Chen, B.K., Chen, M.S., Shoults, R.R., Liang, C.C., 1990, ˜Hybrid Three-phase Load Flow™, IEE

Proceedings on Generation, Transmission and Distribution, Part C 137(3) 177“185.

Chen, M.S., Dillon, W.E., 1974, ˜Power System Modelling™, Proceedings of the IEE 62(7)

901“915.

Harker, B.J., Arrillaga, J., 1979, ˜3-phase a.c./d.c. Load Flow™, Proceedings of the IEE 126(12) 1275“

1281.

Hesse, M.H., 1966, ˜Circulating Currents in Parallel Untransposed Multicircuits Lines™, IEEE Trans.

Power Apparatus and Systems PAS-85(July) 802“820.

Laughton, M.A., 1968, ˜Analysis of Unbalanced Polyphase Networks by the Method of Phase Co-

ordinates™, Proceedings of the IEE 115(8) 1163“1172.

Laughton, M.A., Saleh, A.O.M., 1985, ˜Uni¬ed Phase Coordinate Load-¬‚ow and Fault Analysis of

Polyphase Networks™, Electrical Power & Energy Systems 2(4) 2805“2814.

Smith, B.C., Arrillaga, J., 1998, ˜Improved Three-phase Load Flow using Phase and Sequence

Components™, IEE Proceedings on Generation, Transmission and Distribution, Part C 145(3)

245“250.

266 THREE-PHASE POWER FLOW

Venegas, T., Fuerte-Esquivel, C.R., 2001, ˜Steady-state Modelling of an Advanced Series Compensator

for Power Flow Analysis of Electric Networks in Phase Co-ordinates™, IEEE Trans. on Power

Systems 16(4) 758“765.

Wasley, R.G., Shlash, M.A., 1974a, ˜Newton“Raphson Algorithm for 3-phase Load Flow™, Proceedings

of the IEE 121(7) 630“638.

Wasley, R.G., Shlash, M.A., 1974b, ˜Steady-state Phase-variable Model of the Synchronous Machine

for Use in 3-phase Load-¬‚ow Studies™, Proceedings of the IEE, 121(10) 1155“1164.

Zhang, X.P., Chen, H., 1994, ˜Asymmetrical Three-phase Load-¬‚ow Study based on Symmetrical

Component Theory™, IEE Proceedings on Generation, Transmission and Distribution, Part C 141(3)

248“252.

7

Optimal Power Flow

7.1 INTRODUCTION

Electric power systems have experienced continuous growth in all three sectors of the

business, namely, generation, transmission, and distribution. In the past, transmission

systems were characterised by a low degree of interconnection, hence, it was uncomplicated

to share the load among several generating units. The increase in load sizes and operational

complexity brought about by widespread interconnection of transmission systems, some

encompassing continental distances, introduced major dif¬culties into the operation of

electrical power networks. It became necessary for many electrical utilities to operate their

systems closer to the system operating capacity. It became impractical to determine

appropriate operating strategies based only on observation and the experience of the

operator. The operating philosophy had to be revised, and new concepts based on economic

considerations were adopted. Optimal power ¬‚ow (OPF) solution methods have been

developed over the years to meet this very practical requirement of power system operation

(Alsac et al., 1990; Dommel and Tinney, 1968; El-Hawary and Tsang, 1986; Happ, 1977;

Huneault and Galiana, 1991; Maria and Findlay, 1987; Monticelli and Liu, 1992; Sasson,

1969; Sasson, Viloria, and Aboytes, 1973; Sun et al., 1984; Tinney and Hart, 1967; Wood

and Wollenberg, 1984)

Optimal power ¬‚ows can be more easily understood if one thinks in terms of conventional

power ¬‚ows, where the objective is to determine the steady-state operating conditions of the

power network. Voltage magnitudes and angles at all buses of the network corresponding to

speci¬ed levels of load and generation are determined ¬rst. Power ¬‚ows throughout the

network are calculated afterwards. It is most likely that this solution, although feasible, will

not yield the most economic generating schedule or an operating point where minimum

losses are incurred. The OPF solution, in contrast, includes an objective function that is

optimised without violating the system operating constraints. These include the network

equations, loading conditions, and physical limits on active and reactive power generation.

The selection of the objective function depends on the operating philosophy of each power

system. A common objective function concerns the active power generation cost. The

economic dispatch problem is a particular case of the OPF problem (Wood and Wollenberg,

1984).

FACTS: Modelling and Simulation in Power Networks.

´ ´

Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho

# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2

268 OPTIMAL POWER FLOW

7.2 OPTIMAL POWER FLOW USING NEWTON™S METHOD

7.2.1 General Formulation

OPF solutions are carried out to determine the optimum operating state of a power network

subjected to physical and operational constraints. An objective function, which may

incorporate economic, security, or environmental aspects of the power system, is formulated

and solved using a suitable optimisation algorithm, such as Newton™s method. The

constraints are physical laws that govern power generation and transmission system

availability, the design limits of the electrical equipment, and operating strategies. This kind

of problem is usually expressed as a static, nonlinear programming problem, with the

objective function represented as a nonlinear equation and the constraints represented by

nonlinear or linear equations.

More often than not, the objective function is taken to be the cost of generation, re¬‚ecting

the economic aspects of the electrical power system (Dommel and Tinney, 1968; Maria and

Findlay, 1987; Monticelli and Liu, 1992; Sun et al., 1984). Hence, the mathematical

formulation minimises active power generation cost by suitable adjustment of the control

parameters.

The OPF problem can be formulated as follows:

Minimise f °xÞ subject to h°xÞ ¼ 0 and g°xÞ °7:1Þ

0:

In this expression, x is the vector of state variables, f °xÞ is the objective function to be

optimised, h°xÞ represents the power ¬‚ow equations, and g°xÞ consists of state variable

limits and functional operating constraints.

In general, the aim is to optimise an objective function with the solution satisfying a

number of equality and inequality constraints. Any solution point that satis¬es all the

constraints is said to be a feasible solution. A local minimum is a feasible solution point

where the objective function is minimised within a neighbourhood. The global minimum is

a local minimum with the lowest value in the complete feasible region (Bertsekas, 1982;

Luenberger, 1984).

7.2.1.1 Variables

Variables that can be adjusted in pursuit of the optimal solution are termed control variables,

such as active power generation, taps and phase angles in tap-changing and phase-shifting

transformers, respectively, and voltage magnitudes at the generator buses. The control

parameters are taken to be continuous quantities. Such a representation is well handled by the

OPF formulation and provides a suitable representation of controls with small discrete steps.

Dependent variables are those that depend on the control variables. They can take any

value, within limits, as dictated by the solution algorithm. Examples of dependent variables

are voltage phase angles at all buses, except the slack bus; voltage magnitudes at all load

buses; reactive power at all generation buses; active power generation costs; and active and

reactive power ¬‚ows (network losses) in transmission lines and transformers.

In addition to control and dependent variables, active and reactive power loads and

network topology and data form a set of ¬xed parameters that must be speci¬ed at the outset

of the study.

269

OPTIMAL POWER FLOW USING NEWTON™S METHOD

7.2.1.2 Objective function

The main aim of the OPF solution is to determine the control settings and system state

variables that optimise the value of the objective function. The selection of the objective

function should be based on careful analysis of the power system security and economy.

Arguably, power generation cost is the most popular objective function in OPF studies,

where the thermal generation unit costs are generally represented by a nonlinear, second-

order function (Luenberger, 1984; Sun et al., 1984):

X ÀÁ

ng

FT ¼ Fk Pgk ; °7:2Þ

k¼1

where Fk is the fuel cost of unit k, Pgk is the active power generated by unit k, and ng is the

number of generators in the system, including the slack generator. More speci¬cally,

ÀÁ

Fk Pgk ¼ ak þ bk Pgk þ ck P2 ; °7:3Þ

gk

where ak , bk , and ck are the cost coef¬cients of unit k.

