A New Approach to Secure optimum power generation

7/29/2019 A New Approach to Secure optimum power generation

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A new approach to secure

economic power dispatch

J Z Zhua, M R Irvingb ,* and G Y XuaaDepartment of Electrical Engineering, ChongqingUniversity, Chongqing, P.R. ChinabDepartment of Electrical Engineering and Electronics,Brunel University, Uxbridge, Middlesex UB8 3PH, U.K.

This article presents a new nonlinear convex network flowprogramming model and algorithm for solving the on-lineeconomic power dispatch with N and N 1 security. Basedon the load flow equations, a new nonlinear convex network

flow model for secure economic power dispatch is set up andthen transformed into a quadratic programming model, inwhich the search direction in the space of the flow variablesis to be solved. The concept of maximum basis in a network

flow graph was introduced so that the constrained quadraticprogramming model was changed into an unconstrainedquadratic programming model which was then solved by thereduced gradient method. The proposed model and itsalgorithm were examined numerically with an IEEE30-bus test system on an ALPHA 400 Model 610 machine.Satisfactory results were obtained. 1998 Elsevier Science

Ltd. All rights reserved

Keywords: Economic dispatch control, Security analysis,N 1 security constraints, Convex network flow program-ming, Quadratic programming

I. IntroductionThe aim of real power economic dispatch is to minimize thetotal production cost of supplying all the loads on the powersystem. Security dispatch implies the continuity of supplyservice, even in the event of failure of equipment, and musthandle several physical constraints. Among these post-outage constraints, the line loading constraints should bemet if possible, but may be violated temporarily. Therefore,the security problem is an additional and important aspect ofpower system operation, and has a profound influence on theoverall economic dispatch problem.

It is well known that in real time secure economic powerdispatch, the crux of the matter is to provide speed andprecision in the calculation. A number of methods have beenproposed to solve this problem [14]. The usual methods(linear or nonlinear model) require a large number of AC orDC load flow calculations to be carried out, and seem to beinefficient for large scale power systems. Network flow

programming (NFP) models of security economic power

dispatch were presented in recent years [1,5,6]. NFP is aspecialized form of linear programming. It is characterizedby simple calculation and rapid convergence. In particular,the concept of anN 1 constrained zone was proposed inreference [1] so that the economic power dispatch withNand

N 1 security can be solved by NFP. However, because ofthe adoption of a linear NFP model in [1], the calculationprecision is still not fully satisfactory.

In order to solve the combined problem of speed andprecision in the calculation of secure economic powerdispatch, this article presents a new nonlinear convex net-work flow programming model of on-line economic powerdispatch with Nand N 1 security, which is solved by usinga combination of quadratic programming (QP) and NFP.

Based on the load flow equations, a new nonlinear convexnetwork flow model for secure economic power dispatch isset up and then transformed into a QP model, in which thesearch direction in the space of the flow variables is found.The conceptof a maximum basis in the networkflow graph isintroduced, allowing the constrained QP model to bechanged into an unconstrained QP model, which is thensolved using the reduced gradient method.

In the article, the fast N 1 security analysis method [7,8]is also used to seek out all the binding constraint cases for allpossible single line outages, and then an N 1 constrainedzone is formed which is coordinated with the convexnetwork flow programming model. As a result, the proposedsecure economic dispatch and associated solution method

have both high calculation precision and high calculationspeed.

The proposed model and algorithm are examined numeri-cally with an IEEE 30-bus test system on an ALPHA 400Model 610 machine. Satisfactory results are demonstratedand also compared with the results obtained through theconventional methods.

II. Convex NFP model of secure economicpower dispatch

II.1 Calculation of N 1 security constraints

Economic power dispatch withN 1 security means that theline flows will not exceed the settings of the protectivedevices for the intact lines when any branch has an outage[13]. The usual methods of calculating N 1 security

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Electrical Power & Energy Systems, Vol. 20, No. 8, pp. 533538, 1998 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0142-0615/98/$ - see front matterPII: S0142-0615(98)00019-2

* Corresponding author


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constraints need a large number of AC or DC load flowcalculations to be carried out, and seem to be inefficient foron-line application.

