flujo probabilistico de potencia optimo

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992 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008

Analysis of Probabilistic Optimal Power Flow TakingAccount of the Variation of Load Power

Xue Li, Yuzeng Li, and Shaohua Zhang

AbstractThis paper presents a probabilistic optimal powerflow (POPF) algorithm taking account of the variation of loadpower. In the algorithm, system load is taken as a random vector,which allows us to consider the uncertainties and correlationsof load. By introducing the nonlinear complementarity problem(NCP) function, the KarushKuhnTucker (KKT) conditions ofPOPF system are transformed equivalently into a set of nonsmoothnonlinear algebraic equations. Based on a first-order second-mo-ment method (FOSMM), the POPF model which represents theprobabilistic distributions of solution is determined. Using thesubdifferential, the model which includes nonsmooth functionscan be solved by an inexact LevenbergMarquardt algorithm.The proposed algorithm is verified by three test systems. Results

are compared with the two-point estimate method (2PEM) andMonte Carlo simulation (MCS). The proposed method requiresless computational burden and shows good performance when noline current is at its limit.

Index TermsFirst-order second-moment method, inexactLevenbergMarquardt algorithm, nonlinear complementarityproblem, probabilistic optimal power flow, subdifferential, uncer-tainty and correlation.

I. INTRODUCTION

THE optimal power flow (OPF) has been commonly used

as an efficient tool in the power system planning and op-

erating for many years, and the models of OPF have been gen-

erally addressed as a deterministic optimization problem. How-

ever, many random disturbances or uncertain factors, such as

the variation of nodal load, the change in network configuration

and the measuring or forecasting errors of parameters and input

variables, exist in power system operation. This renders the re-

sults of deterministic OPF, at least to some extent, inaccurate,

which makes it necessary to incorporate uncertainties in OPF

modeling. Therefore, the OPF problem is transformed into the

probabilistic optimal power flow (POPF) problem.

Probabilistic optimization is becoming increasingly con-cerned with taking into account the uncertainties of some

parameters in power systems. For example, bids of market

players in electricity markets are considered uncertain to see

Manuscript received August 22, 2007; revised February 25, 2008. This workwas supported in part by Project 50377023 of the National Natural ScienceFoundation of China, in part by Project 05AZ28 of the Science and TechnologyDevelopment Foundation of Shanghai Municipal Education Committee, and inpart by Project T0103 of the Shanghai Leading Academic Discipline. Paper no.TPWRS-00591-2007.

The authors are with the Key Laboratory of Power Station AutomationTechnology, Department of Automation, Shanghai University, Shanghai, China(e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRS.2008.926437

the impact of participants behavior on electricity prices [1].

The uncertainties in system load are typically modeled as

probabilistic in the POPF problem [2][5]. The system load is

a time dependent variable, which can be better represented by

the load curves.

Since the application of probabilistic analysis to the power

system load flow study was first proposed by Borkowska in

1974 [6], many methods are proposed to account for uncertain-

ties in power systems. Monte Carlo simulation (MCS) method

[7], [8] can provide accurate results but it is computationally

more demanding. A new algorithm combining MCS and multi-

linearized load-flow equations was presented to sufficiently andefficiently evaluate all result quantities in [7]. A direct and effi-

cient approach based on the principle of statistical least square

estimation was used to analyze the effects of nodal data uncer-

tainties on all output quantities in [9]. The conventional convo-

lution technique is also time consuming in order to achieve a

reasonable level of precision. A discrete frequency domain con-

volution technique by applying fast Fourier transforms is used

to reduce the computation time in [10].

More recently, the methods, including the first-order second-

moment method (FOSMM) [3], [4], the cumulant method (CM)

[4], [5], and the two-point estimate method (2PEM) [1], [11]

have been applied to the POPF problem. The main idea behindthese methods is to use approximate formulas for calculating the

statistical moments of a random quantity that is a function of

random variables [1]. It is also pointed out that the 2PEM does

not perform well if the number of uncertain variables is too large

in large systems [1]. The CM and the FOSMM are compared in

[4] and a numerical example shows that the results using the

FOSMM exactly equal the results using the CM.