It should be noted that it is crucial to include the slack generator contribution in the OPF

formulation, Equation (7.1), otherwise the minimisation process will dispatch all the

generating units at their minimum capacity while assigning the rest of the required

generation to the slack generator, which would be seen by the optimisation procedure as

having zero generation cost and in¬nite generation capacity.

7.2.1.3 Equality constraints

The power ¬‚ow equations provide a means for calculating the power balance that exists in

the network during steady-state operation. They must be satis¬ed, unconditionally, if a

feasible solution is to exist (Dommel and Tinney, 1968; Sun et al., 1984), otherwise the OPF

problem is said to be infeasible, with attempts being made to ¬nd a limited but still useful

solution by relaxing some of the network constraints.

The power ¬‚ow equations represent the link between the control variables and the

dependent variables,

Pk °V; Þ þ Pdk À Pgk ¼ 0; °7:4Þ

Qk °V; Þ þ Qdk À Qgk ¼ 0; °7:5Þ

where Pk and Qk are, respectively, the active and reactive power injections at bus k; Pdk and

Qdk are, respectively, the active and reactive power loads at bus k; Pgk and Qgk are,

respectively, the scheduled active and reactive power generations at bus k; V and are,

respectively, the nodal voltage magnitudes and angles.

A generic bus including generation, load, and a transmission line is shown in Figure 7.1.

It should be noted that all equality constraints in the power network are nonlinear.

However, they are incorporated in a linearised form within the OPF formulation

(Luenberger, 1984).

270 OPTIMAL POWER FLOW

Pk(V, q ) + jQk(V, q )

Vk ∠ θk

Pdk + jQdk

Pgk + jQgk

Figure 7.1 A generic bus of the electrical power network

7.2.1.4 Inequality constraints

All variables have upper and lower limits that must be satis¬ed in the optimal solution.

Constraints on control variables re¬‚ect the bounds of the operating conditions of the

equipment used for power dispatch. Arguably, limits on the generated active power and

voltage magnitude at the generating units are the most important of such bounds.

Functional constraints result from the application of limits on control variables, with

constraints on voltage magnitudes at load buses and on active and reactive power ¬‚ows in

transmission lines being the most popular:

Pmax ; k ¼ 1; . . . ; ng ; °7:6Þ

Pmin Pgk

gk gk

Qmax ; k ¼ 1; . . . ; ng ; °7:7Þ

Qmin Qgk

gk gk

Vk ; k ¼ 1; . . . ; nb ; °7:8Þ

min max

Vk Vk

where nb is the total number of buses, ng is the total number of generation buses, and

Qgk ¼ Qk °V; Þ þ Qdk : °7:9Þ

If a reactive power limit violation takes place in a generator bus, it changes to a load bus,

with associated voltage constraints.

It should be mentioned that functional constraints are normally relaxed under

system emergency conditions in order to obtain suboptimal but still technically feasible

solutions.

7.2.2 Application of Newton™s Method to Optimal Power Flow

The ¬rst step towards solving the constrained optimisation problem using Newton™s method

is to convert the problem into an unconstrained optimisation problem. This is achieved by

constructing an augmented Lagrangian function for Equation (7.1), which in generic form

may be written as:

L°x; kÞ ¼ f °xÞ þ kt h°xÞ þ c½g°xÞ; l; °7:10Þ

where k and l are Lagrange multiplier vectors for equality and inequality constraints,

respectively, c½g°xÞ; l is the penalty function of the inequality constraints, and a superscript

271

OPTIMAL POWER FLOW USING NEWTON™S METHOD

t indicates the transpose. In Equation (7.10) there are as many Lagrange multipliers as

number of active constraints. The method for handling functional inequality constraints is

addressed in Section 7.2.6.

In OPF using Newton™s method, the Lagrangian function for active and reactive power

¬‚ows is modelled as an equality constraint, given by the following equation (Luenberger,

1984):

X X

‚ Ã ‚ Ã

nb nb

Lsystem °x; kÞ ¼ FT þ lpk Pk °V; Þ þ Pdk À Pgk þ lqk Qi °V; Þ þ Qdk À Qgk ;

k¼1 k¼1

°7:11Þ

where FT is the objective function, the summations are for the nb buses speci¬ed in the

study, and kpk and kqk are the Lagrange multipliers for the active and reactive power

equations, respectively.

7.2.3 Linearised System Equations

Solution of the Lagrangian function of Equation (7.11) may be ef¬ciently achieved by

solving, by iteration, the following system of linearised equations,

! !

Áx rx

½W ¼ : °7:12Þ

Ák rk

Sometimes, it is more convenient to express the system of Equations (7.12) as follows:

WÁz ¼ Àg; °7:13Þ

where

!

Jt

H

W¼ ; °7:14Þ

J 0

Áz ¼ ½Áx Ákt ; °7:15Þ

g ¼ ½rx rkt ; °7:16Þ

rx ¼ ½rPg rh rVt ; °7:17Þ

rk ¼ ½rkp rkq t ; °7:18Þ

Áx ¼ ½ÁPg Áh ÁVt ; °7:19Þ

Ák ¼ ½Ákp Ákq t : °7:20Þ

Matrix W contains the second partial derivatives of the Lagrangian function L°x; kÞ with

respect to the state variables x and Lagrange multipliers k. Some derivative terms give rise

to the Hessian H whereas others give rise to the Jacobian J or its transposed matrix Jt .

Matrix W is symmetric and has a null submatrix, 0, at its lower right-hand corner, since the

second partial derivatives of the form q2 L°x; kÞ=qlk qlm do not exist.

272 OPTIMAL POWER FLOW

The gradient vector g is rL°x; kÞ, and the ¬rst partial derivatives of g are the second

partial derivatives of the Lagrangian function L°x; kÞ. The Lagrange multipliers are the

incremental costs for active and reactive powers, kp and kq , respectively. Áz is the vector of

correction terms. The state variables are the active power generations, the nodal voltage

magnitudes, and phase angles, Pg , V, and , respectively.

The derivative terms associated with the inequality constraints, c½g°xÞ; l, are not

included at the beginning of the iterative solution. They are incorporated into the linearised

system of Equations (7.12) only after limits become enforced; hence, the Hessian and

Jacobian terms are:

!t

q2 L°x; kÞ q2 f °xÞ q2 h° xÞ

H¼ ¼ þ k; °7:21Þ

qx2 qx2 qx2

q2 L°x; kÞ qh°xÞ

J¼ ¼ : °7:22Þ

qxqk qx

A key property of submatrices H, J, and Jt is that they all have the same sparsity structure as

the nodal admittance matrix (Wood and Wollenberg, 1984).

7.2.4 Optimality Conditions for Newton™s Method

In general, conditions for global optimality (xopt, kopt ) can be checked by assessing the

positiveness of matrix W. However, it is computationally too expensive for large-scale

problems to verify that matrix W is positive de¬nite, and this test is skipped in

most practical problems. Other optimality tests performed involve checking that the gradient

vector is zero and that the Lagrange multipliers for the binding inequalities pass their sign

test (Bertsekas, 1982; Luenberger, 1984).

In practical OPF solutions the following tests are carried out (Sun et al., 1984).

all power mismatches are within a prescribed tolerance;

the inequality constraints are satis¬ed;

the vector gradient is zero;

further reductions in the objective function are possible only if constraints are violated.