A fast and efficient calculation approach for N 1security constraints, which was proposed in references[1,7,8], is adopted in this article. Based on the fast N 1security analysis, all the cases with binding constraints for allpossible single outages can be found. Thus, the maximum

value of the violation in line ij can be determined by thefollowing equations:

DPijmax max1NL

Pij(l) PijM

ijNT1 (1)

DPijmin min1NL

Pij(l) Pijm

ijNT2 (2)

Pijm PijM (3)

where NT1 and NT2 represent the number of lines whichviolate their upper and lower bounds, respectively, for line loutage. NL is the set of single line outages. P ijM is the activepower flow constraint on transmission line ij.

Therefore, an N 1 constrained zone, which is formed

by the intersection of the secure zones for singlecontingencies, can be determined from the followingequations.

DPij DPijmax ijNT1 (4)

DPij DPijmin ijNT2 (5)

PijM Pij PijM ijNTNT1 NT2 (6)

where NT is the total number of transmission lines in thepower network.

It can be observed from the N 1 constrained zone,equations (4)(6) that the number of N 1 constraints is

NT1 NT2 (NTNT1 NT2) NT

This means that the huge number of N 1 securityconstraints can be reduced to the same scale as N securityconstraints due to the adoption of N 1 constrained zone.Therefore, similar to N security, N 1 security can beintroduced into the network flow model.

II.2 Mathematical model

It is well known that the active power flow equations of atransmission line can be written as follows.

Pij V2i gij ViVjgijcosvij ViVjbijsinvij (7)

Pji V2

j gij ViVj( gijcosvij bijsinvij) (8)

where

P ij is the sending end active power on transmission line ij;Pji is the receiving end active power on transmission lineij;Vi is the node voltage magnitude of node i;v ij is the difference of node voltage angles between thesending end and receiving end of the line ij;b ij is the susceptance of transmission line ij;g ij is the conductance of transmission line ij.

In a high voltage power network, the value of v ij is verysmall, and the following approximate equations are easilyobtained

V 1:0 p:u: (9)

sinvij vij (10)

cosvij 1 v2ij=2 (11)

Substituting equations (9)(11) into equations (7) and (8),the active power load flow equations of a line can besimplified and deduced as follows.

Pij bijvij gijv2ij=2 (12)

Pji bijvij gijv2ij=2 (13)

Let

Pij bijvbij (14)

We can obtain

vij Pij=bij (15)

Substituting equations (14) and (15) into equations (12) and(13), we obtain

Pij Pij 1

2

Pij

bij

2gij (16)

Pji Pij

1

2

Pij

bij 2

gij (17)

The active power loss on transmission line ij can be obtainedaccording to equations (16) and (17), i.e.

PLij Pij Pji Pij

bij

2gij (18)

P2

ij

R2ij X

2ij

X2ij

Rij

where

R ij is the resistance of transmission line ij;

Xij is the reactance of transmission line ij.Let

Zij R

2ij X

2ij

X2ij

Rij (19)

the active power loss on the transmission line ij can beexpressed as follows

PLij P2

ij Zij (20)

Therefore, the following nonlinear convex network flowprogramming model, M 1, for real power economicdispatch can be set up.

minF

iNG aiP

2

Gi biPGi ci

h

ijNTP

2

ij Zij (21)

such that

PGi PDi

ji

Pij P

2ij

2b2ijgij

4 5(22)

PGim PGi PGiM iNG (23)

PijM Pij PijM ijNT (24)

where

PGi is the active power of the generator i;PDi is the active power load on the load bus i;P ij is the flow in the line connected to node i, and wouldhave a negative value for a line in which the flow istowards node i;

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a i, b i, c i are the cost coefficients of the ith generator;NG is the number of generators in the power network;NT is the number of transmission lines in the powernetwork;P ijM is the active power flow constraint on transmissionline ij;PLij is the active power loss on transmission line ij;

Zij* is called an equivalent impedance of transmission line

ij, as shown in equation (19);h is the weighting coefficient of the transmission losses;

j i represents node j connected to node i throughtransmission line ij;Subscripts m and M represent the lower and upper boundsof the constraint.

