e. a system must with-stand on its own a loss of any single element. That means that all the line flows mustbe below their limits not only for a given ‘normal’ operating state, but also when anyof the lines is disconnected. Modifications to dispatch increase the system generatingcosts.


3 Network integration

3.1 Drivers and timeframes for change

Climate change mandates reducing carbon flow in the energy system from source toend-use, this is being introduced through legislation. To meet the targets, renewableenergy generation will have to increase from 20 to 150 TWh (5-38%) and conventionalenergy generation to fall from 380 to 250 TWh (95-62%) by 2020.

Many conventional thermal plants will reach their retiral age and fleet of bulk, dispatch-able generation will change. The transmission and distribution networks do what theywere designed to do, very very well. There are however, concerns about the effects of50GW of unconstrained, distributed capacity in the grid.

3.2 Effect of connecting distributed generation

Renewable energy resources are usually remote where the distribution network is radial,passive and unidirectional. The voltage is profiled to ensure quality of customer supplywhere active and reactive power flows outwards.

When DG is connected, active power usually remains outwards but reactive power flowis outwards or inwards, the distribution system now has to deal with an intermettent,bidirectional flow.

The distribution network operator must maintain quality of supply to local load such as:voltage profile, reliability and harmonic content. They must also protect supply assetsby enforcing thmal limits and increasing fault levels.

3.3 Requirements and standards

There are British Standards (BS) and Engineering Recommendations (ER) that definethe standards to which all grid connected machinery must operate. To name a few:

• BS EN 50160 - Voltage Characteristics of Electricity Supplied by Public Distribu-tion Systems, 1995

• BS EN 7671 - Requirements for Electrical Installations, 1992

• ER S5/1 - Earthing Installations in Substations, 1966

• ER P2/5 - Security of Supply, 1978

• ER P26 - The estimation of the maximum PSCC for three phase 415V supplies,1985

There are many more...


3.4 Power-flow, thermal ratings, losses

By connecting generation close to load centres this can reduce losses of transmittingpower down the distribution network and as long as the distributed generation is notexporting more power back through the network than was originally being sent the losseswill be reduced.

3.5 Voltage regulation

If we consider a 2 busbar system where:

Vsend = Vrec + Iline(R+ jX) (3.1)

Then the apparent power on the line will be given by:

Sline = Pline + jQline = VsendI∗line (3.2)

This can be multiplied out to give:

Vsend = Vrec +(RPline +XQline)

V ∗send+ j

(XPline −RQline)V ∗send

(3.3)

This equation is non-linear, since Vsend appears on both sides. Since Vsend is determinedby itself, it can only be solved by iteration from an initial guess for Vsend. ChoosingVsend = V ∗send = 1.0 pu 6 0 is a good starting point. If, and only if, the complex powerreceived and voltage at the receiving end are both known the equation is linear and canbe solved without iteration, otherwise it’s a power flow solution.

∆Vgen =XQgenVgen

if X >> R (3.4)

However in rural networkR > X and voltage rise limits dispatch of active power fromdistributed generation. Too much generation can raise voltage profile outside of statutorylimits which would automatically be disconnected.

3.6 Harmonic content

Rotating machines will comply with British Standards etc. PVs or static conversion willincrease harmonic content and DNOs are not prepared to blindly accept this energy.Directly connected induction generations can balance out-of-balance rural voltages andact as high pass filter to harmonics. Care has to be taken with power factor correctionor self excitation as this can shunt varying amounts of harmonic current.


3.7 Transient stability

Small DGs are unlikely to reduce system stability. A developer must check DG stabilityagainst load changes, faults, switching to remain synchronised - or trip dependably Thisis influenced by:

• Distribution network characteristics and distance

• EG capacity and inertia, stored energy constant

• EG vulnerable to fault in radial feeder

Overall stability must be reviewed as DG capacity increases and displaces conventionalplant, or as thermal plant retires.

3.8 Protection co-ordination and operation

The network has co-ordinated discriminating protection for unidirectional power flow.The presence of DGs can reduce protection stability and predictability as well as causingspurious operation. This requires adjusted settings to accommodate DGs but this canreduce discrimination and levels of protection. The DG is not constrained and it isdifficult to provide equivalent protection ON/OFF.

