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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
Bingyan SUI
Hongjie JIA
Yunfei MU
Xiaodan YU
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)
0 0.5 1.0 1.5
0.1
0.2
0.3
0.4
(a) Scenario 1 (b) Scenario 2
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
References
[1] Stott B (1974) Review of load-flow calculation methods. Proc
IEEE 62(7):916–929
[2] Hou K, Jia H, Xu X et al (2015) A continuous time Markov
chain based sequential analytical approach for composite power
system reliability assessment. IEEE Trans Power Syst
31(1):738–748
[3] Lopes JAP (2002) Integration of dispersed generation on dis-
tribution networks-impact studies. In: Proceedings of IEEE
power engineering society winter meeting, New York, USA,
27–31 January 2002, pp 323–328
[4] Dong Z, Wong KP, Meng K et al (2010) Wind power impact on
system operations and planning. In: Proceedings of IEEE power
& energy society general meeting, Providence, USA, 25–29 July
2010, 5 pp
[5] Borkowska B (1974) Probabilistic load flow. IEEE Trans Power
Appar Syst 93(3):752–759
[6] Conti S, Raiti S (2007) Probabilistic load flow using Monte
Carlo techniques for distribution networks with photovoltaic
generators. Sol Energy 81(12):1473–1481
[7] Carpinelli G, Caramia P, Varilone P (2015) Multi-linear Monte
Carlo simulation method for probabilistic load flow of distri-
bution systems with wind and photovoltaic generation systems.
Renew Energy 76:283–295
[8] Hajian M, Rosehart WD, Zareipour H (2013) Probabilistic
power flow by Monte Carlo simulation with latin supercube
sampling. IEEE Trans Power Syst 28(2):1550–1559
[9] Allan RN, Al-Shakarchi MRG (1976) Probabilistic a.c. load
flow. Proc Inst Electr Eng 123(6):531–536
[10] Allan RN, Grigg CH, Al-Shakarchi MRG (1976) Numerical
techniques in probabilistic load flow problems. Int J Numer
Methods Eng 10(4):853–860
[11] Allan RN, Dasilva AML, Burchett RC (1981) Evaluation
methods and accuracy in probabilistic load flow solutions. IEEE
Trans Power Appar Syst 100(5):2539–2546
[12] Zhang P, Lee ST (2004) Probabilistic load flow computation
using the method of combined cumulants and Gram-Charlier
expansion. IEEE Trans Power Syst 19(1):676–682
[13] Li G, Zhang X P (2009) Comparison between two probabilistic
load flow methods for reliability assessment. In: Proceedings of
IEEE power & energy society general meeting, Calgary,
Canada, 26–30 July 2009, 7 pp
[14] Dong L, Cheng W, Bao H et al (2010) Probabilistic load flow
analysis for power system containing wind farms. In: Proceed-
ings of power and energy engineering conference, Chengdu,
China, 28–31 March 2010, 4 pp
[15] Yuan Y, Zhou J, Ju P et al (2011) Probabilistic load flow
computation of a power system containing wind farms using the
method of combined cumulants and Gram-Charlier expansion.
IET Renew Power Gener 5(6):448–454
[16] Bie Z, Li G, Liu H et al (2008) Studies on voltage fluctuation in
the integration of wind power plants using probabilistic load
flow. In: Proceedings of IEEE power and energy society general
meeting: conversion and delivery of electrical energy in the 21st
century, Pittsburgh, USA, 20–24 July 2008, 7 pp
[17] Fan M, Vittal V, Heydt GT et al (2013) Probabilistic power flow
analysis with generation dispatch including photovoltaic
resources. IEEE Trans Power Syst 28(2):1797–1805
[18] Tang J, Ni F, Ponci F et al (2015) Dimension-adaptive sparse
grid interpolation for uncertainty quantification in modern
power systems: probabilistic power flow. IEEE Trans Power
Syst 31(2):907–919
[19] Ren Z, Li W, Billinton R et al (2016) Probabilistic power flow
analysis based on the stochastic response surface method. IEEE
Trans Power Syst 31(3):2307–2315
[20] Ni F, Nguyen P, Cobben JFG (2017) Basis-adaptive sparse
polynomial chaos expansion for probabilistic power flow. IEEE
Trans Power Syst 21(1):694–704
[21] Zhu X, Liu W, Zhang J (2013) Probabilistic load flow method
considering large-scale wind power integration. Proc CSEE
33(7):77–85
[22] Williams T, Crawford C (2013) Probabilistic load flow model-
ing comparing maximum entropy and Gram–Charlier proba-
bility density function reconstructions. IEEE Trans Power Syst
28(1):272–280
[23] Usaola J (2010) Probabilistic load flow with correlated wind
power injections. Electr Power Syst Res 80(5):528–536
[24] Jaynes ET (1957) Information theory and statistical mechanics.
Phys Rev 106(4):620–630
[25] Shannon CE (1948) A mathematical theory of communication.
Bell Syst Technol J 27(3):379–423
[26] Mohammad-Djafari A (1992) A MATLAB program to calculate
the maximum entropy distributions, vol 50. Springer, Dordrecht,
pp 221–233
[27] Bowden GJ, Barker PR, Shestopal VO et al (1983) The Weibull
distribution function and wind power statistics. Wind Eng
7(2):85–98
[28] Lun IYF, Lam JC (2000) A study of Weibull parameters using
long-term wind observations. Renew Energy 20(2):145–153
[29] Bian Q, Xu Q, Sun L et al (2014) Grid-connected wind power
capacity optimization based on the principle of maximum
entropy. In: Proceedings of IEEE power & energy society
general meeting, National Harbor, USA, 27–31 July 2014, 5 pp
[30] Feijoo AE, Cidras J (2000) Modeling of wind farms in the load
flow analysis. IEEE Trans Power Syst 15(1):110–115
[31] Allan RN, Al-Shakarchi MRG (1977) Linear dependence
between nodal powers in probabilistic AC load flow. Proc Inst
Electr Eng 124(6):529–534
[32] Zimmerman RD, Murillo-Sanchez CE (2015) Title of sub-
ordinate document. In: MATPOWER 5.1 user’s manual.
Available via DIALOG. http://www.pserc.cornell.edu/mat
power/manual.pdf. Accessed 20 March 2015
[33] Zimmerman RD, Murillo-Sanchez CE, Thomas RJ (2011)
MATPOWER: steady-state operations, planning, and analysis
tools for power systems research and education. IEEE Trans
Power Syst 26(1):12–19
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.
123