The second term of the objective function (equation (21))is a penalty on transmission losses based on the systemmarginal cost h (in $ per MWh). Equation (24) is the linesecurity constraint. Equation (23) defines the generatorpower upper and lower limits. Equation (22) is Kirchoffsfirst law (i.e. node current law, KCL).

The model M 1 is similar to a NFP model in form. In

fact, it is a nonlinear convex network flow programming(NLCNFP) model.In the traditional network flow programming model of

economic power dispatch, the total transmission losses arerepresented approximately as the sum of the products of theline resistance and the square of the transmitted power on theline (i.e. PL SP ij

2R ij), and the power loss of an individual

line is assumed to be distributed equally to its ends. Thus, theactive load PDi involved half the transmission losses on allthe lines connected to node i, which had to be estimated a

priori from a power flow calculation [1,5,6]. Obviously, theNLCNFP model of economic power dispatch developedhere, which is shown in equations (21)(24), does notrequire these assumptions, and has higher precision than

the traditional NFP model of economic power dispatch.

II.3 Consideration of KVL

It is well known that Kirchhoffs second law (i.e. loopvoltage law, KVL) has not been considered in the study ofsecure economic power dispatch using general NFP. This iswhy there always exists some modeling error when secureeconomic power dispatch is solved using traditional linearNFP. KVL will be considered in this article.

The voltage drop on the transmission line ij can beapproximately expressed as

Vij PijZij (25)

In this way, the voltage equation of the lth loop can beobtained, i.e.

ij

PijZij

mij, l 0 l 1, 2,,NM (26)

where

NMis the number of loops in the network;m ij,l is the element in the related loop matrix, which takesthe value 0 or 1.

Introducing the KVL equation into model M 1, we canget the following model, M 2.

minF

iNG

aiP2Gi biPGi ci

h

ijNT

P2

ij Zij

such that

PGi PDi

ji

Pij P

2ij

2b2ijgij

4 5

ij

PijZij

mij, l 0 l 1, 2,,NM

PGim

PGi

PGiM iNG

PijM Pij PijM ijNT

In order to obtain the same form as model M 1, thefollowing transformation should be carried out.

The Lagrange function can be obtained from equations(21) and (26).

FL

iNG

aiP2Gi biPGi ci

h

ijNT

P2

ij Zij ll

ij

PijZij

mij, l 27

Therefore, NLCNFP model M 3 for secure economicpower dispatch, which considers KVL, can be written asfollows.

minFL

iNG

aiP2Gi biPGi ci

h

ijNT

P2

ij Zij ll

ij

PijZij

mij, l 28

Subject to equations (22)(24), where l l is the Lagrangemultiplier, which can be obtained through minimizingequation (28) with respect to variable P ij*, i.e.

2hPijZij ll

ijZijmij, l 0 (29)

ll 2hPij=

ij

mij, l (30)

l 1, 2,,NM

If N 1 security constraints are considered, the NLCNFPmodel for economic power dispatch with N 1 security canbe written as follows.

minFL

iNG

aiP2Gi biPGi ci

h

ijNT

P 2ij Zij ll

ij

PijZij

mij, l

Subject to equations (4)(6), (22) and (23), where equations(4)(6) are N 1 security constraints, which include the Nsecurity constraints (equation (24)).

III. Solution of NLCNFP

III.1 NLCNFP model

The model M 3 of real power economic dispatch withsecurity constraints can be changed into a standard model of

NLCNFP, i.e. model M 4minC

ij

c(fij) (31)

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such that

jn

fij fji

ri i n (32)

Lij fij Uij ijm (33)

where

fij is the flow on the arc ij in the network;L ij is the lower bound of the flow on the arc ij in thenetwork;Uij is the upper bound of flow on the arc ij in the network;n is the total number of the nodes in the network;m is the total number of the arcs in the network.