3.9 Synchronous or induction generators?

Synchronous:

• Steady power input

• Self-excited - may be able to sell VArs

• Operates at set PF

• Overexcited > volts rise

• Increases sustained SCFL

• Contributes to stored kinetic energy

• Can reinforce weak net.

• Needs exn and sync equip

• Generally more expensive

Induction:

• Time varying power input

• Excited off network - have to buy VArs


• Operates at varying PF

• Can reduce volts rise

• Increases peak SCFL

• Contributes to stored KE

• Can not reinforce voltage

• No exn or sync equip

• Generally cheaper

3.10 Types of fault

At high voltage transmission network is balanced 3 phase R/Y/B with no 4th or 5th

wire, although overhead line towers are earthed.

Balanced

RedYellowBlue

VryVyb

Vbr

Ir

IyIb

Ir Iy IbVrn

Balanced

Ir Iy Ib

Vrn Vyn Vbn

Zrn Zyn Zbn

Vyn Vbn

Earthed star pointSource Loads

Neutral Neutral

Vrn, Ir

Vyn, Iy

Vbn, Ib

Figure 14: High voltage transmission system - balanced 3 phase

If neutral voltage is zero and neutral current is zero, there is no need for a 4th wire, butthis depends on the source voltages and loads being balanced.

In = Ir + Iy + Ib = 0 (3.5)

Vn = Vrn + Vyn + Vbn = 0 (3.6)


At low voltages (400V and below) loads are seldom balanced and LV cables are requiredto include neutral. VN need not resolve to zero at load and IN is the sum of theunbalanced phase currents that flows back to the supply.

Unbalanced

RedYellowBlue

VryVyb

Vbr

IrIyIb

Ir Iy IbVrn

Balanced

Ir Iy Ib

Vrn Vyn Vbn

Zrn Zyn Zbn

Vyn Vbn

Earthed star pointSource Loads

I neutral

Neutral Neutral

Figure 15: Low voltage transmission system - unbalanced 3 phase

The different types of faults are shown below in figure 16.

Balanced faults are symmetrical and all phase fault currents have equal magnitude.Unbalanced faults are much more common and have asymmetric current flow. Needsymmetrical components for solution. Electrical failure generally occurs in one of twomodes:

• Insulation failure results in a short-circuit condition

• Conducting path failure results in an open-circuit condition, such as ‘downed’overhead lines, or phase failure at switches.

Insulation failure is much more common as a result of mechanical damage or intrusion ofnatural or man-made objects. Insulation materials normally last 20-30 years, and mostdamage is usually accidental. Example causes for certain faults:

• Single phase to earth - Crane jib fouls overhead line.

• Phase to phase - Debris falls onto overhead line.

• Phase to neutral - Tools or components left in equipment that is returned to service.

• Phase to phase to earth - Excavator pulls up an underground cable, or ac machine


Healthy

R

Y

B

E

N

R

Y

B

E

Single phaseto Earth

N

R

Y

B

E

Phase to phase

N

R

Y

B

E

Phase to phaseto Earth

N

R

Y

B

E

Three phase

N

R

Y

B

E

Three phaseto Earth

N

R

Y

B

E

Phase to neutral

N

R

Y

B

E

Neutral to Earth

N

Figure 16: Different types of fault

failure.

• Neutral to earth - Phase fault raises neutral which flashes over in cheap whitegoods.

The factors which affect the severity of a fault, and the determine the potential to causedamage are:

• Source conditions

– Amount and location of all connected rotating plant

– Ties and interconnections with other networks

• Network configuration

– Manner in which network is connected between generation and loads to meetloads, and fulfill voltage profiles from load-flow studies.

• Neutral earthing

– Presence or absence, or value of impedance between source neutrals and earthsrestricts maximum value of earth fault current.


