Maximum entropy based probabilistic load flow calculationfor power system integrated with wind power generation

Bingyan SUI1,2 , Kai HOU2, Hongjie JIA2, Yunfei MU2,

Xiaodan YU2

Abstract Distributed generation including wind turbine

(WT) and photovoltaic panel increased very fast in recent

years around the world, challenging the conventional way

of probabilistic load flow (PLF) calculation. Reliable and

efficient PLF method is required to take into account such

changing. This paper studies the maximum entropy prob-

abilistic density function reconstruction method based on

cumulant arithmetic of linearized load flow formulation,

and then develops a maximum entropy based PLF (ME-

PLF) calculation algorithm for power system integrated

with wind power generation (WPG). Comparing to tradi-

tional Gram–Charlier expansion based PLF (GC-PLF)

calculation method, the proposed ME-PLF calculation

algorithm can obtain more reliable and accurate proba-

bilistic density functions (PDFs) of bus voltages and branch

flows in various WT parameter scenarios. It can solve the

limitation of GC-PLF calculation method that mistakenly

gaining negative values in tail regions of PDFs. Linear

dependence between active and reactive power injections

of WPG can also be effectively considered by the modified

cumulant calculation framework. Accuracy and efficiency

of the proposed approach are validated with some test

systems. Uncertainties yielded by the wind speed varia-

tions, WT locations, power factor fluctuations are

considered.

Keywords Maximum entropy, Probabilistic load flow,

Probability density function, Wind power generation,

Monte Carlo simulation

1 Introduction

Load flow calculation is one of the most important tools

to analyze static characteristics of electric power systems

[1]. Traditional deterministic load flow method calculates

system states and load flows in terms of specific values of

bus injections for a determinate network configuration. As

a result, uncertainties in power systems such as generator

and load fluctuations, component faults [2] cannot be

considered. With increasing concerns on emission reduc-

tion and sustainable utilization of renewable energy, dis-

tributed generation including wind turbine (WT) and

photovoltaic panel developed very fast in recent years all

over the world, challenging the conventional power system

operation and planning [3]. Different from traditional

generations, outputs of wind power generation (WPG) are

usually not controllable due to the wind stochasticity and

intermittency in natural environment. Therefore, the inte-

gration of renewable generation increases the difficulty for

CrossCheck date: 15 November 2017

Received: 24 December 2016 / Accepted: 15 November 2017 /

Published online: 27 February 2018

� The Author(s) 2018. This article is an open access publication

& Kai HOU

hdbhyj@tju.edu.cn

Bingyan SUI

suiby@sepd.com.cn

Hongjie JIA

hjjia@tju.edu.cn

Yunfei MU

yunfeimu@tju.edu.cn

Xiaodan YU

yuxd@tju.edu.cn

1 Shanghai Electric Power Design Institute Co. Ltd., Shanghai

200025, China

2 School of Electrical and Information Engineering, Tianjin

University, Tianjin 300072, China

123

J. Mod. Power Syst. Clean Energy (2018) 6(5):1042–1054

https://doi.org/10.1007/s40565-018-0384-6

grid operators to estimate system reserve, dispatch and

manage system load flow [4].

Probabilistic load flow (PLF) method was first proposed

in 1970s to take into account the facts of unscheduled

outages, load forecast inaccuracies and measurement errors

[5]. PLF can yield the probabilistic density functions

(PDFs) and cumulative distribution functions (CDFs) of

bus voltages and branch flows. PLF results are able to

provide grid operators and planners with an explicit per-

spective on present or future system conditions, helping

them to reduce arbitrariness on decision making. The most

used technique in PLF calculation is Monte Carlo simula-

tion (MCS) [6, 7]. It actually involves repeated load flow

processes with various values of input variables yielded by

their PDFs. Based on the MCS, a Latin supercube sampling

method have been applied to improve the efficiency of

MCS [8]. However, MCS is still remarkably time con-

suming because a huge amount of simulations are required

for an accurate enough outcome. Therefore, MCS is gen-

erally not adequate in practical applications especially for

complex and bulk power system. However, due to its

simple processes and high accuracy, it is usually used as a

baseline for methods comparison.

Different from MCS, the PDF convolution technique is

an analytical PLF calculation method based on the

assumption that all input variables are independent with

each other. By applying DC load flow model [5] or lin-

earized AC load flow model [9], bus voltages and branch

flows can be represented as a linear combination of input

variables, which makes convolution procedure feasible.

However, this technique is still inefficient for real-life

power systems due to its inherent complexity [10]. Fast

Fourier transform based convolution technique [11] was

then proposed to reduce the calculation amount, but it is

still unable to effectively handle the problem.

The combined cumulant (semi-variant) algorithm and

Gram–Charlier (GC) expansion theory can be used to

convert the complicated convolution calculation into a

simple arithmetic process according to the superior prop-

erties of cumulants [12]. This method greatly enhances the

calculation speed and is able to accurately approximate the

PDF or CDF of output variables [13–17]. Based on the GC

theory, several improved approaches were employed to

address PLF analysis, such as the stochastic collocation

interpolation [18], and polynomial chaos expansion

[19, 20]. However, the GC expansion based PLF (GC-PLF)

calculation method may mistakenly obtain negative values

in tail regions of PDF, which limits system planners to

determine line security and maximum power rating

[21, 22]. Reference [21] adopted an improved C-type GC

expansion to avoid gaining negative values results. Refer-

ence [22] introduced the principle of maximum entropy

(ME) from information theory to reconstruct PDF from an

entirely different perspective according to the moments of

output variables, guaranteeing positive PDF results. Work

in [22] showed the superiority of ME to GC in both mean

results and operational limit predictions. However, the

algorithm did not consider the integration of distributed

generation.