In model M 4, the objective function equation (31)corresponds to equation (28) in model M 3. The equalityconstraint equation (32) corresponds to equation (22) inmodel M 3. The inequality constraint equation (33)corresponds to equations (23) and (24) for N securityeconomic dispatch, or equations (4)(6) and (23) for Nand N 1 security economic dispatch.

Equation (32) in the model M 4 can be written asAf r (34)

whereA is a matrix with n (n m), in which every columncorresponds to an arc in the network, and every rowcorresponds to a node in the network.

The matrix A can be divided into a basic sub-matrix andnon-basic sub-matrix, which is similar to the convex simplexmethod, i.e.

A [B, S,N] (35)

where the columns of B form a basis, and both S and Ncorrespond to the non-basic arcs. S corresponds to the non-

basic arcs in which the flows are within the correspondingconstraints. Ncorresponds to the non-basic arcs in which theflows reach the corresponding bounds.

A similar division can be made for the other variables, i.e.

f [fB,fS,fN]

g(f) [gB, gS, gN]

G(f) diag[GB, GS, GN]

D [DB,DS,DN]

where

g(f) is the first-order gradient of the objective function;G(f) is the Hessian matrix of the objective function whichis a block diagonal matrix;

D is the search direction in the space of the flow variables.

III.2 Solution of model

In order to solve model M 4, Newtons method can first beused to calculation the search direction in the space of theflow variables. The idea behind Newtons method is that thefunction being minimized is approximated locally by aquadratic function, and this approximate function isminimized exactly.

Suppose that f is a feasible solution and the search step

along the search direction in the space of flow variablesb 1. Then the new feasible solution can be obtained.

f f D (36)

Substituting equation (36) into equation (31) in the modelM 4, we obtain

C C(f) C(FD) (37)

Near fwe can approximate Cby the truncated Taylor series.

C(D)1

2D

TG(f)D g(f)TD (38)

Substituting equation (36) into equation (32) in the modelM 4, we obtain

AD 0 (39)

Obviously, D ij 0, whenfij L ij and D ij 0, whenfij Uij.In this way, the NLCNFP model M 4 can be changed intothe following QP modelM 5, in which the search directionin the space of the flow variables is to be solved.

minC(D) 12D

TG(f)D g(f)TD (40)

such that

AD 0 (41)

Dij 0, when fij Lij (42)

Dij 0, when fij Uij (43)

Model M 5 is a special quadratic programming modelwhich has the form of network flow. In order to enhance thecalculation speed, we present a new approach, in place of thegeneral quadratic programming algorithm, to solve themodel M 5. The main calculation steps are as follows.

III.2.1 Neglect temporarily equations (42) and (43)This means that L ij fij Uij in this case. Thus DN 0according to the definition of the corresponding non-basic

arc.From equation (41), we know that

AD [B, S,N]

DB

DS

0

PTTR

QUUS 0 (44)

From equation (44), we can obtain

DB B 1

SDS (45)

Then, the search direction in the space of flow variables Dcan be written as follows.

D B

1

SI

0

PTTRQUUSDS ZDS (46)

Substituting equation (46) into equation (40), we get

minC(D) 12(ZDS)TG(f)(ZDS) g(f)

T(ZDS) (47)

Through minimizing equation (47) to variable DS, the modelM 5 can be changed into an unconstrained problem, theoptimization solution of which can be solved from thefollowing equations.

DN 0 (48)

BDB SDS (49)

(ZT

GZ)DS ZT

g (50)



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III.2.2 Introduction of equations (42) and (43)According to equations (48)(50), DS can be solved fromequation (50), and then DB can be solved from equation (49).If DB violates the constraint equations (42) and (43), a newbasis must be sought to calculate the new search direction inthe space of flow variables. This step will not be terminateduntil DB satisfies the constraint equations (42) and (43).

III.2.3 Introduction of maximum basis in networkObviously, the general repeated calculation of DB and DS,which is similar to that of pivoting in linear programming, isnot only time-consuming, but also does not improve thevalue of the objective function. In order to speed up thecalculation, we adopt a new method to form a basis inadvance, so that DB and DS can satisfy the constraintsequations (42) and (43). Therefore, the maximum basis ina network, which consists of as many free basic arcs aspossible, is introduced in the paper.