3.11 Short circuit close to a generator

The most severe transient condition which can occur in a synchronous generator is whenall three line terminals are suddenly shorted. The single phase balanced load equivalentcircuit for a synchronous machine shown below is an internally generated emf Et,in serieswith an armature reactance jXs. The complex current in healthy conditions could becalculated from:

I =Et

jXs + Zload(3.7)

After inception of a three phase terminal fault you would expect the sustained shortcircuit fault current to become but this is not the case,it is given by:

ISC =EtjXs

(3.8)

The phase fault current envelope in one phase of a synchronous generator is shown belowin figure 17 after the occurrence of a balanced three phase short circuit. The amplitudeof the current envelope, or decrement curve varies with time yet the internal emf Etbehind the reactance in series with the short circuit is not externally changed, since theAVR does not change excitation at this time. Further, the envelope is asymmetric, sothe above model no longer represents the transient conditions correctly.

Figure 17: Asymmetric fault current

If the generator is operating on open circuit with an induced Et = 1.0 pu the rotor,airgap and stator are magnetised from the rotor side and the machine inductances are


all charged with stored magnetic energy. If the generator is synchronised to a networkand is overexcited on load the stator inductance can be thought to be storing from andreturning to the network magnetic energy. In the event that a fault takes place thereis a significant component of initial current added as the inductances are discharged.This increases and offsets the transient currents that will feed the fault. They decayexponentially with two time constants.

Figure 18: Short circuit close to a generator

For a short time, typically a few cycles of generated emf, the current produced is partlydetermined by the flux linking the rotor and stator windings at the instant of the fault.The high alternating current which flows in each phase of the stator during this periodis called the sub-transient current denoted I”.

The reactance which at that time limits the flow of sub transient current is termed thesub-transient reactance Xd” and is heavily influenced by the rotor pole construction,and the presence or absence of a damping winding in the pole-tips. The instantaneouscurrent I” flowing during the subtransient period may be over 10 x the steady statefault current. This is the current used to calculate the fault contribution in fault levelstudies.

3.12 Fault level studies

Power flow studies examine the steady state flow of real and imaginary power in a healthypower system. Fault level studies determine the short circuit fault currents that flowinto electrical faults in the network, at the point of the fault and everywhere else. Powersystem fault may be defined as the electrical failure of primary equipment operating atnetwork voltages (generators, motors, transformers, busbars, overhead lines and cables)


sub-transient period

transient period

steady state period

t’’ t’

Isc

I’

I’’

time

Figure 19: Simplified current following a fault

which causes the abnormal flow of power and current Equipment must be specified tosafely withstand (carry), or in the case of switchgear, interrupt these currents. Protectionengineers use the calculated fault currents to ensure that overcurrent protection operatesCBs in rapid, safe and systematic way.

Sustained overloads (say 1-2 x normal current) may be accommodated for a modestperiod. Generators, transformers and cables have sufficiently high thermal capacity andlong thermal time constants to enable them to pass shortlived overcurrent that (say 2-5 xnormal). Higher over-currents (say 5-10 x normal or greater) can cause electromechanicalstress between conductors and lead to insulation failure and/or mechanical damage,which is likely to occur very rapidly. Continuing high current will lead to thermal damagedue to the I2t ohmic heating effect. The worst-case fault is 3-phase short-circuit fault.Clearly such faults must be detected and immediately interrupted by the over-currentprotection tripping the circuit breakers.

Short-circuit fault level changes with network conditions between two extreme values,all plant connected and minimum plant connected.

3.13 Per-unit equivalent impedance of a network

Where a network operates at 1.0 pu voltage, has a finite source impedance, Xgrid and apredicted 3 phase symmetrical short circuit fault level MVAgrid then:

Xgrid(pu) =1

MVAgrid(3.9)

Changing to a common MVAbase for fault calculation:

Xgrid(pu) =MVAbaseMVAgrid

(3.10)


to a chosen MVAbase.

Analysis of balanced faults analysed consists of three parts:

1. The system with its fault condition is represented by its positive sequence singlephase equivalent circuit

2. The equivalent circuit is solved in terms of per-unit quantities

3. The resulting per-unit quantities are converted to actual values

Simplified fault analysis makes the following assumptions:

• Power system balanced before and during fault and fault is symmetrical across 3phases.

• Generated voltages are not affected by fault and remain constant in magnitudeand phase.

• The pre-fault voltage at any busbar is 1.0 pu - generators operate at rated terminalvoltage and all transformers are set to nominal tap position.

• Synchronous motors and generators are modelled by emf and subtransient reac-tance only.

• Synchronous motors ”generate” into fault - stored kinetic energy keeps shaft spin-ning, and rotor field induces Et in stator.