It should be noticed that the stochastic output of WPG is

not a normal distribution and has a complex relationship

with wind speed which usually follows Weibull distribu-

tion. Therefore, WPG is completely different from tradi-

tional generation units, and may cause severe fluctuation

and skew on PDF curves of bus voltages and branch flows.

So far, most researches obtained the PDF of WPG based on

an assumption that the relationship between wind speed

and WPG is linear [14–16]. This can introduce unaccept-

able errors for the cumulant calculation of WPG. In addi-

tion, GC PDF reconstruction method is applied directly to

obtain the PDF of bus voltages and branch flows, but the

results are inaccurate in some cases. Lastly, dependence

between active and reactive power injections of WPG

caused by constant power factor control strategy is still not

well considered, which also introduces extra errors to PDF

calculations.

In this paper, a ME based PLF (ME-PLF) calculation

algorithm for power systems integrated with WPG is pro-

posed to improve the PDF accuracy of bus voltages and

branch flows. The proposed approach can also deal with

linear dependence between active and reactive power

injections of WPG by the modified cumulant calculation

framework.

The rest of the paper is organized as follows. In Sect. 2,

the ME reconstruction method along with probability the-

ories are reviewed. In Sect. 3, the mathematical models of

the basic loads and WPG, the modified cumulant calcula-

tion based on linearized load flow formulation and the

procedure of ME-PLF calculation algorithm are detailed.

Case studies are presented in Sect. 4. Finally, Sect. 5 gives

the conclusions.

2 ME reconstruction method

In this section, theoretical background including

moment and cumulant is first introduced. Then, the prin-

ciple of ME and the model of ME reconstruction method

are presented to find the most unbiased probability

distribution.

2.1 Moment and cumulant

Defining p(x) as the PDF of a continuous random vari-

able x. For a positive integer n, the function xn is integrable

with respect to p(x) over (-?, ??), the integral shown in

Maximum entropy based probabilistic load flow calculation for power system integrated with… 1043

123

(1) is called the nth raw moment of p(x) [12]. If the p(x) of

the random variable x is unknown, the raw moment an canbe gained from its historical data by statistic way, as shown

in (2).

an ¼Z þ1

�1xnpðxÞdx ð1Þ

an ¼ E xnð Þ ð2Þ

where E denotes the calculation of expectation.

Defining characteristic function /(t) as the Fourier

transform of p(x):

/ðtÞ ¼Z þ1

�1ejtxpðxÞdx ð3Þ

where j ¼ffiffiffiffiffiffiffi�1

p.

If the Nth raw moment of p(x) exists, the characteristic

function /(t) can be developed in MacLaurin’s series for

small value of t, as follows:

/ðtÞ ¼ 1þXNn¼1

ann!ðjtÞn þ OðtNÞ ð4Þ

The nth order cumulants cn are then defined by:

ln/ðtÞ ¼XNn¼1

cnn!ðjtÞn þ OðtNÞ ð5Þ

The cumulants and raw moments of a random variable

can be converted to each other [23] by:

cn ¼ an n ¼ 1

cnþ1 ¼ anþ1 �Pnk¼1

Cknakcn�kþ1 n� 2

8<: ð6Þ

If a random variable Y has a linear relationship with

other m independent random variables, e.g.,

Y = a0 ? a1X1 ? ��� ? amXm, then the cumulants of

Y can be obtained by:

cY ;n ¼a0 þ a1cX1;n þ � � � þ amcXm;n n ¼ 1

an1cX1;n þ an2cX2;n � � � þ anmcXm;n n� 2

(ð7Þ

where cY,n denotes the nth order cumulant of Y; cX1;n, cX2;n,

…, cXm;n denote the nth order cumulants of X1, X2, …, Xm,

respectively; amn denotes am to the nth power. This calcu-

lation property makes cumulants much simpler than raw

moments in theoretical deduction.

2.2 ME reconstruction method

The principle of ME [24] states that subject to the given

information, the probability distribution which best repre-

sents the current state of knowledge is the one with max-

imum Shannon entropy. It conveys an idea that the ME

estimate is maximally noncommittal with respect to

unknown information, any other estimations will lead to

biased results.

In detail, ME is a nonlinear probability density function

reconstruction method that maximizes Shannon entropy

H of the probability distribution [25] shown in (8) while at

same time satisfying the constraints shown in (9).

maxH ¼ �Z

pðxÞ ln pðxÞdx ð8ÞZ

pðxÞunðxÞdx ¼ an n ¼ 0; 1; . . .; N ð9Þ

where a0 = 1; a1, a2, …, aN are known raw moments of

p(x); u0(x) = 1; un(x) = xn, n = 1, 2, …, N.

The entropy distributions will be of the following form

through Lagrange multiplier method:

pðxÞ ¼ exp �XNn¼0

knunðxÞ !

ð10Þ

where kn is Lagrange multiplier, which can be solved by

(11), and is obtained by substituting (10) into (9).

ZunðxÞ exp �

XNn¼0

knunðxÞ !

dx ¼ an n ¼ 0; 1; . . .;N

ð11Þ

In the numerical calculation for solving (11), Newton

iteration method is used [26], and the upper and lower

limits of integral are described by interval (l-r, l?r),where l and r are the mean and standard deviation of

random variable x; is an arbitrary real number which

determines the length of integral interval. The values of land r can be obtained in cumulant calculation process,

which will be discussed in the next section. In this paper,

the value of is selected with 6 considering the efficiency

and accuracy of PLF calculation.