It should be emphasised that in general optimisation problems, the solution has to satisfy

a number of equality and inequality constraints. Inequality constraints are made active by

changing them into equality constraints. Hence, the general optimisation problem is to ¬nd

the optimum of a function subjected to a set of equality constraints.

7.2.5 Conventional Power Plant Modelling in Optimal Power Flow

Superposition is used to construct the linearised system of Equations (7.12) at each iterative

step. The plant components of the power system are modelled independently and their

individual entries placed in W and g. The bus number to which the plant component is

connected determines the location of the individual Hessian and Jacobian terms in the

overall W and g structures.

273

OPTIMAL POWER FLOW USING NEWTON™S METHOD

7.2.5.1 Transmission lines

The positive sequence representation of the nominal p-circuit shown in Figure 3.1 is used to

derive the transmission line power ¬‚ow equations required by the OPF formulation in a

similar manner to the procedure carried out in Section 4.2.1 for the case of a series

impedance.

The Lagrangian function associated with the power mismatch equations at buses k and m

is:

L ¼ Ltrans-line °x; kÞ ¼ pk °Pk þ Pdk À Pgk Þ þ qk °Qk þ Qdk À Qgk Þ

°7:23Þ

þ pm °Pm þ Pdm À Pgm Þ þ qm °Qm þ Qdm À Qgm Þ:

The ¬rst partial derivatives of the Lagrangian function in Equation (7.23), with respect to the

voltage magnitudes and phase angles at buses k and m, and the four Lagrange multipliers,

are used as entries in the gradient vector g. The individual entries of matrix W are the

second derivative terms of the Lagrangian function with respect to voltage magnitudes and

phase angles at buses k and m, and the four Lagrange multipliers in Equation (7.23). These

terms are given explicitly in Appendix B, Section B.1.

The contribution of a transmission line to the overall linearised system of Equations (7.12)

is:

! ! !

Ázk

Wkk Wkm g

¼À k °7:24Þ

Ázm

Wmk Wmm gm

where

2 3

q2 L q2 L qPk qQk

6 7

qk qVk qk qk 7

6 q2

6 7

k

62 7

6 qL qQk 7

q2 L qPk

6 7

6 qV q qVk 7

qVk

qVk2

¼6 k k 7; °7:25Þ

Wkk 6 7

6 qPk 7

qPk

6 07

0

6 q 7

qVk

6 7

k

6 7

4 qQk 5

qQk

0 0

qk qVk

2 3

qL qL qPm qQm

2 2

6 7

6 qk qm qk qVm qk qk 7

62 7

6 qL qQm 7

q2 L qPm

6 7

6 qVk 7;

¼ 6 qVk qm qVk qVm qVk 7 °7:26Þ

Wkm 6 qP 7

qPk

6 7

k

6 07

0

6 qm 7

qVm

6 7

4 qQk 5

qQk

0 0

qm qVm

274 OPTIMAL POWER FLOW

2 3

q2 L q2 L qPk qQk

6 qm qk qm qVk qm 7

qm

6 7

62 7

6 qL qQk 7

q2 L qPk

6 7

6 7

6 qV q qVm qVk qVm qVm 7

7; °7:27Þ

Wmk ¼ 6 m k

6 qP 7

qPm

6 07

m

6 7

0

6 qk qVk 7

6 7

4 qQ 5

qQm

m

0 0

qk qk

2 3

q2 L q2 L qPm qQm

6 q2 qm 7

qm qVm qm

6 7

6 7

m

6 q2 L qQm 7

q2 L qPm

6 7

6 qVm 7;

Wmm ¼ 6 qVm qm qVm qVm 7

2

°7:28Þ

6 7

6 qPm 7

qPm

6 07

0

6 qm 7

qVm

6 7

4 qQm 5

qQm

0 0

qm qVm

‚ Ãt

Ázk ¼ Ák ÁVk Ápk Áqk ; °7:29Þ

‚ Ãt

Ázm ¼ Ám ÁVm Ápm Áqm ; °7:30Þ

t

gk ¼ ½ rk rVk rpk rqk ; °7:31Þ

‚ Ãt

gm ¼ rm rVm rpm rqm : °7:32Þ

These terms are systematically placed in W and g to make them correspond to the locations

of buses k and m.

7.2.5.2 Shunt elements

In electrical power systems, nodal voltages are markedly affected by load variations and by

network topology changes. The voltage drops when the network operates under heavy

loading, and, conversely, when the load level is low overvoltages can arise owing to the

capacitive effect of transmission lines. Such voltage variations are not conducive to good

operation, and voltage regulation is enforced by controlling the production or absorption of

reactive power at key locations in the network. Shunt capacitors and shunt reactors are used

for such a purpose. Shunt compensators are either permanently connected to the network or

are switched on or off according to requirements (Wood and Wollenberg, 1984).

A way to include a purely reactive shunt element in the OPF formulation is shown below.

If the shunt element is connected at bus k, the Lagrangian function is given by:

L ¼ Lshunt °x; kÞ ¼ lqk °Qshunt;k Þ ¼ lqk °ÀVk Bshunt Þ: °7:33Þ

2

275

OPTIMAL POWER FLOW USING NEWTON™S METHOD

The shunt element contribution to the overall linearised system of equations is:

! ! !

ÁVk À2qk Vk Bshunt

À2qk Bshunt À2Vk Bshunt

¼À : °7:34Þ

Áqk ÀVk Bshunt

À2Vk Bshunt 2

0

7.2.5.3 Synchronous generators

In addition to providing the active power demanded by the system, synchronous generators

also control the production or absorption of reactive power, aimed at maintaining a constant

voltage magnitude at their terminals. In the OPF formulation, the active power“cost

characteristics of steam generators are included in the problem formulation whereas hydro

generators are assumed to operate at a ¬xed active power generation while contributing fully

to the production or absorption of reactive power.

The generator representation in the OPF formulation may be based on a quadratic

expression of the active power“cost characteristic. The Lagrangian function of a generator

supplying active power to a bus k is given by:

L ¼ Lgen °x; kÞ ¼ ak þ bk Pgk þ ck P2 : °7:35Þ

gk

Its contribution to the overall linearised system of equations is:

! ! !

2ck À1 ÁPgk bk þ 2ck Pgk À pk

¼À : °7:36Þ

Ápk

À1 0 0

7.2.6 Handling of Inequality Constraints

The set of equality constraints included in the Lagrangian function at any stage of the

iterative process is called the active set (Bertsekas, 1982; Luenberger, 1984). The set of

inequality constraints that are active when the optimum is reached is called the binding set,

and the optimal solution does not necessarily require all the inequality constraints to be

binding. The binding set is not known a priori, and it is the task of the optimisation

algorithm to determine it as well as to enforce it. The inequalities that become active during

the solution process are changed to equalities and included in the active set. The problem is

then to minimise the Lagrangian function for the newly updated active set.

7.2.6.1 Handling of inequality constraints on variables

The inequality constraints are handled in the OPF formulation by means of the multiplier

method, as opposed to the penalty function method (Bertsekas, 1982; Luenberger, 1984).

The inequality constraints, when made active, are changed to equality constraints. This has

the effect of a restraining force that pulls the inadmissible points back into the admissible

region. In the multiplier method, a penalty term is added to the Lagrangian function L(x,k),

thus forming an augmented Lagrangian function, given by Equation (7.10). The

minimisation of the Lagrangian function is carried out by using Newton™s method only

for the primal variables (state variables). The dual variables, l, are updated at the end of

276 OPTIMAL POWER FLOW

each global iteration. Multipliers (dual variables) are checked for limit violations, and

variables within bounded limits are ignored.