• Transformers normally represented by their series reactances only.

• Overhead lines and cables represented by series reactance - neglect shunt terms.

• The impedance of the fault is negligible.

• The network impedance is purely reactive.

• Pre-fault currents are ignored because the load current is mainly active and thefault current is much larger and mainly reactive.

This allows a fault anywhere in network is reduced to a single source of emf and a singleimpedance up to the point of the fault

3.14 Fault level changes due to new DG

If new DG is close to a primary substation, the system fault level is high and this makesthe DG and plant vulnerable. DG switchgear must be system fault rated.

If new DG is remote from a primary substation, the weak rural system has a high Zso a lower fault level. The DG will increase fault levels everywhere depending on type(S/I), reactance, excitation system. This makes system switchgear vulnerable. This


may involve the cost of upgrading the network to be included in the connection charge.Protection settings must be reappraised before the DG is connected.

4 The UK electricity market

4.1 UK & National Grid

The UK has a complex administrative structure with devolved administrations but thesedo not control Energy policy. Generation in the UK is mostly thermal generation withDC connections to France and Northern Ireland. Transmission is at 400 and 275 kV,with distribution at 132, 66, 33 & 11 kV. At the commercial / domestic level it is at400/230 V.

Prior to 1990 the entire market was state owned. This was broken up with the electricitypool in 1990, replaced with the New Electricity Trading Agreements (NETA) in 2001 andthese were extended to Scotland under the British Electricity Transmission & TradingArrangements in 2005.

4.2 Privatisation and unbundling

The then Thatcher Government had a privatisation programme where competition throughthe market was seen as ‘good’ and money from sales were used to fund tax cuts. TheArea boards became Regional Electricity Companies (RECs).

Aimed to achieve competition in the electricity industry by breaking down the monopolyinto competing parts (unbundling).

4.3 Electricity markets

Markets trade commodities that can be stored (wheat, crude oil, gas). Electricity is acommodity with unique characteristics:

• Cyclical demand means price should change with the load (time-varying prices)

• Electricity cannot be stored economically to use at times of peak demand and thiscauses price spikes

• Integrated nature of electrical system means a change of generation at any locationinfluences power flows in all lines

– Generation has to be coordinated to avoid transmission overloads.

– Price should vary with location

– Security has to be considered system-wide to avoid blackouts


When the laws of physics collide with the laws of economics, physics wins every time.

Trading must be followed by delivery as the demand-supply balance must be maintainedat any time at (almost) any cost. Electricity produced by different generators must bepooled where:

• Electricity is indistinguishable

• Electricity flows around the network according to physical, rather than economiclaws

• Generators cannot choose whom they supply physically

• Consumers cannot choose who supplies them physically

• Electricity cannot be stored

This defines a need for a SPOT MARKET or TRADING POOL.

4.4 The electricity pool

The electricity pool operated between 1990 and 2001 as a day ahead spot market whereplants submit bids and a least cost schedule is determined.

Figure 20: Clearing in the pool

Market Clearing Price (System Marginal Price): intersection between the supply anddemand curves System Marginal Price SMP is $16/MWh. This is the price paid by theconsumers and paid to producers for all electricity traded up to 450 MWh.

This mechanism of paying (and charging) everyone the same clearing price (SystemMarginal Price SMP) is often referred to as a uniform price auction as all the winnersget the same market price, no matter what their bids were. Note that generators arefree to submit bids that are not derived from their true marginal costs. If the bids areequal to marginal costs, the resulting dispatch is optimal as it maximises social welfare.If bids are not equal to marginal costs, the resulting dispatch may not be optimal.

A uniform price auction is meant to encourage generators to bid their true marginalcosts, and ensure the resulting dispatch is optimal (assuming perfect competition). ThePool determines the market clearing (marginal) price based on the bids and determines


who is scheduled to run. Advantage of Pool trading are small transaction costs - no needto arrange separate contracts as the Pool is counterparty to all deals. Generators andsuppliers may prefer to arrange bilateral contracts rather than depend on the uncertainPool price. This can be arranged as contracts for difference, through which price fixingwas a major problem with the major generators who had large market power.