3 ME-PLF calculation algorithm

In this section, the probabilistic model of basic loads and

WPG injections are first introduced. Then, the linearized

load flow formulation and modified cumulant calculation

process considering linear dependence between active and

reactive power injections of WPG are elaborated. Lastly,

the calculation procedure of ME-PLF calculation algorithm

for power system integrated with WPG is presented.

3.1 Probabilistic model of power injections

In this paper, two significant uncertainties in power

systems are considered, i.e. load fluctuations and WPG

injections.

1044 Bingyan SUI et al.

123

For active load PD and reactive load QD, normal dis-

tribution is commonly used to describe their probabilistic

characteristics [5]. The PDFs of PD and QD can be for-

mulated as:

f ðPDÞ ¼1ffiffiffiffiffiffi

2pp

rPD

exp �PD � lPD

� �22r2PD

" #

f ðQDÞ ¼1ffiffiffiffiffiffi

2pp

rQD

exp �QD � lQD

� �22r2QD

" #

8>>>>><>>>>>:

ð12Þ

where lPDand r2PD

are the expectation and variance of PD;

lQDand r2QD

are the expectation and variance of QD. So,

their raw moments can be immediately obtained by (1).

For WPG, uncertainties mainly result from the inter-

mittent characteristic of wind speed, which can be descri-

bed by Weibull distribution [27]. In this paper, the PDF for

wind of speed is assumed to be a Weibull distribution [28]

as follows:

f ðtÞ ¼ k

c

tc

� �k�1

exp � tc

� �k� �ð13Þ

where t is the wind speed; k is the shape parameter; c is a

scale parameter which indicates the average wind speed.

Generally, k and c are set to be 2.0 and 8.5 [21, 29], or

other values [14, 16]. These will be further compared in

case studies.

The WPG can be calculated from the wind speed by the

wind power curve [29], which is formulated as follows:

PwðtÞ ¼0 t\tci or t[ tco

PR

t3R � t3ciðt3 � t3ciÞ tci � t\tR

PR tR � t� tco

8><>: ð14Þ

where Pw is the active power output of WT; PR is the rated

output power; tci is the cut-in wind speed; tR is the rated

wind speed; tco is the cut-out wind speed. The parameters

can be selected in terms of [29].

In this paper, the active power generation moments of

WT can be calculated from the historical data record

generated by (13) and (14). It is simpler than the mathe-

matical approach of gaining WPG distribution and com-

puting its moments.

Asynchronous generator model is common applied in

most WTs. WT usually absorbs a large amount of reactive

power to provide the excitation current for establishing the

magnetic field while generating active power. So WT can

be simplified as PQ bus and have a constant power factor

through the automatic capacitor switching [30]. The reac-

tive power of WT can be formulated as:

Qw ¼ Pw tanðdÞ ð15Þ

where tan(d) is the proportional coefficient between active

and reactive power. Obviously, there is a linear dependence

relationship between the active and reactive power injec-

tions of WPG.

3.2 Modified cumulant calculation

The original AC load flow can be formulated as:

W ¼ FðVÞZ ¼ GðVÞ

(ð16Þ

where W is the bus power injection vector; V is the bus

voltage vector; Z is the branch power flow vector; F and

G denote the power injection function and line flow func-

tion, respectively.

Expand (16) at the operating point in terms of Taylor’s

series and neglect the high order items, as follows:

DW ¼ J0DV

DZ ¼ G0DV

(ð17Þ

where J0 = qF(V0)/qV; G0 = qG(V0)/qV.Equation (17) can also be expressed as:

DX ¼ J�10 DV ¼ S0DV

DZ ¼ G0S0DV ¼ T0DV

(ð18Þ

where S0 is the inverse matrix of J0, which is called the

sensitivity matrix; T0 = G0S0 is another sensitivity matrix

[11].

So the calculation of (V, Z) can be solved in three steps,

first is to calculate the operating point (V0, Z0) through

deterministic load flow calculation, then is to calculate the

indeterminate item (DV, DZ) by PLF calculation method,

and last is to add the two parts together, as follows:

V ¼ V0 þ DV ¼ V0 þ S0DW

Z ¼ Z0 þ DZ ¼ Z0 þ T0DW

(ð19Þ

From (19), if the power system bus number is R, the

state variables of bus voltage and branch flow (i.e. vi and zi)

can be formulated as:

vi ¼ vi0 þXRr¼1

sir0Dwr

zi ¼ zi0 þXRr¼1

tir0Dwr

8>>>><>>>>:

ð20Þ

According to (20), the output variables are actual the

linear sums of the power injections of which the weight

coefficients are corresponded to the sensitivity elements

assuming all the injections are independent from each

Maximum entropy based probabilistic load flow calculation for power system integrated with… 1045

123

other. Therefore the cumulant method can be adopted to

obtain the numerical characteristics of the output

variables.

For bus i integrated with WT, some power injections

become dependent according to (15), so that (20) should be

modified to consider the linear dependence between

WPG’s active power injection Dpwi and reactive power

injection Dqwi.The power injection of bus i is corresponded to two

items Dwk and Dwm in (20), as follows:

Dwk ¼ DpDi � DpwiDwm ¼ DqDi � Dqwi

(ð21Þ

where * represents convolution calculation; DpDi and DqDiare the probabilistic part of basic active and reactive loads.