The FOSMM has less computational burden than other

methods by taking into account more initial operating states in

one numerical calculation. In [3], FOSMM was used to find

the statistical characteristics of random variables. Then the

formulated POPF model was solved by the Newtons method in

association with a combined penalty and Lagrange multiplierapproach to handle inequality constraints.

In this paper, FOSMM is employed to account for the un-

certainties of load power, which is taken as a general vector

of correlated random variables in the POPF problem. After the

original POPF model is formulated, the KarushKuhnTucker

(KKT) conditions of POPF system are transformed equivalently

into a set of nonsmooth nonlinear algebraic equations by in-

troducing nonlinear complementarity problem (NCP) function.

Then, FOSMM is used to find the mean values and standard

deviations of the random variables. Finally, using the subdiffer-

ential, the POPF model, which includes nonsmooth functions,

can be solved by an inexact LevenbergMarquardt algorithm.

0885-8950/$25.00 2008 IEEE


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LIet al.: ANALYSIS OF PROBABILISTIC OPTIMAL POWER FLOW 993

The algorithm, taking into account both the Newton and gra-

dient search directions, could improve the stability of algorithm

and be suitable for the large-scale case [12].

The paper is organized as follows: An overview of the

FOSMM is described and the POPF problem is formulated

in Section II. An inexact LevenbergMarquardt algorithm is

presented to the solution of POPF model in Section III. Theresults obtained on a five-bus system, the IEEE 30-bus system

from MATPOWER and the IEEE 118-bus system are presented

and discussed in Section IV. Conclusions are presented in

Section V.

II. PROBABILISTICOPTIMALPOWERFLOW

A. First-Order Second-Moment Method

Generally it is easy to obtain the first origin moment and

the second center moment of samples, i.e., mean and variance.

FOSMM uses exactly a first-order Taylor series approximation

to compute first-order and second-order statistical information.The method can be applied to the POPF.

The derivation of the FOSMM begins with a nonlinear system

(1)

where is the output vector, is a non-

linear vector function and is an input random vector

with mean and covariance matrix .

The general nonlinear model (1) can be linearized using a

first-order Taylor series expansion for around , i.e.,

(2)

where is the Jacobian matrix of about at the point

. Taking expectation in both sides of (2), the approximations

for the mean vector and covariance matrix of can be obtained

(3)

(4)

For the certain probabilistic distribution, for example normal

distribution, FOSMM can use the first-order and second-order

information to obtain the probabilistic characteristic of random

variables.

B. POPF Problem Formulation

The uncertainties and correlations of random load are consid-

ered in the paper, and the FOSMM is used to account for prob-

abilistic characteristics of load. The POPF can be shown as the

following stochastic nonlinear programming problem:

(5)

where is the vector ofsystem variables,

is t he o bjective f unction, represents t he p owerflow equations excluding bus injection quantities, is

a random vector that represents the nodal injection with mean

vector and covariance matrix , and rep-

resents the equipment and system inequality constraints. The

Appendix details the entire formulation. Note that when the un-

certainties of load are incorporated, the generation powers, the

voltages and the transformer ratios are also modeled as proba-

bilistic.The Lagrange function for (5) can be written as follows:

(6)

where and are the vectors of Lagrange multipliers about

equality constraints and inequality constraints, respectively. The

Lagrange multipliers of equality constraints have the same eco-

nomical significance with the spot prices.

is a vector of primal and dual variables.

The KarushKuhnTucker (KKT) condition of optimality for

(5) can be written as the following equations and complemen-

tarity conditions:

(7)

(8)

(9)

To deal with a set of complementarity conditions in (9), a

nonlinear complementarity problem (NCP) function

is introduced as follows:

(10)

The function satisfies the basic property

(11)

Using (11), the complementarity conditions (9) can be ex-

pressed as the following set of nonlinear equations:

(12)

After the NCP function is applied, (7)(9) can be equivalently

reformulated as the nonlinear system

(13)

where is a semismooth system, and

the nonsmooth points exist if [13].