4.5 New Electricity Trading Arrangements (NETA)

NETA was brought in 2001 because centralised price setting was identified as ineffec-tive and flawed. In the new market there is no centralised scheduling or price setting,prices are determined through bilateral contracts (like financial markets). The balancingMechanism is run by the National Grid (only about 5% of energy).

Dual settlement prices mean that:

• Difference between System Buying Price (SBP) and System Selling Price (SSP)forces players into bilateral contracts

• SBP can be much higher and volatile, but is only available in ‘short’ deals doneclose to needs and settlement

• SSP less volatile, available in longer term and closer to UK Price Index UKPX(Spot prices based on Reference Price Data) but may be negative

• Most companies prefer to be ‘long’

NETA had the effect of a 40% drop in wholesale prices - looks great, but

• Reduction of capacity payments to ensure margins - 27% of price drop

• Reduction of spare capacity - 12% of price drop

• Industrial and commercial customer prices fell by 17%

However...

• Domestic prices fell by only 8% - households slow to switch

• A lot of plant was mothballed - and costly to restore

• British Energy nearly went bankrupt

• NETA may have forced uneconomic plant & players out

• Many argue that the same benefits could available by restructuring the Pool andrunning it better more cheaply


4.6 British Electricity Transmission and Trading Arrangements (BETTA)

NETA was extended to Scotland from April 2005 as BETTA (British Electricity Tradingand Transmission Arrangements). The whole transmission network managed by NationalGrid. This has been very challenging for generation particularly in Scotland in terms ofconnection and transmission use of system charges.


5 Ancillary services

Ancillary services can be broadly divided into control (scheduling, voltage control, mon-itoring and control in real time, stability) and reserve (security). The FERC defines 12ancillary services:

• System control• Regulation• Spinning reserve• Backup supply• Real power loss replacement• Dynamic scheduling• System black start

• Network stability• Voltage control• Load following• Supplemental reserve• Energy imbalance

5.1 System control

System Operator (SO) functions:

• Scheduling of generators, transmission resources and transactions before delivery.

• Monitoring and control in real time to maintain reliability.

• Accounting, billing (settlement)

Well defined and understood

5.2 Voltage control

Use of generating and transmission-system equipment to inject/absorb reactive powerto maintain voltages on the transmission system within required ranges. Increased con-sumption of reactive power causes voltages to drop

Qr =VrVsX

cos δ − V 2r

X≈ VrVs

X− V 2

r

X(5.1)

Voltage control must be dispersed and close to loads as reactive power flows cause hightransmission losses and voltage drops.

5.3 Cost of producing reactive power

Voltage and reactive power can be controlled by both static assets (capacitors, reactors,tap changing transformers) and excitation (field current) control of generators:

Q =EfV

Xscos δ − V 2

XsQ =

VsVrX

cos δ − V 2r

X(5.2)


The emf, Ef is proportional to the field current If (Faraday’s Law). Hence excitationcontrol = voltage (reactive power) control.

The System Operator (SO) usually uses only static elements as these only have a cap-ital cost and no operating cost. Generators are the best way to control voltage (andreactive power). The actual cost of producing reactive power (increasing field current)is negligible, however there is an opportunity cost.

Consider armature (stator) current heating limit of a generator:

Smax = V Imax =√P 2 +Q2 =

√(V Imax cosφ)2 + (V Imax sinφ)2 (5.3)

Generator produces max real power at unity power factor.

5.4 Reactive power markets

Difficult, if impossible, to create a market for reactive (imaginary) power. Real power isa commodity, imaginary isn’t

• Real power can be pooled wherever it is generated (subject to transmission con-straints)

• Reactive power cannot travel far because it disturbs the voltage profiles - and youcannot pool it

• Must be dispersed geographically

• If a market was created, there would be many pockets of market power

In the UK there is an auction twice a year where capacity (£/MVAr) & utilisation(£/MVArh) bids from non-participating generators get default payments.Phasing out ofcapacity element to force generators into the market.

5.5 Frequency regulation

Frequency control is the use of connected generators (and spinning reserve) with au-tomatic generation control and that can change output quickly (MW/minute) to trackand balance the moment-to-moment fluctuations in customer loads and unintended fluc-tuations in generation to maintain frequency, minimise differences between actual andscheduled flows between control areas and match generation to load within control ar-eas.