According to (7), (21) can be converted to a much

simpler calculation formulation for cumulants, as follows:

DwðnÞk ¼ DpðnÞDi þ DpðnÞwi

DwðnÞm ¼ DqðnÞDi þ DqðnÞwi

(ð22Þ

where DpðnÞDi and DqðnÞDi are the nth cumulant of basic active

and reactive loads, respectively; DpðnÞwi and DqðnÞwi are the nth

cumulant of WPG’s active and reactive power injections,

respectively; DwðnÞk and DwðnÞ

m are the nth cumulant of Dwk

and Dwm.

Considering the linear dependence relationship as

shown in (23), Dpwi and Dqwi should be put together into a

power injection group (1 ? tan(di))Dpwi, which is consid-

ered to be independent of all other independent power

injections [31]. So (22) can be modified into two forms

considering the sensitivity coefficients, as shown in (24)

and (25).

Dqwi ¼ tanðdiÞDpwi ð23Þ

DwðnÞvk ¼ DpðnÞDi þ 1þ sim0 tanðdiÞ

sik0

DpðnÞwi

DwðnÞvm ¼ DqðnÞDi

8<: ð24Þ

DwðnÞzk ¼ DpðnÞDi þ 1þ tim0 tanðdiÞ

tik0

DpðnÞwi

DwðnÞzm ¼ DwðnÞ

vm ¼ DqðnÞDi

8<: ð25Þ

Equations (24) and (25) represent the modification for

the power injections in case of computing the bus voltage viand branch flow zi, respectively. Thus the reactive power

injection of WPG is eliminated and the power injection of

WPG can be only corresponded to active power injection.

So the probabilistic power injection matrix DW can be

modified into two forms in terms of (24) and (25), as

follows:

DWv ¼ Dw1 � � � Dwvk � � � Dwvm � � � DwR½ �T ð26Þ

DWz ¼ Dw1 � � � Dwzk � � � Dwzm � � � DwR½ �T ð27Þ

For other buses integrated with WTs, the relevant

elements of DW can be modified according to (21)-(27)

similarly. The cumulants of output variables can be gained

by (7), as follows:

DVðnÞ ¼ Sn0DWðnÞv

DZðnÞ ¼ Tn0DW

ðnÞz

(ð28Þ

where DV(n) and DZ(n) are the nth order cumulants of DV

and DZ, respectively; DWðnÞv and DWðnÞ

z are the nth order

cumulants of DWv and DWz, respectively; S0n and T0

n denote

elements of S0 and T0 raised to the power n.

3.3 Procedure of ME-PLF calculation algorithm

According to (28), the cumulants of DV and DZ from the

1st to Nth order can be calculated, these cumulants can then

be converted to raw moments in terms of (6). The raw

moments are regarded as the constraints for ME recon-

struction model (8) and (9), so the PDFs of DV and DZ can

be obtained by solving (11). So far, we get the PDFs of the

probabilistic part of V and Z. Finally, add them to V0 and

Z0 (i.e. the expectations of V and Z) so as to get the PDFs

of V and Z.

The flow chart of ME-PLF calculation algorithm is

shown in Fig. 1.

4 Case studies

In this section, IEEE 118-bus test system integrated with

WPG is first studied to demonstrate the accuracy and

efficiency of the proposed method. Essential differences

between ME and GC reconstructions are analyzed in detail.

Properties of the accuracy of ME-PLF calculation algo-

rithm with different orders are also studied with different

WT locations. In addition, the impact of power factors on

PLF results is elaborated. The 1999 to 2000 winter peak

Polish 2383-bus system [32] is then introduced to confirm

the effectiveness of the proposed method in a large scale

power system. All case studies are performed with

MATLAB platform. The programs for solving ME model

are produced referring to [26], and power flow simulation

are completed by MATPOWER [33] (Intel i5-4460

3.20 GHz, 8 GB, Windows 7).

Assume that the variances of all bus active and reactive

basic loads are 10% of their expectations. The capacity of

each WT is set to be 1.5 MW, with the cut-in wind speed

tci = 5 m/s, the rated wind speed tR = 15 m/s and the cut-

out wind speed tco = 25 m/s. The proportional coefficient

1046 Bingyan SUI et al.

123

tan(d), shape parameter k and scale parameter c of Weibull

distribution for wind speed are set in three scenarios, as

shown in Table 1. As the comparative baseline, maximum

sampling number of the MCS is set to be 50000.

4.1 Test case of IEEE 118-bus system

4.1.1 Accuracy and efficiency of ME-PLF calculation

algorithm

In this case, bus 101 is integrated with 20 9 1.5 MW

WTs based on IEEE 118-bus system. Wind speed param-

eters k and c in Scenarios 1 and 2 are set to be different to

study the effects of wind power characteristics on accura-

cies of ME-PLF calculation algorithm and GC-PLF cal-

culation method. Previous works on PLF calculation

integrated with WTs presented GC effectiveness in terms

of Scenario 1 [14, 16]. However, [27] and [28] note that the

shape parameter k should be a value between 1.8 * 2.3.

Generally, it is set to be 2.0 [21, 29]. So wind speed

parameters in Scenario 2 is more practical than those in

Scenario 1. The discrete PDFs for WPG of a WT in Sce-

narios 1 and 2 are shown in Fig. 2.