Note that the uncertainties of load render all output variables

uncertain as well, i.e., the variables of vector , including primal

and dual variables, are uncertain and their statistic properties can

be described by the numerical characteristics.

The above mentioned FOSMM is employed to the POPF

problem with randomly varying node load. Then the nonlinear

system (13) can be linearized using a first-order Taylor series

expansion around the mean points , i.e.,

(14)


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where includes a set of nonsmooth equations,

is either element among B-subdifferential

of at .

The expression of in (14) can be shown in detail

(15)

Given , and are the corre-

sponding vectors of different system operating states. Mean

of the vectors is , so the following

equation can be determined:

(16)

Taking expectation in both sides of (14) and using (15) and

(16), the mean expression can be obtained

(17)

where , is the gradient

of the primal objective function at the mean , and are the

Jacobian matrices of and about , respectively.

From (14) and (17), the covariance matrix can be derived

as follows:

(18)

Using (15) and (17), can be determined as

(19)

The POPF model is reformulated by (17) and (18) with the

following relationship:

(20)

(21)

where the covariance matrix can be derived from (19), and

the statistic property of random vector can be described by the

mean vector and the covariance matrix of nodal injection

powers. is composed of the variances of all nodal powers

along the diagonal and the covariances between nodal powers

in the off diagonal positions.

The semismooth function exists in the POPF model, so

inexact LevenbergMarquardt algorithm attempts to solve the

semismooth system (20) and (21). At the solution point of POPF

problem, we are able to compute the mean and variance of each

variable as well as the covariances between all variables. The

probabilistic information is of increasing interest in market sys-tems. For example, the fluctuation of electricity price due to

many factors such as fuel costs, bidding strategies and consumer

behavior can be embodied by its probabilistic characteristics.

In [3], the success of the solution method largely depends on

the ability to find the binding inequalities efficiently, and the

method cannot work well when iterative equations are ill-condi-

tioned. Compared with the solution method in [3], the proposed

method can easily handle the inequality constraints of the POPFmodel and does not require to identify the binding constraints.

Another advantage of the proposed method is that it avoids the

ill-conditioning by using the well-conditioned iterative coeffi-

cient matrix.

III. SOLUTIONMETHOD

The inexact LevenbergMarquardt algorithm is based on the

recently developed theory for solving semismooth systems for-

mulated by the NCP. The algorithm falls into the Newton-type

method and is proven to be quadratically convergent on the so-

lution of the NCP [13]. Due to the nonsmooth function of

(17), the notion of subdifferential is introduced in this algorithm

to determine the search direction and only the approximate so-

lution of a linear system is required at each iteration of Lev-

enbergMarquardt algorithm, which makes the algorithm quite

applicable to the large-scale cases. Note that the symbol in-

dicates the Euclidean vector norm or its associated matrix norm.

The natural merit function is defined as

(22)

In spite of the nonsmoothness of , the function turns out

to be continuously differentiable. A nonmonotone line search to

minimize is performed to globalize the local method [14].

A. Computational Procedure

The procedure of the inexact LevenbergMarquardt algo-

rithm for solving the POPF problem can be summarized in the

following steps.

Step 1) Initialization: Set , , , ,

, choose a starting point , and compute

the covariance matrix by the standardized daily

operating curves.

Step 2) Stopping criterion: If , compute the

covariance matrix and stop; otherwise, go to step

3).

Step 3) Search direction calculation: Select an element

and then find a solution of

the system

(23)

where is the LevenbergMarquardt pa-

rameter and is the residual vector. Set

if the following condition (24) is not sat-

isfied:

(24)


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Step 4) Nonmonotone line search: Find the smallest

such that

(25)

where , ,

.

Step 5) Variables update: The variable is updated as

follows:

(26)

then go to step 2).

B. Specifying

The gradient of merit function is shown as

(27)

where is either element among B-subdifferential

of at . How an element of can be obtained is ex-

plained in [12]. Let be the index set.