The maximum basis in a network can be obtained bysolving the following model M 6.

max

B

ij

dijAij (51)

where

Suppose basisB is the maximum basis from equation (51), inorder to satisfy equation (39), if the flow on a free non-basicarc needs to be adjusted only the flows on the free arcs inbasis B need to be adjusted [9].

The introduction of the maximum basis indicates thedirection of flow adjustment, i.e. the change of flow is carriedout according to the maximum basis. Through selecting themaximum basis, equations (42) and (43) in model M 5 can

always be satisfied in the calculation of the search directionin the space of the flow variables. Therefore, the QP modelM 5 is equivalent to the unconstrained problem equations(48)(50).

In order to enhance the calculation speed further,equations (48)(50) can be solved by the reduced gradientmethod [10].

IV. Test examplesThe proposed economic power dispatch with N and N 1security, including the model and its algorithm, are examinedwith the IEEE 30-bus test system on an ALPHA 400 Model610 computer. The system consists of six generators, 21 loads

and 41 transmission/transformation branches. The basicnumerical data and parameters are taken from reference [3].The security constraints of transmission lines and real powerconstraints of generators used in the paper are listed inTables 1 and 2, respectively. The test results are given inTables 3 and 4, where the proposed NLCNFP economicdispatch method with N and N 1 security is identified asNLCNFP, and the other methods are identified by theirreference. All values of power in Tables 14 are in p.u.,and the base value is 100 MVA. Table 5 is the resultsshowing the effect of KVL on solution accuracy and speed.

From Tables 3 and 4, the results by the proposed NLCNFPmethod are coincident with those obtained from the conven-

tional methods. It can be observed from Tables 35 that the

Secure economic power dispatch: L. Z. Zhu et al 537

dij

1,when arc ij is a free one,

i:e: the flow in arc ij is within its bounds

0,

when arc ij is not a free one,

i:e: the flow in arc ij reaches its upper or

lower bounds

VbbbbbbbbX

Aij 1, when arc ij is in the basis B

0, when arc ij is not in the basis B

@

Table 1. Security limits of transmission lines for IEEE 30-bus system

Line P ijM Line P ijM Line P ijM

1 1.3000 15 0.6500 29 0.32002 1.3000 16 0.6500 30 0.16003 0.6500 17 0.3200 31 0.16004 1.3000 18 0.3200 32 0.16005 1.3000 19 0.3200 33 0.16006 0.6500 20 0.1600 34 0.16007 0.9000 21 0.1600 35 0.16008 0.7000 22 0.1600 36 0.65009 1.3000 23 0.1600 37 0.1600

10 0.3200 24 0.3200 38 0.160011 0.6500 25 0.3200 39 0.160012 0.3200 26 0.3200 40 0.320013 0.6500 27 0.3200 41 0.320014 0.6500 28 0.3200

Table 2. Generator data for IEEE 30-bus system

Gen. No a i b i c i PGim PGiM

PG1 37.50 200.00 0.00 0.50 2.00PG2 175.00 175.00 0.00 0.20 0.80PG5 625.00 100.00 0.00 0.15 0.50

PG8 83.40 325.00 0.00 0.10 0.35PG11 250.00 300.00 0.00 0.10 0.30PG13 250.00 300.00 0.00 0.12 0.40


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proposed approach has very small solution error, 0.0226%compared with the exact method. The solution error will beraised to 0.1860% if KVL is neglected in the proposedNLCNFP method. However, the calculation speed of theproposed approach is far faster than that of the exact method.

It means that the proposed NLCNFP method has not onlyhigh calculation precision, but also fast calculation speedcompared with the conventional methods. Therefore, theproblem obtaining the requisite speed and accuracy in thecalculation of secure economic power dispatch can beeliminated by using the proposed NLCNFP model and thecorresponding algorithm.