The inequality constraints used in Equation (7.10) are handled by using the following

generic form:

8

> k ½gk °xÞ À "k þ 2 ½gk °xÞ À "k ;

g2 if k þ c ½gk °xÞ À "k ! 0;

c

g g

<h i h i2 h i

ck ½gk °xÞ; k ¼ k g °xÞ À gk þ c g °xÞ À gk ; if k þ c g °xÞ À gk 0;

>

: k k k

2

0 otherwise;

°7:37Þ

"

where g and g are limits on state variables as well as functional constraints.

At a given iteration, (i þ 1), the multipliers are adjusted according to the following

criteria:

8 °iÞ ‚ Ã ‚ Ã

°iÞ

> k þ c°iÞ gk °x°iÞ Þ À "k ; if k þ c°iÞ gk °x°iÞ Þ À "k ! 0;

g g

>

< h i h i

°i þ 1Þ

¼ °iÞ þ c°iÞ gk °x°iÞ Þ À gk ; if °iÞ þ c°iÞ gk °x°iÞ Þ À g

k °7:38Þ

0;

>k

> k k

:

0 otherwise;

where 0 < c°iÞ < c°iþ1Þ .

Upon convergence, l satis¬es the optimality conditions as given by Kuhn and Tucker

(Bertsekas, 1982; Luenberger, 1984). In such a case, all the state variable increments are

smaller than a pre-speci¬ed tolerance and no limit violations occur.

The multiplier method provides an ef¬cient way to handle the binding and nonbinding

constraints. After moving a variable to one of its limits, the algorithm holds it there for as

long as it is required, otherwise the variable is freed.

Equation (7.37) satis¬es the Kuhn and Tucker conditions (Bertsekas, 1982; Luenberger,

1984):

g°xÞ ¼ 0; ! 0: °7:39Þ

For any given constraint, if the product g°xÞ is equal to zero, either is equal to zero or

g°xÞ is nonbinding; if > 0, then g°xÞ must be zero. Equation (7.37) provides a means to

indicate whether or not a constraint is binding.

At the end of each iteration, all variables are checked according to Equation (7.37) and

updated according to Equation (7.38). Equation (7.37) is used to evaluate the gradient vector

and matrix W. Hence, the ¬rst and second derivatives of Equation (7.37) are required. The

¬rst derivative is added to the gradient vector g and the second derivative to matrix W. It

should be noted that when a variable is within limits, the derivatives are null.

Successful initialisation and updating of the penalty parameter c is largely dependent on

the kind of system being solved and on experience, but the following practical conditions

should be observed:

the initial parameter c(0) should not be too large to the point that the unconstrained

minimisation becomes ill-conditioned;

the parameter c(i) should not be increased too fast to the point that the unconstrained

minimisation becomes numerically unstable;

the parameter c(i) should not be increased too slowly to the extent that the multiplier

iterations have a poor rate of convergence.

277

OPTIMAL POWER FLOW USING NEWTON™S METHOD

Experience shows that an effective evaluation of the penalty parameter is achieved by giving

c(0) a value determined by experimentation, with subsequent evaluations of c(i) based on the

following monotonic increases: c°i þ 1Þ ¼ c°iÞ , where is a scalar greater than one.

7.2.6.2 Handling of inequality constraints on functions

Arguably, the most important functional inequality constraints are those corresponding to

controllable sources of reactive power (Wood and Wollenberg, 1984). Reactive power

generator limits are checked at the end of each global iteration. It should be pointed out that

there are computational advantages gained by including explicitly the reactive power

equation of a generator in matrix W. If the generator operates within reactive power limits, a

large number in the diagonal element associated with q is used to nullify the reactive power

equation. However, the large number is removed when the functional inequality constraint

becomes activated, in order to enforce either an upper or a lower reactive power limit.

The penalty function technique may be used either to activate or to deactivate the

equations corresponding to generator buses. Quadratic penalty functions are used since they

have ¬rst and second derivatives. The form of the penalty function for the reactive power

constraint at a generator bus k is (Bertsekas, 1982; Luenberger, 1984):

1

Eqk ¼ S2 : °7:40Þ

2 qk

The ¬rst and second derivatives are:

dEqk

¼ Sqk ; °7:41Þ

dqk

d2 Eqk

¼ S; °7:42Þ

d2 qk

where S is a large, positive penalty weighting factor.

Adding the ¬rst and second derivatives of the penalty function to the elements associated

with qk in g and W deactivates the reactive power ¬‚ow equation of the generator bus k. In

such a situation, qk has a zero value. When one of the reactive power limits is violated, the

derivatives are removed from W and g, and the bus changes from being a generator bus to a

load bus. Hence, qk changes its value from zero to nonzero.

The sign in qk indicates whether or not the reactive power has returned within limits as

indicated by the criteria given in Table 7.1.

Constraints on reactive power injections

Table 7.1

qk < 0 qk > 0

Limit

Upper Add penalty term Remove penalty term

Lower Remove penalty term Add penalty term

278 OPTIMAL POWER FLOW

7.3 IMPLEMENTATION OF OPTIMAL POWER FLOW USING

NEWTON™S METHOD

The mathematical formulation for active power optimisation has been dealt with above

(Bertsekas, 1982; Luenberger, 1984). Practical aspects of computer implementation are now

presented, with three main steps identi¬ed in the ¬‚ow diagram of Figure 7.2: (1)

initialisation of control variables; (2) the outer (main) iteration loop; and (3) the inner

´

iteration loop, which corresponds to the actual Newton process (Ambriz-Perez, 1998).

The ¬rst step comprises initialisation of variables and a lossless economic dispatch (Wood

and Wollenberg, 1984). In the main iterative loop, the state variables x are checked to assess

Start

State variable initialisation

Economic dispatch based

on equal incremental

generation costs

Newton process: assembly

and solution of the linearised

system of equations

Update of multipliers and

penalty weighting factors

Stop

Are all variables

Yes

within limits?

i=i+1

No

i > imax

No Yes

Figure 7.2 Flowchart for active power optimisation

279

IMPLEMENTATION OF OPTIMAL POWER FLOW USING NEWTON™S METHOD

whether or not they are within bounds. The inequality constraints either are activated or

inactivated according to the criteria established in Equation (7.37). The multipliers and

penalty weighting factors are updated by using Equation (7.38). At a given iteration, (i), if

no change in the inequality constraint set takes place at the end of the main loop then the

optimisation process terminates.

The Newton process takes place in the inner iteration loop, a process characterised by

¬xed values of the multipliers and penalty weighting factors. The linearised system of

equations for minimising the active power generation cost is solved in this loop. Once the

linearised system of equations has been assembled then a sparsity-oriented solution is

carried out. This process is repeated until a small, prespeci¬ed, tolerance is reached.

Normally, a tight convergence criterion is adopted for the mismatch gradient vector (i.e.

1e À 12).

7.3.1 Initial Conditions in Optimal Power Flow Solutions

All the state variables and Lagrange multipliers must be given an initial value at the

beginning of the solution procedure. The initial values should be selected by following good

engineering judgement in order to ensure an acceptable rate of convergence. In this

application, nodal voltages are initialised in a way similar to that of the power ¬‚ow problem

[i.e. 1 p.u. magnitude and 0 phase angle for all buses]. This provides a suitable starting

condition. Engineering experience indicates that, for most problems, departure from the

unitary voltage magnitude and zero phase angle is not too large (i.e. 0.95 Vk 1.05, and

À 10 k 10 .

7.3.2 Active Power Schedule

A lossless economic dispatch, as opposed to a power ¬‚ow solution, is used to provide good

starting conditions for the OPF application. The equal incremental cost criterion may be

used for this purpose. Different variants of the method are available in the open literature,

but the one recommended here is to take the generator limits into consideration, since this

yields more realistic starting conditions (Wood and Wollenberg, 1984).