The voltage magnitudes V101 and V102, the branch active

load flows P100-101 and P101-102 are calculated by GC-PLF

calculation method with the constraint order of 8th (GC-8th)

and ME-PLF calculation algorithm with the constraint

orders from 4th to 8th (ME-4th to ME-8th). In order to

compare the results, the average root mean square (ARMS)

[12] is calculated using MCS as a reference. ARMS is used

for comparing the accuracies of different calculation

methods. The value of ARMS is defined as:

ARMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPMi¼1

PLFi �MCSið Þ2s

Mð29Þ

where PLFi is the ith point’s value on the cumulative dis-

tribution curve calculated using the ME-PLF calculation

algorithm or GC-PLF calculation method; MCSi is the ith

point’s value on the CDF calculated using the MCS;

M denotes the number of points. M = 1000 is chosen to

ensure legible plots and with the same M used for all

comparative calculations.

Input data

Start

Calculate the raw moments ofbasic loads and WPG

Convert the moments tocumulants ∆W

Run the Newton-Raphsonalgorithm and obtain the

operating point (V0, Z0) and thesensitivity matrices S0, T0

Modify cumulants ∆W to ∆Wvand ∆Wz

Calculate the cumulants of ∆V and ∆Z

Convert the cumulants of ∆V and ∆Z tomoments

Obtain the PDFs of ∆V and ∆Z by ME-PLF

End

Calculate the expectationsof basic loads and WPG

Add the PDFs of ∆V and ∆Z to V0 and Z0

Fig. 1 Flow chart of proposed ME-PLF calculation algorithm

1.50 0.5 1.0

0.1

0.2

0.3

0.4

WPG (MW)

PDF

0 0.5 1.0 1.5

0.1

0.2

0.3

0.4

(a) Scenario 1 (b) Scenario 2

PDF

WPG (MW)

Fig. 2 Discrete PDF for WPG of a WT

Table 2 ARMS results comparison in Scenario 1

Method ARMS (%)

V101 V102 P100–101 P101–102

GC-8th 0.0232 0.0181 0.0276 0.0222

ME-4th 0.0234 0.0175 0.0244 0.0153

ME-5th 0.0230 0.0178 0.0208 0.0148

ME-6th 0.0230 0.0179 0.0126 0.0089

ME-7th 0.0229 0.0179 0.0115 0.0091

ME-8th 0.0229 0.0181 0.0115 0.0087

Table 1 Wind parameter scenarios

Scenario k c tan(d)

Scenario 1 3.97 10.7 - 0.30

Scenario 2 2.00 8.5 - 0.30

Scenario 3 2.00 8.5 - 0.98

Maximum entropy based probabilistic load flow calculation for power system integrated with… 1047

123

Table 2 and Table 3 show the ARMS results of V101,

V102, P100-101 and P101-102 in Scenario 1 and Scenario 2,

respectively. Figure 3 compares ME-PLF calculation

algorithm and GC-PLF calculation method for V101 and

P101-102 particularly. From Fig. 3, Tables 2 and 3, it is

found that GC-PLF calculation method and ME-PLF cal-

culation algorithm get approximate ARMS for voltage

magnitudes. GC-8th has a larger ARMS than ME-4th for

branch flows calculation in Scenario 2, while in Scenario 1

the differences are slight. ME-4th can give satisfactory

results, and an apparent decline in ARMS emerges with

ME-6th. It is worth noting that the decline of ARMS is non-

monotonic with the increase of order of ME-PLF calcula-

tion algorithm, because the cumulant calculation errors

exist resulted from the linearized AC load flow model.

Figures 4 and 5 show the PDF curves of V101 and

P101–102 in Scenario 1 and Scenario 2, respectively. GC-8th

gains negative PDF with the tail regions in branch flow

curves, and has a considerable distortion with the MCS

baseline in Scenario 2. ME-6th gives a reliable result and

ME-8th obtains a more accurate PDF curves refer to MCS.

Despite of the change of scenarios, ME-PLF calculation

algorithm always gets accurate PDF curves.

Obviously, the accuracy of GC-PLF calculation method

can be influenced by the change of scenarios, because the

two scenarios adopt different sets of wind speed parame-

ters, which makes a direct impact on wind power injection

characteristics shown in Fig. 2. GC-PLF calculation

method can give reliable calculation results in Scenario 1,

but in most cases, the wind speed is with the parameters of

Scenario 2, which will lead to significant errors by GC-PLF

calculation method. ME-PLF calculation algorithm gives

accurate PDF curves in both scenarios.

Table 4 shows the 9th to 12th cumulants relative errors of

P100-101 and P101-102 PDF curves obtained by ME-PLF

calculation algorithm and GC-PLF calculation method with

MCS baseline based on the same 1st to 8th order cumulants

constraints. The result illustrates that the PDF results

gained by ME-PLF calculation algorithm can give a better

4th 5th 6th 7th 8th0

0.03

0.06

0.09

Constraint order

ARMS

(%)

V101 in Scenario 1 with ME-PLF

P101-102 in Scenario 1 with ME-PLF

V101 in Scenario 2 with ME-PLF

P101-102 in Scenario 2 with ME-PLF

V101 in Scenario 1 with GC-PLF

P101-102 in Scenario 1 with GC-PLF

V101 in Scenario 2 with GC-PLF

P101-102 in Scenario 2 with GC-PLF

Fig. 3 Comparison of ARMS for V101 and P101–102

0.986 0.988 0.990 0.992 0.994 0.9960

100

200

300

Prob

abili

ty d

ensi

ty

-40 -35 -30

Prob

abili

ty d

ensi

ty

-29 -280.01

V101 (p.u.)(a) Bus voltage

P101-102 (MW)

