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ORIGINAL RESEARCH Open Access
Analysis of voltage stability uncertaintyusing stochastic
response surface methodrelated to wind farm correlationZhaoxing
Ma1* , Hao Chen2 and Yanli Chai3
Abstract
Wind speed follows the Weibull probability distribution and wind
power can have a significant influence on power systemvoltage
stability. In order to research the influence of wind plant
correlation on power system voltage stability, in this paper,the
stochastic response surface method (SRSM) is applied to voltage
stability analysis to establish the polynomial relationshipbetween
the random input and the output response. The Kendall rank
correlation coefficient is selected to measure thecorrelation
between wind farms, and the joint probability distribution of wind
farms is calculated by Copula function. Adynamic system that
includes system node voltages is established. The composite matrix
spectral radius of the dynamicsystem is used as the output of the
SRSM, whereas the wind speed is used as the input based on wind
farm correlation. Theproposed method is compared with the
traditional Monte Carlo (MC) method, and the effectiveness and
accuracy of theproposed approach is verified using the IEEE 24-bus
system and the EPRI 36-bus system. The simulation results also
indicatethat the consideration of wind farm correlation can more
accurately reflect the system stability.
Keywords: Power system, SRSM, Correlation, Wind farm
1 IntroductionAround the world, power systems have witnessed
increasedamount of renewable and dispersed generation,
especiallywind power and solar power. Renewable energy is a
usefulsupplement to traditional energy sources, but is
differentfrom the traditional form of energy because of its
uncer-tainty and intermittency. The renewable energy is con-nected
to power grid by concentrated form or distributedform, bringing
many uncertainties to power system voltagestability as well as new
problems and challenges to re-searchers. If system voltage
stability is evaluated in the mostsevere working model for
studying, the results are often tooconservative, and therefore, a
new effective way should beinvestigated. This paper propose a
method to investigatethe impact of stochastic uncertainty of
grid-connected windpower generation on power system voltage
stability bystructure dynamic systems that include node
voltages.The impact of stochastic power injections on power
flows
and voltage profile is a widely studied topic since the
1970s
[1]. The probabilistic analysis was firstly introduced
intostudying power system small signal stability by Burchettand
Heydt in [2]. A series of work later on [3–6] have fur-ther
improved the various aspects of the analytical methodof power
system probabilistic small signal stability. In [7, 8],a method of
probabilistic analysis was proposed to directlycalculate the
probabilistic density function of critical eigen-values of a large
scale power system from the probabilisticdensity function of gird
connected multiple sources of windpower generation to investigate
the impact of stochastic un-certainty of grid-connected wind
generation on power sys-tem small-signal stability [9]. Reference
[10] presented acomparative analysis of the performance of three
efficientestimation methods when applied to the probabilistic
as-sessment of small-disturbance stability of uncertain
powersystems. In [11], an analytical approach was proposed to
in-volve the effects of correlation of wind farms in probabilis-tic
analytical multi-state models of wind farms outputgeneration.
Reliability models of wind farms consideringwind speed correlation
are proposed in [12].In this paper, power system voltage stability
is analyzed
using the stochastic response surface method (SRSM).The
algebraic equations that contain the voltages are
* Correspondence: [email protected] of Automation
Engineering, Qingdao University of Technology,Qingdao 266033,
ChinaFull list of author information is available at the end of the
article
Protection and Control ofModern Power Systems
© The Author(s). 2017 Open Access This article is distributed
under the terms of the Creative Commons Attribution
4.0International License
(http://creativecommons.org/licenses/by/4.0/), which permits
unrestricted use, distribution, andreproduction in any medium,
provided you give appropriate credit to the original author(s) and
the source, provide a link tothe Creative Commons license, and
indicate if changes were made.
Ma et al. Protection and Control of Modern Power Systems (2017)
2:20 DOI 10.1186/s41601-017-0051-3
http://crossmark.crossref.org/dialog/?doi=10.1186/s41601-017-0051-3&domain=pdfhttp://orcid.org/0000-0003-4833-6630mailto:[email protected]://creativecommons.org/licenses/by/4.0/
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converted into a differential system with system nodevoltages.