The Lagrangian of the lossless economic dispatch may be expressed by

!

X

ng

L ¼ Lgen °x; kÞ ¼ FT þ Pd À Pgk : °7:43Þ

k¼1

Necessary conditions to minimise active power generation cost is that the ¬rst derivative of

the cost function, with respect to each one of the variables involved, is zero and that the

balance between the generation and the demand be met:

qL dFk °Pgk Þ

¼ À pk ¼ 0; °7:44Þ

qPgk dPgk

Xng

Pgk ¼ Pd : °7:45Þ

k¼1

280 OPTIMAL POWER FLOW

Moreover, the inequality constraints given by Expression (7.6) have to be satis¬ed. If this

is not the case, an economic dispatch is carried out and the inequality constraints are

handled by means of the multiplier method.

The following set of equations is formed when Newton™s method is applied to the lossless

economic dispatch problem.

22 32 3 2 3

qL

ÁP1

d F1

ÁÁÁ À1 7

6 dP2 0 0

76 7 6 7

qPg1

61

76 ÁP2 7 6 7

6

76 7 6 7

qL

d2 F 2

6

À1 76 7 6 7

60 ÁÁÁ 0

76 7 6 7

6 qPg2

2

76 . 7 6 7

dP2

6 6 . 7 ¼ À6 7: °7:46Þ

6. . 76 . 7 .

. .. . 6 7

.

. . . . 76

6. .

. . . .7 7 6 7

6

76 7 6 7

6

76 ÁPng 7 6 7

qL

6 2

d Fng

À1 76 7 6 7

60 ÁÁÁ 54 5 4 5

0 qPng

4 2

dPng Pn g

Pd À k ¼ 1 Pgk

Á

À1 À1 Á Á Á À1 0

If only quadratic cost functions are used and no limits violations take place then the optimal

solution is reached in just one iteration. Otherwise, if generation limits are violated then

such limits are enforced and a new iteration is started.

7.3.3 Lagrange Multipliers

The Lagrange multipliers for active and reactive power ¬‚ow mismatch equations are

initialised at the p value given by the lossless economic dispatch and q equal to zero,

respectively. Experience shows that these values give rise to very robust iterative solutions.

7.3.4 Penalty Weighting Factors

There is general agreement that the multiplier method is more effective than the penalty

function method to deal with inequality constraints (Bertsekas, 1982; Luenberger, 1984).

The former is a less empirical method, but a great deal of experimentation is still needed

to select suitable values for the weighting parameter c. For instance, a value of c°0Þ ¼ 1000

is recommended for voltage magnitude constraints, whereas for active power constraints a

good value to choose is the largest quadratic coef¬cient of the cost curves multiplied by

´

1000 (Ambriz-Perez, 1998).

In subsequent iterations, the parameter c(i) is increased by a constant factor . Values of

¼ 1:3 produce reliable solutions. Larger values of may lead to ill-conditioned situations

whereas smaller values of may lead to a slow rate of convergence.

The weighting factor S in Equations (7.40)“(7.42) is a positive parameter as large as

1e þ 10. It provides an effective enforcement of the functional inequality constraints for

controllable sources of reactive power.

7.3.5 Conjugated Variables

The voltage magnitude and reactive power generation at a given bus are strongly interlinked.

If a pair of such variables is simultaneously outside limits during the solution process, only

281

IMPLEMENTATION OF OPTIMAL POWER FLOW USING NEWTON™S METHOD

one of them will be made active in the ¬rst instance. The voltage magnitude is bounded ¬rst,

that is, reactive power generation is not made active if its associated voltage magnitude is

outside limits. Likewise, if these variables are bounded and they are about to be released at

´

the end of a main iteration, only one of them will be released at the time (Ambriz-Perez,

1998).

7.3.6 An Optimal Power Flow Numerical Example

The ¬ve-bus test network (Stagg and El-Abiad, 1968) used in Section 4.3.9 to illustrate the

use of the conventional power ¬‚ow Newton“Raphson method is also used in this section to

illustrate the use of the Matlab1 OPF computer program and associated data, given in

Appendix C.

The maximum and minimum voltage magnitude limits at all buses are taken to be 0.9 p.u.

and 1.1 p.u., respectively, except at North, where the maximum limit is set at 1.5 p.u. The

cost coef¬cients of the two generating units are taken to be: a ¼ 60 $ hÀ1 ,

b ¼ 3:4 $ MWÀ1 hÀ1 , and c ¼ 0:004 $ MWÀ2 hÀ1 . The maximum and minimum generator

active power limits are set at 200 MW and 10 MW, respectively, whereas the maximum and

minimum reactive power limits are set at 300 MVAR and À 300 MVAR, respectively.

The resulting power ¬‚ows are shown in Figure 7.3, and the nodal voltages and Lagrange

multipliers at the optimum operating point are given in Table 7.2.

40 + j5

45 + j15

0.29

80.15

Lake Main

North 32.94 14.87

32.23 14.89

47.20 1.08 1.37 5.20 2.02

4.29

5.01 2.16

30.14

27.66

4.85

5.50

5.15

46.84

30.61

28.06

1.96 5.00 3.56

1.53

Elm

South 56.06 55.00

20 + j10 60 + j10

6.44

6.07

87.89 14.41

Five-bus test network and optimal power ¬‚ow results

Figure 7.3

282 OPTIMAL POWER FLOW

Nodal parameters for the ¬ve-bus system

Table 7.2

Bus

””” ”””””””””””

”””””””””””””””

Elm Main Lake South North

Voltage:

Magnitude (p.u.) 1.0726 1.0779 1.0784 1.1000 1.1096

À 4.42 À 3.85 À 3.62 À 1.31

Phase angle (deg) 0.00

lp °$ MWÀ1 hÀ1 Þ 4.2639 4.2341 4.2232 4.1032 4.0412

It can be observed from the results presented in Table 7.2 that all nodal voltages edge

towards the high side. However, they serve the purpose of the OPF solution in this example,

where limit violations take place during the iterative process and the multiplier method

handles the violations very ef¬ciently. For example, the voltage magnitude in South is

bounded at its upper limit of 1.1 p.u. at the end of the solution process. All other nodal

voltages are well within their permitted range. It should be mentioned that selection of a

more stringent voltage range (e.g. 100 Æ 6 %) poses no problem in Newton™s method.

The results in Table 7.2 also show that the largest nodal Lagrange multiplier is at Elm,

which, incidentally, is the most remote bus in the network. The nodal Lagrange multipliers

´

are closely connected with the cost of supplying nodal load (Ambriz-Perez, 1998).

It may be argued that the active power ¬‚ows shown in Figure 7.3 are not markedly

different from those given by the conventional power ¬‚ow solution, presented in the

numerical example in Section 4.3.9, except for the active and reactive power ¬‚ows in line

North“South. It was remarked in Section 4.3.9 that these power ¬‚ows were quite high and

that the line incurred high power losses: 2.5 MW and 1.12 MVAR. This is in contrast to the

values provided by the OPF solution, where the new active power ¬‚ow is 47.2 MW and the

transmission line generates reactive power. The active power loss reduces to 0.36 MW.

The powers produced by the two generators in the OPF solution are very different from

those obtained in the conventional power ¬‚ow solution. In the case of the OPF solution the

production or absorption of reactive power is an intrinsic function of the optimisation

algorithm, thus avoiding the undesirable situation that arises in the case of the conventional

power ¬‚ow solution, where one generator is set to generate a large amount of reactive power

only for the second generator to absorb slightly more than 60 % of that power. In the OPF

solution of this example, the two generators tend to share as evenly as possible the system

active power requirements because both generators have been given equal cost functions.