(b) Branch flow

400

58.458.658.8

0.988858.2

0.9889

-45

0

0.04

0.08

0.12

0.16

-27

0.02

0.03

-25

MCS; GC-8th; ME-6th; ME-8th

0.06

0.10

0.14

0.02

-0.02

Fig. 4 Probability density curves comparison in Scenario 1

Table 3 ARMS results comparison in Scenario 2

Method ARMS (%)

V101 V102 P100–101 P101–102

GC-8th 0.0140 0.0134 0.0810 0.0654

ME-4th 0.0138 0.0129 0.0403 0.0195

ME-5th 0.0135 0.0137 0.0313 0.0165

ME-6th 0.0128 0.0133 0.0148 0.0128

ME-7th 0.0128 0.0136 0.0129 0.0101

ME-8th 0.0094 0.0132 0.0123 0.0069

1048 Bingyan SUI et al.

123

estimation of the unknown higher cumulants information

than those of GC-PLF calculation method, indicating that

ME-PLF calculation algorithm is able to get more accurate

PDF results from another point of view.

From the study, accuracy of GC-PLF calculation method

is much more susceptible than ME-PLF calculation algo-

rithm with different wind speed parameters, because the

error of GC reconstruction mainly caused by truncation,

which is substantially different from ME in dealing with

the unknown information. The truncation with GC expan-

sion means a biased estimation for the true distribution by

partly discarding the uncertainties of unknown information.

Whereas ME aims at finding an estimation that maximizes

PDF entropy which is maximally noncommittal with

regard to unknown information so as to avoid an arbitrary

reconstruction.

For time consumptions, 50000 times MCS is 242.6 s.

Both GC-8th and ME-8th methods are more efficient, with

0.96 s and 1.04 s, respectively.

4.1.2 Impacts of WT location

So far, ME-4th can obtain preferable PDF curves than

GC-8th, and ME-6th is adequate to achieve accurate PDF

curves in most situations. In the following study, buses

101, 102, and 106 are integrated with 20 9 1.5 WM,

20 9 1.5 WM and 40 9 1.5 WM WTs, respectively. The

WT parameters are formed with Scenario 2.

Table 5 shows the ARMS of GC-PLF calculation

method and ME-PLF calculation algorithm, Fig. 6 shows

the P101-102 and P100-106 PDF curves of the two methods. It

is noted that ME-4th gets a worse curve than GC-8th for

P101-102, and ME-6th obtains an approximate PDF curve

with GC-8th, both of which are inadequate to reach accu-

rate enough results comparing with MCS.

The reason for this phenomenon is that both sides of the

branch from bus 101 to bus 102 are integrated with wind

power injections, which can form a symmetry effect on the

branch flow distribution. This feature makes the skew and

higher odd orders cumulants of the branch flow distribution

sharply decreased, meanwhile rises the higher even orders

cumulants. Therefore, the PDF of branch flow is tend to be

an approximate normal distribution, which makes the GC-

0.986 0.988 0.990 0.992 0.994 0.9960

100

200

300

V101 (p.u.)

Prob

abili

ty d

ensi

ty

0.9890 0.9891

60.260.661.0

-45 -40 -35 -30 -25

0.050

0.100

0.150

0.200

P101-102 (MW)

Prob

abili

ty d

ensi

ty

(b) Branch flow

(a) Bus voltage

400

0.9892

61.4

59.8

0

-0.025

0.075

0.125

0.175

0.225

0.025

-34 -32 -300

0.02

0.04

MCS; GC-8th; ME-6th; ME-8th

Fig. 5 Probability density curves comparison in Scenario 2

Table 5 Maximum ARMS results comparison of multi-bus WTs

case

Method ARMS (%)

V101 V102 V106 P100-101 P101–102 P100–106

GC-8th 0.0142 0.0278 0.0432 0.0339 0.0201 0.0961

ME-4th 0.0159 0.0329 0.0429 0.0205 0.0560 0.0526

ME-5th 0.0146 0.0289 0.0437 0.0138 0.0409 0.0382

ME-6th 0.0158 0.0293 0.0441 0.0168 0.0224 0.0106

ME-7th 0.0158 0.0281 0.0434 0.0181 0.0132 0.0118

ME-8th 0.0160 0.0281 0.0432 0.0167 0.0120 0.0115

Table 4 Comparison of cumulant estimation error results

Scenario Order Cumulant estimation

error of P100–101 (%)

Cumulant estimation

error of P101–102 (%)

GC-PLF ME-PLF GC-PLF ME-PLF

Scenario 1 9th 5.6 0.20 0.7 3.7

Scenario 1 10th 12.9 0.02 14.4 4.4

Scenario 1 11th 19.7 0.80 11.2 6.0

Scenario 1 12th 81.7 0.60 67.2 6.7

Scenario 2 9th 9.3 15.10 15.1 10.2

Scenario 2 10th 24.0 3.10 26.1 11.7

Scenario 2 11th 26.8 4.10 28.9 14.5

Scenario 2 12th 126.3 5.90 95.9 14.6

Maximum entropy based probabilistic load flow calculation for power system integrated with… 1049

123

PLF calculation method effectual in this case. From Fig. 7,

ME-8th obtains the most accurate PDF for P101-102 com-

paring with others.

4.1.3 Dependence between wind active and reactive power

associated with wind power factor

The constant wind power factor control leads to a linear

dependence relationship between active and reactive power

injections of WPG, and the factor is usually negative. Some

studies consider the wind reactive power injection absor-

bed by compensators so that wind power integration buses

only correspond with the active power injection, which can

avoid discussing the dependence issue. This treatment is

not practical, because it cannot consider wind power factor

fluctuation and introduces extra errors to PDF calculation

results.