Then, the matrix is defined by

(28)

where , ,

, and are diagonal matrices

whose th diagonal elements are given, respectively, by

if

if

(29)

if

if

(30)

where , .

IV. NUMERICALEXAMPLES

A computer program was implemented in MATLAB to solve

the POPF problem. The proposed method was tested on three

systems, i.e., a five-bus system, the IEEE 30-bus, and IEEE

118-bus systems. The tests were conducted on an advanced

micro devices (AMD) 1.80 GHz with 480 Mbytes of RAM. We

set , , , and .

In order to demonstrate accuracy and efficiency of the pro-

posed method, the comparisons with the 2PEM and MCS were

performed. In the 2PEM, every uncertain variable is replaced

with only two deterministic points placed on each side of the

corresponding mean, which enables the use of the deterministicOPF [1].With the buspower injections uncertainties considered,

TABLE ICOMPARISONS OFCOMPUTATIONALPERFORMANCE

main procedures for the 2PEM, MCS and proposed method

were listed as follows.

1) The means and standard deviations of random load and

generation powers were obtained from the nodal power op-

erating curves [15]. Then, the 2PEM was used to account

for the uncertainties and the POPF problem was solved.

2) In all cases, assuming normal distribution with the means

and standard deviations provided by (1), the random

samples were generated. Subsequently, MCS with 10 000

samples was implemented to solve the POPF problem,

repeating the process of deterministic OPF calculation.

3) The covariance matrix of nodal powers was obtained from

the random samples, and the proposed method was carried

out.

Using the results obtained from MCS method as the basis, two

performance indices, denoted as , , are used to ascertain the

performance of the proposed method. The relative errors ,

for the mean and standard deviation, respectively, are defined as

(31)

where indicates the absolute value. , and

, are the means and standard deviations for

the MCS and proposed method, respectively.

Table I shows the comparisons of computational performance

about the OPF and POPF on three test systems. Compared with

the deterministic OPF, the POPF has the slightly extra com-

puting about the covariance matrices in (21), and the solution

of the means in (20) is not different from the deterministic OPF.

Thus, POPF does not consume the more computational time.

Due to the same initial operating state applied, the OPF and

POPF have the identical iteration.

A five-bus and the IEEE 30-bus systems were used as thesample systems to verify the proposed method. Without spec-

ification, all data were taken as per-unit value and the fiducial

value was 100 MVA.

A. Five-Bus System

A five-bus system is shown in Fig. 1. Bus 5 is the slack bus.

Standardized daily operating curves shown in Figs. 2 and 3 were

used to obtain the means and standard deviations of the uncer-

tain power injections. Then the 2PEM can be executed. Using

the means and standard deviations, random samples are gener-

ated to perform the MCS. The covariance matrix of nodal

powers in (19) can be obtained from the random samples to im-plement the proposed method.


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Fig. 1. Five-bus system.

Fig. 2. Daily curves of load powers.

Fig. 3. Daily curve of G4 generation power.

Two different cases, i.e., line current limits are not reached

and are reached, are considered to analyze the influence of the

maximum line current constraints on spot prices, respectively.

Case 1: In the first case, line current limits are not reached

with a maximum current constraint of 2.5 at line 12. According

to the performance indices in (31), Table II summarizes the error

comparisons of spot prices obtained from the proposed method

and 2PEM for five-bus system. Table III shows the executiontimes of different methods.

Fig. 4. PDF of spot price at bus 1.

From Table II it can be seen that the errors in the standard

deviations obtained for the proposed method and 2PEM with

respect to the MCS results are well below 1%, whereas the errors

in the means never exceed 0.2%. The proposed method has the

roughly similar results with the 2PEM. The errors for standard

deviation obtained from the proposed method and 2PEM deviate

slightly from the results of the MCS with 10 000 samples. It

can be seen from Table III that the proposed method has less

execution time than the 2PEM and is about 800 times faster than

the MCS. The results from Tables II and III also indicate that

the proposed method and 2PEM have the similar accuracy and

require approximate computational effort.