Table 7.3 summarises the key parameters generated by the OPF solution, such as active

power generation cost and active power loss.

Optimal power ¬‚ow solution for the ¬ve-bus

Table 7.3

system

Quantity Value

Active power generation cost ($ hÀ1 ) 747.98

Active power loss (MW) 3.05

Active power generation (MW) 168.05

Reactive power generation (MVAR) 14.71

283

LOAD TAP-CHANGING TRANSFORMER

7.4 POWER SYSTEM CONTROLLER REPRESENTATION IN

OPTIMAL POWER FLOW STUDIES

Building on the basic theory and practice of OPF using Newton™s method, covered in

Section 7.2, extensions are now made to study the representation of controllable equipment

found in electrical power networks, such as the well-established tap-changing transformer

and the new breed of power electronic controllers generically known as FACTS equipment

´

(IEEE/CIGRE, 1995).

The following controllers are studied in the remainder of the chapter: the tap changer, the

phase shifter, the static VAR compensator (SVC), the thyristor-controlled series com-

pensator (TCSC), and the uni¬ed power ¬‚ow controller (UPFC). The nature and control

characteristics of each of these controllers differ from one another, and their modelling

within the OPF solution re¬‚ects these facts; hence, they are addressed separately.

In general, an augmented Lagrangian function is established for each controller, in the

form of Equation (7.11), which serves the basis for establishing a linearised equation, in the

form of the system of Equation (7.12). The state variables of a given power system

controller are combined with the network nodal voltage magnitudes and phase angles in a

single frame of reference for a uni¬ed optimal solution using Newton™s method. The

controller state variables are adjusted automatically to satisfy speci¬ed power ¬‚ows, voltage

magnitudes, and optimality conditions, as given by Kuhn and Tucker (Bertsekas, 1982;

Wood and Wollenberg, 1984).

Once the equation has been assembled and combined with matrix W and gradient vector g

of the entire network, a sparsity-oriented solution is carried out. This process is repeated

until a small, prespeci¬ed, tolerance is reached for all the variables involved.

7.5 LOAD TAP-CHANGING TRANSFORMER

Load tap-changing (ltc) transformers regulate nodal voltage magnitude by varying auto-

matically the transformer tap ratio under load. Their representation in system application

studies is a matter of paramount importance that has received a great deal of research

attention over many years. Nowadays, the problem is well understood and a variety of ltc

´

models are available in the literature (Acha, Ambriz-Perez, and Fuerte-Esquivel, 2000). A

case in point is the simple and yet ¬‚exible power ¬‚ow ltc model derived in Section 4.4.1. We

now turn our attention to the more involved problem of load tap changer representation in

OPF studies.

7.5.1 Load Tap-changing Lagrangian Function

The nodal power equations required in this OPF application are the same as those derived in

Section 4.4.1 for the power ¬‚ow ltc model, namely, Equations (4.52)“(4.55). These

equations are used in the Lagrangian function associated with the active and reactive power

mismatches at buses k and m, which can be expressed by:

Lltc °x; kÞ ¼ pk °Pk þ Pdk À Pgk Þ þ qk °Qk þ Qdk À Qgk Þ

þ pm °Pm þ Pdm À Pgm Þ þ qm °Qm þ Qdm À Qgm Þ: °7:47Þ

284 OPTIMAL POWER FLOW

In this expression, k is the vector of Lagrange multipliers, and the state variable vector x

includes Pg , V, , and Tk. If the tapping facilities are on the secondary winding, as opposed

to the primary winding, then Um replaces Tk as state variable.

7.5.2 Linearised System of Equations

Representation of the ltc transformer in the OPF algorithm requires that matrix W be

augmented by one row and one column. Furthermore, Tk or Um becomes an extra state

variable in the OPF formulation.

Application of Newton™s method to the case when the LTC taps are on the primary

winding yields the following linearised system of equations:

2 32 3 23

Ázk

Wkk Wkm WkT gk

4 Wmk 54 Ázm 5 ¼ À4 gm 5: °7:48Þ

Wmm WmT

ÁzT

WTk WTm WTT gT

In this expression, the structure of the matrix and vector terms: Wkk, Wkm, Wmk, Wmm, Ázk,

Ázm, gk, and gm is given by Equations (7.24)“(7.32), respectively. The additional matrix

terms in Equation (7.48) re¬‚ect the contribution of Tk, the ltc state variable. These terms are

giving explicitly by the following matrix and vector terms:

!

q2 L q2 L qPk qQk

WtkT ¼ WTk ¼ ; °7:49Þ

qk qTk qVk qTk qTk qTk

!

q2 L q2 L qPm qQm

¼ WTm ¼ ; °7:50Þ

WtmT

qm qTk qVm qTk qTk qTk

!

q2 L

¼ ; °7:51Þ

WTT

qTk2

ÁzT ¼ ½ÁTk ; °7:52Þ

gT ¼ ½rTk : °7:53Þ

If the LTC taps are on the secondary winding rather than the primary winding the state

variable Um replaces Tk in Equations (7.49)“(7.53).

It should be noted that the ¬rst and second partial derivatives for the various entries in

Equation (7.48) are derived from the Lagrangian function of Equation (7.47), given in

Appendix B, Section B.1.2. The derivative terms corresponding to inequality constraints are

entered into matrix W only if limits are enforced as a result of one or more state variable

having violated limits.

If the LTC is set to control voltage magnitude at a speci¬ed value at either bus k or bus m

then WTT in Equation (7.51) is modi¬ed by adding the second derivative term of a large

quadratic penalty function. Furthermore, the ¬rst derivative term of the quadratic penalty

function is entered into the gradient element gT in Equation (7.53).

The initial values of the primary and secondary taps are set to 1. The experience gained

with the OPF using Newton™s method indicates that the algorithm is highly reliable towards

´

convergence (Ambriz-Perez, 1998).

285

LOAD TAP-CHANGING TRANSFORMER

7.5.3 Load Tap-changing Transformer Test Cases

The ¬ve-bus test network used in the numerical example in Section 7.3.6 is modi¬ed to

include LTC-1 in series with transmission line Lake“Main, and LTC-2 and LTC-3, in

parallel, connected in series with transmission line Elm“Main. Two dummy buses, namely

LakeLTC and ElmLTC, are used to connect the three LTCs. The topology of the upgraded

network is shown in Figure 7.4, where none of the three LTCs is set to maintain voltage

magnitude at a speci¬ed value. The LTC taps are assumed to be on the primary windings

and are initiated at 1 p.u. The impedances are taken to be on the secondary winding, having

zero resistance and 0.05 p.u. inductive reactance. The OPF algorithm takes four iterations to

converge.

The nodal voltages and active and reactive powers dispatched by the generators and

Lagrange multiplier at each bus are given in Table 7.4. The power ¬‚ows and tap positions as

a function of iteration number are shown in Figure 7.4 and Table 7.5, respectively. It should

be noted that the algorithm updates the taps of both parallel LTCs identically, something

expected as these two LTCs have identical parameters. Experience with the OPF algorithm

shows that Newton™s method can handle any number of parallel transformers with ease

´

(Ambriz-Perez, 1998). This applies whether or not the transformers have different

parameters or tap position limits. If an LTC hits one of its limits then the multiplier method

is used to enforce that limit (Bertsekas, 1982; Wood and Wollenberg, 1984).

40 + j5

45 + j15

0.24

80.14

North Lake Main

Tv = 1.002 p.u. 12.52

32.02 31.34 12.51

LakeLTC

1.24

48.11 1.49 1.49

5.43 3.77 4.05 1.94

26.18 31.54

4.04 3.79

5.86 4.54

4.95

47.74

ElmLTC

32.06

26.54

2.19

Tv = 1.001 p.u.