This case will discuss the effects of wind power factor

change on PDFs of voltages and branch flows, so it is

necessary to consider the reactive power injection. If we

regard the active and reactive injections as dependent

random variables, the PDF curves should be different from

the true ones.

Figure 8 shows the PDF curves of V101 and P101-102 in

case of Scenario 2 (20 9 1.5 WM WTs are integrated into

bus 101) considering two different assumptions to the

correlation of wind active and reactive power injections,

i.e., independent and linear dependent. The results show

-50 -45 -40 -35 -30 -25 -20

0.04

0.08

0.12

Prob

abili

ty d

ensi

ty

30 400

0.04

0.08

0.12

0.16

Prob

abili

ty d

ensi

ty

P101-102 (MW)(a)

P100-106 (MW)(b)

38 40 42 44 46 48

0.10

0.02

0.06

0.14

0

-32 -30 -28 -260.010.020.030.04

7050 60 80

00.010.02

MCS; GC-8th; ME-6th; ME-8th

Fig. 6 Probability density curves comparison of branch flows

-45 -40 -35 -30 -25 -20

0.04

0.08

0.12

Prob

abili

ty d

ensi

ty

P101-102 (MW)

-25 -24 -230

MCS; ME-6th; ME-8th

-50

0.14

0.02

0.06

0.10

0

0.004

0.008

Fig. 7 Probability density curves comparison of P101–102

0.982 0.986 0.990 0.994 0.998

100

200

300

400

Prob

abili

ty d

ensi

ty

-44 -40 -36 -32 -28 -24

0.04

0.08

0.12

0.16

0.20

Prob

abili

ty d

ensi

ty

-28.0 -27.00.01

0.02

0.03

V101 (p.u.)(a) Bus voltage

P101-102 (MW)(b) Branch flow

0

0

0.24

-28.5 -27.5 -26.5

MCS dependent; MCS independent; ME-8th

Fig. 8 Probability density curves comparison considering

dependence

1050 Bingyan SUI et al.

123

that the correlation of power injections mainly influences

the shapes of PDF curves of voltage magnitude. Figure 8

also demonstrates that ME-8th can consider the linear

dependence between active and reactive power injections

of WPG comparing with MCS baseline. So when the wind

power factor alters, the PDFs change of voltages and

branch flows can be researched through the proposed ME-

PLF calculation algorithm.

Figure 9 shows the PDF curves of V101 and P101-102 in

Scenario 3 (20 9 1.5 WM WTs are integrated into bus

101). The proportional coefficient tan(d) alters from - 0.3

to - 0.98. It can be seen that PDF shapes of voltage

magnitudes have an evident change comparing with Sce-

nario 2, while little on branch flows. ME-8th can give

reasonable PDF results in this case while GC-8th is still

inadequate to calculate the PDFs accurately.

4.2 Test case of Polish power system

A 2383-bus test case from MATPOWER platform [29]

is utilized to confirm the effectiveness of ME-PLF calcu-

lation algorithm in a large scale power system. This system

represents the 1999 to 2000 winter peak Polish power

system. It has 2383 buses, 327 generators and 2898 bran-

ches [28]. In this case, buses 2015 and 1684 are both

integrated with 25 9 1.5 MW WTs. The WT parameters

are formed with Scenario 2. PDFs of V1640, V1684, V2015,

P2015-1640, P2015-1684 and P1684-1683 are calculated by GC-

8th and ME-4th to ME-8th.

Table 6 shows the ARMS for PDFs of voltage magni-

tudes and branch flows. Figure 10 shows the PDF curves of

V2015, P2015-1640 and P2015-1684. From ARMS and PDF, it

can be seen that both ME-PLF calculation algorithm and

GC-PLF calculation method can keep a constant error level

with the rapid increase of power system scale. The char-

acteristics of PDFs for bus voltages and branch flows are

similar with the study of IEEE 118-bus case. ME-PLF

calculation algorithm still gives better results than GC-PLF

calculation method.

For time consumptions, MCS takes up a tremendous

time, with 4909 s, GC-8th and ME-8th are both efficient,

with 27.07 s and 24.12 s, respectively. In conclusion, time

consumption of ME-PLF calculation algorithm increases

slowly with the enlargement of system scale without sac-

rificing the superiority of calculation accuracy against GC-

PLF calculation method, which demonstrates the practi-

cality of ME-PLF calculation algorithm facing PLF cal-

culation of large scale power system.

Table 6 Comparison of cumulant estimation error results in Polish system

Method Cumulant estimation error (%)

V1640 V1684 V2015 P2015–1640 P2015–1684 P1684–1683

GC-8th 0.0405 0.0293 0.0334 0.0483 0.0326 0.0197

ME-4th 0.0401 0.0382 0.0327 0.0323 0.0714 0.0201

ME-5th 0.0405 0.0327 0.0334 0.0227 0.0449 0.0093

ME-6th 0.0408 0.0326 0.0338 0.0168 0.0309 0.0077

ME-7th 0.0410 0.0306 0.0341 0.0137 0.0203 0.0089

ME-8th 0.0405 0.0319 0.0334 0.0124 0.0147 0.0097

0.970 0.980 0.990 1.000

0

40

80

120

160

200

Prob

abili

ty d

ensi

ty

-45 -40 -35 -30 -25

Prob

abili

ty d

ensi

ty

V101 (p.u.)(a) Bus voltage

P101-102 (MW)

(b) Branch flow

0.975 0.9775

15

25

MCS; GC-8th; ME-6th; ME-8th

20

-20

60

100

140

180

220

0.975 0.985 0.995 1.005

0.973

-20-0.025

0.025

0.075

0.125

0.175

0.225

0.01

0.02

0.03

-32 -31 -30

Fig. 9 Probability density curves comparison in Scenario 3

Maximum entropy based probabilistic load flow calculation for power system integrated with… 1051

123

5 Conclusion

This paper proposes a ME-PLF calculation algorithm for

power system integrated with WPGs. The proposed algo-

rithm gives more reliable and accurate PDF results of bus

voltages and branch flows comparing with GC-PLF cal-

culation method in scenarios with different WT parameters.