Then, the output of the SRSM is the output ofthe composite matrix
spectral radius of the dynamicsystem is constructed, and is used to
judge the stabilityof the system voltage. The IEEE 24-node system
and theEPRI 36-node system with wind power are considered astwo
examples to verify the accuracy of the proposedanalysis method.
2 Discussion2.1 System model analysisIn power system stability
analysis, a power system ischaracterized by the set of nonlinear
dynamic equationsas:
_x ¼ f x; y; τð Þ ð1Þ
0 ¼ g x; y; τð Þ ð2Þ
where f and g express the system dynamic equations andthe
algebraic equations, respectively. x represents thestate variables,
y represents the algebraic variables of thenode voltage magnitudes
and angles, and τ is the controlparameters.Neglecting the
resistances, the algebraic equations that
consist of the algebraic variables of the node voltagemagnitudes
and angles can be shown as:
PLi þXmþnj¼1
BijV iV j sin αi−αj� � ¼ ypi ¼ 0 ð3Þ
−QLi þXmþnj¼1
BijV iV j cos αi−αj� � ¼ yQi ¼ 0 ð4Þ
i ¼ nþ 1;⋯; nþm:
where PLi and QLi are the ith node active and reactivepower,
respectively. Vi and Vj are the voltage of the ithnode and the jth
generator bus, respectively. Bij repre-sents the reactance of the
admittance matrix, and αi isthe ith bus phase angle. If 1 ≤ j ≤ n,
then αj = δj, where δjis the generator rotor angle of the jth
machine. n is thenumber of generator buses and m is the number of
loadbuses. PLi and QLi are functions of Vi and αi.Equations (3 and
4) represents a pure dynamic system
and taking derivative of (3, 4) leads to the dynamicquantity _V
i and _αi [12, 13] as:
∂yPi�
∂t¼∂yPi
�∂V
_V þ∂yPi�
∂α_αþ
∂yPi�
∂δ_δ¼0
ð5Þ
∂yQi�
∂t¼∂yQi
�∂V
_V þ∂yQi
�∂α
_αþ∂yQi
�∂δ
_δ¼0
ð6Þi ¼ nþ 1;⋯; nþm:
It is more convenient to represent the generatordynamic
equations as follows:
_δ i ¼ ωi ð7Þ
Mi _ωi ¼ Pmi− MiMT PCOI−Bi;iþnEgiV iþn sin δi−ψiþn� �
ð8Þ
where δi ¼ δ i−δ0 , ωi ¼ ωi−ω0 , ψi ¼ ψ i−δ0 , MT
¼Xni¼1
Mi , δ0 ¼ 1MTXni¼1
Miδ i , ω0 ¼ 1MTXni¼1
Miωi , PCOI
¼Xni¼1
Pmi−XnþNi¼nþ1
PLi . PLi is the active load at each node
and Pmi is the input mechanical power. δ i and ωi arethe rotor
angle and angular speed of the ith machine,respectively. δ0 and ω0
are the centers of angle andangular speed, respectively. Mi and Egi
are the ithmachine inertia and internal voltage, respectively.
Brepresents the reactance of the admittance matrix. nis the number
of generators, Vi+n and ψ iþn are thegenerator bus voltage and
phase angle, respectively. Nis the number of non-generator buses in
the powersystem.From (5, 6), there is
∂yPi�
∂V_V þ
∂yPi�
∂α_α¼−
∂yPi�
∂δ_δ
ð9Þ
∂yQi�
∂V_V þ
∂yQi�
∂α_α¼−
∂yQi�
∂δ_δ
ð10ÞFrom (9, 10), it yields
A τð Þ B τð ÞC τð Þ D τð Þ
� �_V_α
� �¼ − E τð Þ _δ
F τð Þ _δ:� �
ð11Þ
where, A τð Þ ¼∂yP�
∂V , B τð Þ ¼∂yP�
∂α , C τð Þ ¼∂yQ�
∂V ,
D τð Þ ¼∂yQ�
∂α , E τð Þ ¼∂yP�
∂δ and F τð Þ ¼∂yQ�
∂δ . De-fine τ = (δ′, ω′, V′, α′).