1.35

South Elm

57.05 55.95

20 + j10 60 + j10

6.21

5.96

87.91 14.55

Modi¬ed ¬ve-bus system with three load tap changers (LTCs) and the optimal power ¬‚ow

Figure 7.4

solution

286 OPTIMAL POWER FLOW

Table 7.4 Nodal voltages in the modi¬ed ¬ve-bus system with three load tap changers (LTCs)

Bus

””” ”””””””””””””””””””””””””

” ”””””””””””

LakeLTC ElmLTC Elm Main Lake South North

Voltage:

Magnitude (p.u.) 1.077 1.072 1.072 1.077 1.078 1.100 1.109

À 3.815 À 4.457 À 4.508 À 4.013 À 3.505 À 1.332

Phase angle (deg) 0.000

lp °$ MWÀ1 hÀ1 Þ 4.2247 4.2640 4.2645 4.2352 4.2222 4.1033 4.0411

Table 7.5 Load tap changer (LTC) tap positions in the

¬ve-bus system

Iteration LTC-1 LTC-2 LTC-3

0 1.000 1.000 1.000

1 1.007 1.007 1.007

2 1.001 0.998 0.998

3 1.003 1.001 1.001

4 1.002 1.001 1.001

It may be observed that this OPF solution changes little compared with the base OPF case

presented in Section 7.3.6, where no LTCs are used. This may be explained by the fact that

the solution achieved in Section 7.3.6 was already a very good solution and that the OPF is

¬xing the taps of all three LTCs to be fairly close to their nominal value of 1 p.u. (i.e. the

three LTCs are operating as conventional transformers).

7.6 PHASE-SHIFTING TRANSFORMER

The OPF implementation of the advanced transformer model derived in Section 3.3.4, with

reference to its phase-shifting capability, is addressed in this section. The OPF uses

Newton™s method as its optimisation engine, enabling an OPF phase-shifter model that is

´

both ¬‚exible and robust towards convergence (Acha, Ambriz-Perez, and Fuerte-Esquivel,

2000). It can be set to simulate a wide range of operating modes with ease. The power ¬‚ow

Equations (4.76)“(4.79) provide the starting point for the derivation of the phase-shifter

OPF formulation.

7.6.1 Lagrangian Function

The main aim of the optimisation algorithm described in this chapter is to minimise the

active power generation cost in the power system by adjusting suitable controllable

287

PHASE-SHIFTING TRANSFORMER

parameters. For a phase-shifter model with phase-shifting facilities in the primary winding,

the Lagrangian function may be expressed by:

L°x; kÞ ¼ f °Pg Þ þ kt h°Pg ; V; ; t Þ: °7:54Þ

In this expression, f (Pg ) is the objective function to be optimised; the term h(Pg , V, , t)

represents the power ¬‚ow equations; x is the vector of state variables, k is the vector of

Lagrange multipliers for equality constraints; and Pg , V, , and t are the active power

generation, voltage magnitude, voltage phase angle, and phase-shifter angle for tapping

position t, respectively. The inequality constraints, h(Pg , V, , t) < 0, are not shown in

Equation (7.54) because they are included only when variables are outside limits.

The Lagrangian function of the power ¬‚ow mismatch equations at buses k and m is incor-

porated into the OPF formulation as an equality constraint, given by the following equation:

Lkm °x; kÞ ¼ pk °Pk þ Pdk À Pgk Þ þ qk °Qk þ Qdk À Qgk Þ

þ pm °Pm þ Pdm À Pgm Þ þ qm °Qm þ Qdm À Qgm Þ: °7:55Þ

In this expression, Pdk, Pdm, Qdk, and Qdm are the active and reactive power loads at buses k

and m; Pgk, Pgm, Qgk, and Qgm are scheduled active and reactive power generations at buses

k and m; and pk, pm, qk, and qm are Lagrange multipliers for active and reactive powers

at buses k and m.

A key function of the phase-shifting transformer is to regulate the amount of active power

that ¬‚ows through it, say Pkm. In the OPF formulation this operating condition is expressed

as an equality constraint, represented by the following Lagrangian function:

Lflow °x; kÞ ¼ flow--km °Pkm À Pspecified Þ: °7:56Þ

In this expression, ¬‚ow“km is the Lagrange multiplier associated with the active power

¬‚owing from bus k to bus m; Pspeci¬ed is the required amount of active power ¬‚ow through

the phase-shifter transformer.

The overall Lagrangian function of the phase shifter, encompassing the individual

contributions, is:

Lps °x; kÞ ¼ Lkm °x; kÞ þ Lflow °x; kÞ: °7:57Þ

7.6.2 Linearised System of Equations

Representation of the phase-shifting transformer in the OPF algorithm requires that matrix

W be augmented by one row and one column, with t becoming the state variable.

Furthermore, if the phase shifter is set to control active power ¬‚ow then the dimension of

matrix W is increased further by one row and one column. Hence, for each phase shifter

involved in the OPF solution the dimension of W is increased by up to two rows and

columns, depending on operational requirements.

If the two-winding transformer has phase-shifting facilities in the primary winding, the

linearised system of equations for minimising the Lagrangian function using Newton™s

method is:

2 32 3 23

Ázk gk

Wkk Wkm Wk

4 Wmk Wmm Wm 54 Ázm 5 ¼ À4 gm 5: °7:58Þ

Áz g

Wk Wm W

288 OPTIMAL POWER FLOW

In this expression, the structure of matrix and vector terms Wkk, Wkm, Wmk, Wmm, Ázk,

Ázm, gk, and gm is given by Equations (7.24)“(7.32), respectively. The additional matrix

terms in Equation (7.58) re¬‚ect the contribution of t, the phase shifter state variable. These

22 3

terms are given explicitly by:

qL q2 L qPk qQk

6 q q qV q q q 7

6kt t7

k t t

Wk ¼ Wk ¼ 6 2 7; °7:59Þ

t

4 qL 5

qL2

0 0

qk q qVk q

2 3

q2 L q2 L qPm qQm

6 q q qt 7

qVm qt qt

6mt 7

Wtm ¼ Wm ¼6 2 7; °7:60Þ

4 qL 5

q2 L

0 0

qm q qVm q

2 3

q2 L q2 L

6 q2 qt q 7

6 7

¼6 2t 7; °7:61Þ

W

4 qL 5

0

q qt

‚ Ãt

Áz ¼ Át Á ; °7:62Þ

‚ Ãt

g ¼ rt r : °7:63Þ

If the phase-shifting mechanism is on the secondary winding rather than the primary

winding, the state variable u replaces t in Equations (7.59)“(7.63).

It should be noted that the ¬rst and second partial derivatives for the various entries in

Equation (7.58) are derived from the Lagrangian function of Equation (7.57), and given in

Appendix B, Section B.2. The derivative terms corresponding to inequality constraints are

entered into matrix W only if limits are enforced as a result of one or more state variables

having violated limits.

The procedure described by Equations (7.58)“(7.63) corresponds to a situation where the

phase shifter is set to control active power ¬‚owing from buses k to m, which is the phase-

shifter standard control mode. However, in OPF solutions the phase shifter variables are

normally adjusted automatically during the solution process in order to reach the best

operating point of the electrical power system. In such a situation, the phase shifter is not set

to control a ¬xed amount of active power ¬‚owing from buses k to m, and matrix W is

suitably modi¬ed to re¬‚ect this operating condition. This is done by adding the second

partial derivative term of a large (in¬nite), quadratic penalty function to the diagonal

location in the matrix in Equation (7.61) corresponding to the Lagrange multiplier km. The

¬rst derivative term of the function is added to the corresponding gradient element in

Equation (7.63).