5.6 Epochs of response

Regulation service handles rapid fluctuations in generation/demand balance. Some gen-erators must be partly loaded (spinning reserve) and operate under Automatic Genera-


tion Control (AGC). The turbine governor has a sloping characteristic so that reductionin frequency results in increase of generation. Part loading requires appropriate com-pensation.

Load following handles slower changes of demand whereas reserve services cover large andunpredictable power deficits - units usually not connected to the grid but waiting. Allinfluence generation so generators must be engaged and paid for services provided.

5.7 Market for ancillary services

Usually implemented through a mixture of administrative and market-based measures.For example administrive measures through the grid code such as

• Regulation: e.g. all generators must be equipped with 4% droop coefficient

• Reactive power: e.g. all generators must be equipped with Automatic VoltageRegulator (AVR) and be capable of operating at pf from 0.85 lead to 0.9 lag

Administrative means are not necessarily the most economically effective, not all units,due to their different technical and economical characteristics, should equally contributeto frequency and voltage control. Also without a market, there is no drive to improveservices and no compensation for losses. Market solutions are better but more difficultto devise and administer. Market power may be a problem if the number of providers issmall, e.g. separation due to transmission constraints, voltage support in remote areas.Demand-side provision of AS improves competition.

Energy and reserve markets are inter-related as provision of reserve requires part-loadingof cheaper units and scheduling of more expensive ones. Some markets: successiveclearing, i.e. first cleared energy and then reserve market:

• Generators submit bids to successive markets and could transfer resources fromone to another market

• Inefficient, complicated trading, and abandoned (California)

Both energy and reserve markets should be cleared simultaneously.


6 Transmission networks and locational marginal pricing

6.1 Transmission Network Congestion

So far we have assumed that all generation can be absorbed by the network (no trans-mission congestion). In practice almost all networks are to some extent congested, if anetwork is not congested at all, it has been over designed. The optimal condition existswhen the marginal cost of congestion = marginal cost of network reinforcement to getrid of congestion.

The Pool mechanism has to be modified to take congestion into account. Producers andconsumers submit bids and offers to a central market, the ISO (Independent SystemOperator) selects the winning bids and offers in a way that optimally clears the market(i.e. selects generation that satisfies demand) and respects network security constraints(maximum flows, voltage limits, etc). If there is no congestion or losses the same uniformprice applies everywhere determined by the marginal generator. If there is congestionor losses, the price depends on a location (Locational Marginal Prices LMPs).

6.2 Interconnected Systems

An interconnected system allows countries with excess generation to sell this and inconstrained countries to buy cheaper generation from the interconnector. The economiesof both countries benefit as economic dispatch maximises social welfare. In particular thegenerators in the cheaper country and the consumers in the country with more expensivegeneration. However the expensive generators lose out and the customers which now haveto pay more lose out and the losers usually shout louder than winners.

6.3 Constrained Connection

Congestion on interconnection reduces the benefits of the interconnection. It allows thepossibility of gaming as congestion separates the markets at both ends of the intercon-nector. The more expensive generators can raise their prices above marginal costs ascheaper generation from across the interconnector is constrained.

Note that congestion means that payments made by consumers (Etotal) is higher thanrevenues of generators (Rtotal). This is termed congestion surplus.

6.4 Effects of Network Congestion

This congestion surplus is collected by the Market Operator (MO) in Pool trading, itshould not be managed by MO or System Operator (SO) as it gives perverse incentives.


The higher congestion, the higher surplus. There are ways of dealing with congestionsurplus.

Reducing network access charges: Network Operator is a monopoly so it receives arevenue (fixed by the Regulator) to cover its costs. This revenue is paid by grid ac-cess charges of generators and consumers. Congestion surplus could be used to reduceuniformly grid access charges.

Financing network expansion (used in South America and Australia): A private investorcan upgrade the interconnector and recoup the cost from congestion rents (equivalentto road tolls). Note that the investment will not remove congestion completely as thenthe congestion rents would be zero. The remaining congestion should be such that theexpected congestion rents should cover the cost of investment. This is the investmentoptimality condition: marginal cost of investment = marginal cost of congestion. LMPsare used in a number of markets worldwide, most notably in the US.