V. ConclusionsA new NLCNFP model of on-line economic power dispatchwith N and N 1 security, which is solved by using acombined method of QP and NFP, has been presented. Basedon the load flow equations, a new nonlinear convex networkflow model for secure economic power dispatch is set up, andthen transformed into a QP model, in which the searchdirection in the space of the flow variables is to be solved.The concept of maximum basis in a network flow graph wasintroduced so that the constrained QP model was changedinto an unconstrained QP model, which is then solved by thereduced gradient method. Resulting from the adoption ofN 1 constrained zone as well as consideration of KVLin the new NLCNFP solution method, the problem of speedand accuracy in the calculation of N and N 1 securityeconomic dispatch is solved in the article. The test results

and comparison show that the proposed model and itsalgorithm are feasible and effective.

VI. Acknowledgements

Dr Jizhong Zhu wishes to acknowledge The Royal Societyfor the award of a Royal Fellowship.

VII. References1. Zhu, J Z and Xu, G Y, A new economic power dispatch method with

security. Electric Power Systems Research, 1992, 25, 915.2. Li, W Y, An on-line economic power dispatch method with security.

Electric Pow er Systems Research, 1985, 9, 173181.3. Alsac, O and Stott, B, Optimal load flow with steady-state security.

IEEE Tra ns., PAS, 1974, 93, 745751.4. Irving, M R and Sterling, M J H, Economic dispatch of active power

with constraint relaxation. IEE Proceedings. C, 1983, 130, 172177.5. Lee, T H, Thorne, D H and Hill, E F, A transportation method for

economic dispatchingApplication and comparison. IEEE Trans.,

PAS, 1980, 99, 23722385.6. Hobson, E, Fletcher, D L and Stadlin, W O, Network flow linearprogramming techniques and their application to fuel scheduling andcontingency analysis. IEEE Trans., PAS, 1984, 103, 16841691.

7. Zhu, J Z and Xu, G Y, Approach to automatic contingency selection byreactive type performance index. IEE Proceedings. C, 1991, 138, 6568.

8. Zhu, J Z and Xu, G Y, A unified model and automatic contingencyselection algorithm for the P and Q sub-problems. Electric PowerSystems Research, 1995, 32, 101105.

9. Smith, D K Network Optimization Practice. Ellis Horwood,Chichester, UK (1982).

10. Luenberger, D G Introduction to Linear and Nonlinear Programming.Addison-wesley, Reading, MA, USA (1984).


Table 3. Results and comparison of economic dispatchwithout N 1 security

Gen. no. NLCNFP Ref. [3]

PG1 1.7595 1.7626PG2 0.4884 0.4884PG5 0.2152 0.2151PG8 0.2229 0.2215

PG11 0.1227 0.1214PG13 0.1200 0.1200Total generation 2.9286 2.9290Total active power losses 0.0946 0.0948Total generation cost ($) 802.3986 802.4000

Table 4. Results and comparison of economic dispatch with N 1 security

Gen. no. NLCNFP Ref. [1] Ref. [2] Ref. [3]

PG1 1.41270 1.40625 1.41108 1.38540PG2 0.50080 0.60638 0.58172 0.57560PG5 0.24060 0.25513 0.26183 0.24560PG8 0.35000 0.30771 0.30114 0.35000

PG11 0.20110 0.17340 0.14871 0.17930PG13 0.19910 0.16154 0.20208 0.16910Total generation 2.90434 2.91041 2.90656 2.9050Total active power losses 0.07034 0.07641 0.07256 0.0711Total generation cost ($) 813.2135 813.44 813.34 813.74Computer type ALPHA 400/610 M-340 IBM 370/168 CDC 7600CPU time (s) 0.140 0.308 0.45 14.30

Table 5. Effect of KVL on solution accuracy and speed

Method NLCNFP withoutKVL

NLCNFP withKVL

Solution error 0.1860% 0.0226%Improved accuracy 0.0 0.1634%Power loss (MW) 7.401 7.034Reduced loss (MW) 0.0 0.367

Computer type ALPHA 400/610 ALPHA 400/610CPU time (s) 0.125 0.140Added time (s) 0.0 0.015

A New Approach to Secure optimum power generation

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