The initial conditions given to all variables involved in the study impact signi¬cantly the

convergence pattern. Experience has shown that the phase-shifter model is very robust

towards convergence when the phase-shifting angle is initialised at 0 . State variables are

initialised similarly to the power ¬‚ow problem (i.e. 1 p.u. voltage magnitude and 0 voltage

angle for all buses). The Lagrange multiplier for the power ¬‚ow constraint, ¬‚ow“km, is set to

´

zero. These values enable very robust iterative solutions (Ambriz-Perez, 1998).

289

PHASE-SHIFTING TRANSFORMER

40 + j5

45 + j15

0.38

80.15

Lake Main

North 32.95 32.23 14.90

14.92

LakePS

47.20 1.28 1.66 1.40

5.48 3.66

tv = 1 p.u.

φt =’ 0.34˚ 5.02 1.95

27.68 30.12

5.95 4.43

4.95

46.82

30.59

28.09

2.41 5.01 3.77

1.09

Elm

South 56.04 54.98

20 + j10 60 + j10

6.22

5.85

87.90 14.42

Five-bus network with one phase shifter, and optimal power ¬‚ow solution

Figure 7.5

7.6.3 Phase-shifting Transformer Test Cases

The ¬ve-bus system given in Section 7.3.6 is used to illustrate the performance of the

phase-shifter model. One phase shifter is connected in series with the transmission line

Lake“Main. An additional bus, termed LakePS, is used for the purpose of incorporating

the phase shifter, as shown in Figure 7.5. Two different modes of phase-shifter operation are

considered in this test case: (1) no active power ¬‚ow regulation and (2) active power ¬‚ow

regulation at LakePS.

The phase-shifter primary and secondary windings contain no resistance, and 0.05 p.u.

inductive reactance. The phase-shifting control is assumed to be located in the primary

winding and having phase angle limits of Æ10 . For both test cases, the primary complex tap

is initialised at 1¬0 , and convergence is obtained in four iterations to a tolerance of 1e À 9.

7.6.3.1 Case 1: no active power ¬‚ow regulation

The OPF solution for the unregulated case is shown in Figure 7.5. This case enables the OPF

solution to ¬nd the optimum amount of power transfer between buses Lake and Main, which

is calculated to be 14.92 MW. This power ¬‚ow value yields minimum fuel cost and active

power system losses (i.e. 747.98 $ hÀ1 , and 3.052 MW, respectively). The voltage magni-

tudes and phase angles, active and reactive powers dispatched by the generators, and

Lagrange multipliers are given in Table 7.6.

290 OPTIMAL POWER FLOW

Table 7.6 Nodal voltages in the ¬ve-bus network with one phase shifter: case 1

(no active power ¬‚ow regulation)

Bus

”””

””””””””””” ””””””””””

” ””””””””””””””

LakePS Elm Main Lake South North

Voltage:

Magnitude (p.u.) 1.079 1.072 1.078 1.077 1.100 1.109

À 3.632 À 4.424 À 3.864 À 3.610 À 1.306

Phase angle (deg) 0.000

lp °$ MWÀ1 hÀ1 Þ 4.223 4.26 4.234 4.223 4.103 4.041

It should be noted that, in this case, the OPF solution forces the phase angle of the phase-

shifter transformer to be small, yielding a very similar power ¬‚ow distribution to that

produced when no phase shifter is used in the network, which is the case presented in

Section 7.3.6. The slight differences between the two solutions can be traced to the fact that,

in the modi¬ed network, the inductive reactance of the original transmission line Lake“

Main may be seen as having increased by approximately 10 %.

7.6.3.2 Case 2: active power ¬‚ow regulation at LakePS

Information similar to that given for case 1 is presented in Table 7.7 for the case when the

phase shifter is set to regulate active power ¬‚ow through LakePS at 25 MW.

The phase shifter is set to control active power ¬‚ow at a level different from the one that

yields an optimum solution; hence, the fuel cost and network losses are bound to increase.

The solution given by the OPF algorithm gives an active power generation cost of

748.33 $ hÀ1 , and transmission losses are 3.143 MW.

It is interesting to note that the 40 % increase in active power ¬‚ow through LakePS is

achieved with a relatively modest increase in total cost, calculated to be below 0.05 %, but

the active power loss increases more markedly, calculated to be just under 3 %. This test

case indicates that the great operational ¬‚exibility brought about by power system

controllers may come at a price. It should be remarked, however, that this is a small network

Table 7.7 Nodal voltages in the ¬ve-bus network with one phase shifter: case 2

(active power ¬‚ow regulation at LakePS)

Bus

”””

””””””””””” ””””””””””

” ””””””””””””””

LakePS Elm Main Lake South North

Voltage:

Magnitude (p.u.) 1.079 1.073 1.079 1.076 1.100 1.109

À 2.705 À 4.097 À 3.102 À 4.098 À 1.193

Phase angle (deg) 0.000

lp °$ MWÀ1 hÀ1 Þ 4.182 4.251 4.201 4.251 4.101 4.044

291

STATIC VAR COMPENSATOR

Phase-shifter angles in the ¬ve-bus test system

Table 7.8

t (deg)

””””””””””””””

Iterations Case 1 Case 2

0 0.000 0.000

À 0.325 À 1.874

1

À 0.363 À 2.122

2

À 0.346 À 2.009

3

À 0.346 À 2.010

4

Note: case 1, no active power ¬‚ow regulation; case 2, active power

¬‚ow regulation at LakePS.

and no general conclusions can be drawn for practical utility networks, but this comparative

study does indicate that copious OPF studies and trade-offs may become necessary

when dealing with large-scale power systems and a large number of power system

controllers.

The phase-shifter angles for both test cases are shown in Table 7.8, highlighting the strong

convergence characteristics of OPF using Newton™s method. Owing to the two very different

operational requirements on the phase shifter, its phase angles reach quite distinct values

(i.e. À 0.346 and À 2.01 ). The larger value corresponds to the regulated case, where a

larger amount of active power passes through the phase-shifter transformer.

7.7 STATIC VAR COMPENSATOR

This section focuses on SVC models suitable for OPF solutions using Newton™s method

´

(Ambriz-Perez, Acha, and Fuerte-Esquivel, 2000). The modelling approach taken is to

assume that the SVC acts as a continuous, variable shunt susceptance, which adjusts

automatically in order to ensure that a target nodal voltage magnitude at the SVC terminal is

met, while satisfying network constraint conditions.

Two different ˜¬‚avours™ of the SVC model are presented in this section: (1) the ¬ring-

angle model and (2) the shunt susceptance model.

A linearised SVC model suitable for OPF iterative solutions using Newton™s method is

described below. The SVC state variable is combined with the network state variables for a

´

uni¬ed, optimal solution using Newton™s method (Ambriz-Perez, 1998).

7.7.1 Lagrangian Function

The constrained optimisation problem stated in Equation (7.1) is transformed into an

unconstrained optimisation problem by forming the augmented Lagrangian function of an

SVC model expressed in the form of an adjustable shunt susceptance:

L°x; kÞ ¼ f °Pg Þ þ kt h½Pg ; V; ; B°Þ: °7:64Þ

292 OPTIMAL POWER FLOW

In this expression, f °Pg Þ is the objective function; h°Pg ; V; ; B°Þ represents the power

¬‚ow equations; x is the vector of state variables; k is the vector of Lagrange multipliers for

equality constraints; Pg , V, , and B() are the active power generation, voltage magnitude,

voltage phase angle, and SVC shunt susceptance, respectively. The inequality constraint