7 UK renewables & support mechanisms

7.1 UK climate & renewable energy targets

The UK Government has set a challenging long term aim of 60% cuts in CO2 emissionsby 2050. The UK has a Kyoto Protocol commitment for 12.5% reduction below 1990levels by 2008-2012 and an additional target of a 20% cut in CO2 by 2010 which hasbeen missed. The Government expects renewable energy to play a vital role in meetingthese targets.

EU Renewables Directive requires UK renewables to meet 10% of electricity demandby 2010 with an aspiration of 20% by 2020 and a view of 30-40% by 2050. This hasbeen broken down into seperate targets within the devolved administrations. Achiev-ing these targets and aspirations will require a massive increase in capacity from allrenewables.

7.2 Recent history of UK renewables

In Scotland there was extensive development in hydro and more recently the main newsource of renewable generation is from onshore wind. Onshore wind growth is very highand offshore wind consented in 2nd & 3rd rounds. There have also seabed leases for 1.2GW of wave and tidal energy just announced around Orkney.

7.3 Support

Range of schemes to support UK renewables since electricity industry privatisation in1990:

• The Non-Fossil Fuel Obligation

• The Climate Change Levy

• The Renewables Obligation

The Non-Fossil Fuel Obligation (NFFO) provided a competitive mechanism for non-fossil generation where renewable developers bid a price they could generate at. Fundedby levy paid by suppliers (i.e. consumers), this was originally 10% (dropped to 2.2%).Renewable NFFO by-product of a mechanism for supporting nuclear and vast majorityof the subsidy went to nuclear.

The Climate Change Levy (CCL) was a tax on the energy used in industry, commerceand the public sector according to the type of energy used. It was designed to encourageenvironmentally friendly fuel sources to help meet emissions targets. It was financiallyneutral to the UK Treasury through cuts in Employers National Insurance contributions


for registered companies. Energy-intensive industries can see an 80% reduction in CCLrates by implementing agreed energy efficiency schemes.

The Renewables Obligation (RO) requires all licensed electricity suppliers in England andWales to supply a specific proportion of their electricity from renewables. It is in placeuntil 2027 to provide a stable and long-term market with annual obligations originallyspecified up to 2010/11 but extended through to 2015/16. Generators receive ROCs forelectricity produced from renewable sources and are able to sell excess certificates.

7.4 Challenges

Despite the financial incentives there are still challenges in meeting UK renewables tar-gets, namely:

• Financing of projects

• Size

• Renewables Obligation itself

• Network issues

• UK electricity market

• Planning issues

ROC value and buy-out premium will vary with supply and demand for renewables.So there are no price / revenues guarantee, as supply exceeds demand, ROC value willfall. Difficulties for project finance as lenders mark down ROC value . This introducesdifficulties for project finance as lenders mark down ROC value.

UK transmission network flows are primarily from Scotland/northern England to theSouth East, exploitation of renewables especially hydropower will tend to accentuatethis. The transmission network is constrained and a £1.5 billion upgrade is required toship extra (wind) power.

Transmission use charges will be higher in the North and as most renewables connect atdistribution level with impacts that tend to make connections expensive, this requiresnew incentives for Network Operators to connect and empathetic planning and ‘activenetworks’. Generators pay the market rate to cover anyshortfall and this penalisesintermittent / less controllable generation and makes it less valuable.

Despite these challenges there have been significant increases in capacity.


Perc

enta

ge o

f UK

gene

ratio

n ex

cept

whe

re s

how

n 6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.01990 1992 1994 1996 1998 2000 2002 2004 2006 2008

All renewablesHydro

All renewables ObligationWind, wave, solar and biofuels (% of UK electricity sales)

Figure 21: Increase of renewable capacity

7.5 Conclusions

The UK has long if chequered history in developing and supporting renewables, albeitoriginally in some cases as a by-product of a nuclear subsidy. The RO set challengingrenewable targets and uses a form of ‘green’ certificate, but it is stimulating investmentin the renewable sector. Difficulties with the RO and financial / technical issues haveacted to constrain development. Banding looks like an interesting prospect but will ithave the desired effect?

power systems engineering and economics - · pdf file3.10 types of fault ... connects...

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