Table II also shows that bus 2 and 4 have the same errors.Since the line 24 only includes the transformer with the reac-

tance, which leads to the same incremental transmission losses

between bus 2 and 4, and then the same spot prices. Bus 3 and

bus 5 have the similar case.

The probability density function (PDF) of spot price at bus

1, obtained from the proposed method, 2PEM and MCS, is rep-

resented and shown in Fig. 4. The solid curve is the result of

the proposed method. The dash-dotted curve is the result of the

2PEM and the dotted curve is the curve fitted to the MCS results

with a normal distribution (MCS fit). The histogram is a result

of the MCS. From Fig. 4 it can be seen that different methods

yield almost the same PDF and the proposed method gives goodresults.

Case 2: In the second case, line current limit is reached. In the

proposed method, the maximum current limit of line 12 is de-

creased to 2.036 and the current of line 12 is exactly computed

as 2.036. Thus, the limit is reached and the transmission conges-

tion occurs, which leads to the spot price of bus 1 increasing. In

the MCS, the maximum current limit of line 12 is also set as

2.036. The PDF of spot price at bus 1 obtained from MCS is

shown in Fig. 5.

It can be seenfrom Fig. 5 that the results of the MCS havetwo

spikes which are caused by 10 000 different random samples.

The first spike is taken as the results of part of samples which

give the constant spot prices without limit reached, whereas thesecond spike is generated by other samples which give the rising


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TABLE IVPERFORMANCECOMPARISONS OFDIFFERENTMETHODS FOR30-BUSSYSTEM

TABLE VEXECUTIONTIMES OFDIFFERENTMETHODS FOR30-BUSSYSTEM

TABLE VICORRELATION COEFFICIENTS

of percentages of the actual values obtained from the MCS, are

taken among the active powers corresponding to the system gen-erator buses considering parameters mean and standard de-

viation . Average relative errors are taken among the

reactive powers corresponding to the system generator buses.

Average relative errors , , are taken among the trans-

former ratios, the real and the imaginary of voltage phasors cor-

responding to all buses, respectively.

Tables IV and V show the performance comparisons of dif-

ferent methods. In this test, the statistical characteristics of all

control variables and state variables for 30-bus system obtained

from different methods are used for computing the performance

indices. It can be seen from Tables IV and V that the proposed

method has smaller errors in estimating POPF solution than the2PEM and is about 40 times faster than the 2PEM. Due to the

large number of uncertain variables, the 2PEM do not perform

well. Taking into account more initial operating states in one

numerical calculation, the proposed method is also the fastest

one among all methods with being about 500 times faster than

the MCS with 1000 samples and about 5000 times faster than

the MCS with 10 000 samples. The MCS with 1000 samples has

more computational burden although it has smaller errors. The

results from Tables IV and V also indicate that the proposed

method has a more accurate result and has less computational

effort than the 2PEM.

To obtain more accurate results, the Taylor series expansion

with higher-order items can be used to compute the covari-ances between the uncertain variables. However, the nonlin-

earity caused by higher-order Taylor series certainly will signif-

icantly increase the computational expense. How to reach the

balance between the computational precision and time will be

researched as the future work.

According to the covariance matrix in (21), the correla-

tion matrix of all output variables can be obtained. The corre-

lation coefficient , which explains the linear relation be-tween the variable and the variable , can be computed as

(32)

where is the covariance between and , and

are the variances of and , respectively. Table VI

shows the correlation coefficients between some variables.

In Table VI, and are the active and reactive powers of

bus 2, is the transformer ratio located between the bus 6 and

9, and are the real and the imaginary of voltage phasors of

the bus2, and and arethe active and reactive spotprices

of the corresponding buses . Table VI means the correlation co-

efficients between the left column variables and the upper linevariables. The correlation coefficients absolute value which is

equal or equal approximately one indicates that two variables

are well related linearly. If the correlation coefficients absolute

value is close to zero, two variables are weakly related linearly.

For example, shows that and are

correlated whereas shows that and

are badly correlated. The correlation analysis between the spot

prices and other variables is useful for the suppliers to ensure

the economical operating of the system according to the infor-

mation provided.