It can also avoid gaining negative values in tail regions of

PDFs, whereas GC-PLF calculation method sometimes

mistakenly gets negative values. PLF can be analyzed

along with the changes of wind power factor by the mod-

ified cumulant calculation framework, which can take into

account linear dependence between active and reactive

power injections of WPGs. The conclusions of case studies

can be summarized as follows:

1) With the same input information, ME-PLF calculation

algorithm can obtain preferable PDFs in scenarios with

all WT parameters against GC-PLF calculation

method. It is because ME-PLF calculation algorithm

not only satisfies the given constraints, but also gives a

maximally noncommittal estimation of the unknown

information through maximizing the Shannon entropy

of PDF. So the proposed method can output better

PDFs than GC-PLF calculation method with limited

information. Besides, it can also deal with the

limitation of GC-PLF calculation method that mistak-

enly gaining negative values in tail regions of PDFs.

2) ME-6th is always more accurate than GC-8th, and is

adequate to achieve accurate PDFs in most situations.

For some special circumstances, ME-8th is requisite

for a reliable outcome. Further increase of order of

ME-PLF calculation algorithm will no longer improve

the accuracy due to cumulant calculation errors

introduced by linearized model of AC load flow.

3) Linear dependence between active and reactive power

injections of WPGs greatly affects PDF curves of

voltage magnitudes. When the proportional coefficient

tan(d) alters from -0.3 to -0.98, PDF curves of voltage

magnitudes get wider. In another word, the voltage

fluctuation becomes larger.

4) For large scale power systems, ME-PLF calculation

algorithm keeps the advantage of gaining correct and

accurate PDF against the GC-PLF calculation method.

In addition, both ME-PLF calculation algorithm and

GC-PLF calculation method consume much less time

than MCS without sacrificing accuracy. Therefore, the

proposed method is able to deal with PLF calculation

for practical systems reliably and effectively.

Acknowledgements This work was supported by National Natural

Science Foundation of China (No. 51625702, No. 51377117, No.

51677124) and National High-tech R&D Program of China (863

Program) (No. 2015AA050403).

Open Access This article is distributed under the terms of the

Creative Commons Attribution 4.0 International License (http://

creativecommons.org/licenses/by/4.0/), which permits unrestricted

use, distribution, and reproduction in any medium, provided you give

appropriate credit to the original author(s) and the source, provide a

link to the Creative Commons license, and indicate if changes were

made.

0.976 0.978 0.980 0.982 0.984 0.9860

100

200

300

400

Prob

abili

ty d

ensi

ty

0.978727.528.0

28.5

-25 -15 -5 5 15

0.02

0.04

0.06

0.08

0.10

Prob

abili

ty d

ensi

ty

-55 -45 -35 -25

0.02

0.06

0.10

Prob

abili

ty d

ensi

ty

-30 -26 -22

V2015 (p.u.)(a)

P2015-1640 (MW)(b)

P2015-1684 (MW)(c)

2 6 100.004

0.012

MCS; GC-8th; ME-6th; ME-8th

0.9786

0-0.01

0.01

0.03

0.05

0.07

0.09

0.11

25

0.020

0 4 8

-15

0.04

0.08

0.12

0.14

0

0

0.01

0.02

0.9785

Fig. 10 Probability density curves comparison in Polish system

1052 Bingyan SUI et al.

123

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Bingyan SUI received his B.S. degree in electrical engineering from

Zhejiang University, Hangzhou, China in 2014. He received his M.S.

degree in electrical engineering from Tianjin University, Tianjin,

China in 2017. He is currently a system planning engineer in

Shanghai Electric Power Design Institute Co. Ltd. His research

interests include probabilistic load flow, reliability assessment of

power system and smart grids.

Kai HOU received his Ph.D. degree in electrical engineering from

Tianjin University, Tianjin, China in 2016. He is currently a lecturer

at Tianjin University. His research interests include reliability and risk

assessments of power system, integrated energy system and smart

grids.

Hongjie JIA received his Ph.D. degree in electrical engineering in

2001 from Tianjin University, China. He became an associate

professor at Tianjin University in 2002, and was promoted as

professor in 2006. His research interests include power reliability

Maximum entropy based probabilistic load flow calculation for power system integrated with… 1053

123

assessment, stability analysis and control, distribution network

planning and automation and smart grids.

Yunfei MU received his B.S., M.S. and Ph.D. degrees in electrical

engineering from Tianjin University, China, in 2007, 2009, and 2012

respectively. He is an associated professor of Tianjin University. His

research interests include power system stability analysis and control,

renewable energy integration and electrical vehicles.

Xiaodan YU received her B.S., M.S. and Ph.D. degrees in the

electrical engineering from Tianjin University, Tianjin, China. She is

currently an associate professor at Tianjin University. Her research

interests include power system stability, integrated energy system,

electric circuit theory and smart grids.

1054 Bingyan SUI et al.

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