Complicated correlation indeed exists between nodal powers

for various reasons. For example, there is a certain degree ofdependence between a group of bus/area loads. A linear relation,

which is assumed for this statistical dependence, can simplify

the formulated model and make it easier to solve the model.

Using the results of the proposed method, the interval es-

timators for output variables can be performed. For example,

the confidence intervals for the spot prices can be found for the

certain confidence probability. The probabilistic analysis of the

spot prices can truly reflect the electric power supply cost and

the demand information, and provide the price signal for the

customers and suppliers. Consequently, it is significant in elec-

tricity market to guide the customers to adopt a reasonable elec-

tric consume pattern and optimize the plant output and the net-

work operation.

V. CONCLUSION

This paper has presented a formulation of a POPF problem

using the FOSMM to account for the uncertainties and correla-

tions of the system load. By introducing the NCP function, the

KKT conditions of POPF system were transformed equivalently

into a set of nonsmooth nonlinear algebraic equations. Using the

subdifferential, the nonsmooth functions can be solved by an in-

exact LevenbergMarquardt algorithm. The proposed method

was tested on a five-bus system, the IEEE 30-bus and IEEE

118-bus systems.

The proposed method shows good performances providedthat no transmission congestion occurs in test systems. If the


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case appears, the proposed method may not be sufficiently ac-

curate due to the exceptional spot prices caused by the conges-

tion.

Other than the 2PEM, due to taking into account more initial

operating states in one numerical calculation, the proposed

method has minor computational expense regardless of the

number of uncertain variables. The proposed method is alsocomputationally significantly faster than the MCS. Numerical

examples show the proposed method is feasible and effective.

It should be noted that the proposed method could be ex-

tended to the real size power networks, using sparsity techniques

[17], [18] storing the nodal admittance matrix and the coeffi-

cient matrix in (23) to save the memory and improve the com-

putational efficiency. In addition, other than load uncertainties,

considering the uncertainties of other parameters in power sys-

tems could be a interesting subject for future research.

APPENDIX

Considering a -bus system, the objective function of the

POPF problem is formulated as the minimization of the total

fuel cost for generation

(33)

where is the set of power generation, , and are

the generation cost coefficients. The equality constraints of the

POPF problem are the power flow equations in the rectangular

coordinates

slack (34)

where and are the active and reactive generation out-

puts, respectively, and are the active and reactive load,

and the nodal powers and are expressed as the function

of the real and the imaginary of voltage phasors and the

transformer ratios . Real and reactive power generations, ratios,

voltage amplitudes and line currents are limited due to equip-

ment and system constraints

(35)

where , and arethe set of the transformers, the system

nodes and the restricted line, respectively.

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Elect. Power Syst. Res., vol. 46, no. 8, pp. 221227, Sep. 1998.[16] R. Zimmerman and D. Gan, MATPOWER: A Matlab Power System

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Xue Li received the B.Eng. degree and the M.Eng. degree in electrical engi-neering from Zhengzhou University, Zhengzhou, China, in 2002 and 2006, re-spectively. She is currently pursuing the Ph.D. degree at Shanghai University,Shanghai, China.

Her research interests are in power market and power system analysis andoperation.

Yuzeng Li graduated from Fudan University, Shanghai, China, in 1968. Hereceived the M.S. degree from the Mathematics and Physics Department,

Academia Sinica, Shanghai, in 1982 and the Ph.D. degree in electrical engi-neering from Hong Kong Polytechnic in 1993.He is currently a Professor at Shanghai University. His research interests in-

clude game analysis for power markets, power system economics, and powersystem analysis and operation.

Shaohua Zhangreceived the B.Eng. degree from Xian Jiao Tong University,Xian, China, in 1988, the M.Eng. degree from Shanghai University of Tech-nology, Shanghai, in 1991, and the Ph.D. degree from Shanghai University in2001, all in electrical engineering.

He is currently a Professor at Shanghai University. His research interests arein power system restructuring, pricing, and reliability.

flujo probabilistico de potencia optimo

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