Ma et al. Protection and Control of Modern Power Systems (2017)
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Substituting (7, 8) into (11), and solving the dynamicquantity
_V i and _αi yield:
_V_α
� �¼ − A τð Þ B τð Þ
C τð Þ D τð Þ� �−1
E τð ÞωF τð Þω
� �ð12Þ
where V = (Vn+1, …, Vn+m)′, α = (αn+1, …, αn+m)′, ω = (ω1,…,
ωn)′, δ = (δ1, …, δn)′. The equation can also beexpressed as:
_x ¼ f x;ωð Þ ð13Þ
where x = (V, α)′.Using (13), a dynamic system can be
constructed that
contains power system node voltages. A node voltagestate
equation and its Jacobian matrix can then be estab-lished and used
to meet the uncertainty of wind powergeneration. The uncertain
elements that are included inthe wind power are considered as the
input, and a singleelement that can measure the system voltage
stability isconsidered as the output. After the application of
“blackbox algorithm”, it can analyze the influence of the
uncer-tainties on system voltage stability.Setting J the Jacobian
matrix of system (13) at the
equilibrium. If the real parts of all the eigenvalues of
theJacobian matrix J is negative, the system (13) is
stable;otherwise, the system can become unstable. Thus, a newsimple
and effective lemma [14] is given as follows:Lemma 1 If the
spectral radius ρ(J) of matrix J satisfies
ρ(J)
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In the following, a 2-rank expansion with n randomvariables is
taken as an example for illustration. The2-rank expansion is given
as:
y2 φ1;φ2ð Þ ¼ a0 þ a1φ1 þ a2φ2 þ a3 φ21−1� �
þa4 φ22−1� �þ a5φ1φ2 ð20Þ
For (16), six certainty coefficients need be solved in2-rank
output model y2. Thus, the number of the
certainty coefficients ai be solved in the 2-rank modelis 1 + 2n
+½(n(n-1)) with n random variables.In applying SRSM, the most
important task is to solve
the unknown coefficient ai. Both the probability distri-bution
method and the efficient regression method canbe used to solve the
unknown coefficient, although, theefficiency of the efficient
regression method is better andthe results are more robust. As the
input variable quan-tities are φ1i and φ2i, (16) can be represented
as:
y2i φ1i;φ2ið Þ ¼ a0 þ a1φ1i þ a2φ2i þ a3 φ21i−1� �
þa4 φ22i−1� �þ a5φ1iφ2i
ð21ÞTo calculate the unknown coefficients ai, some sample
points with forms as (φ1m, φ2m) are required to beselected. In
this paper, φ describes the wind speedfollowing probability
distribution, and y expresses thedegree of voltage instability by
(16) and (17). Thus, therelation that is related to power system is
established foranalysis.Equation (17) is the 2-rank expansion
model, and the
roots (0,ffiffiffi3
p, −
ffiffiffi3
pof the 3-rank Hermite polynomials
can be selected with a total of nine sample points. If thenumber
of random variables is more than 3, the numberof sample points is
two times larger than that of the un-known factor, and thus large
amount of calculation isrequired. However, the selected sample
points are in thestandard normal distribution space, and therefore,
it isnecessary to convert them to the original space.
Thetransformation of the original space sample points corre-sponds
to the real response value, and the unknowncoefficients ai can be
obtained using the least squaremethod for solving linear
equations.
2.3 Copula theory correlation analysis2.3.1 Copula function
definitionAssuming H(·,·) is the joint distribution function of
F(·)and G(·) with marginal distribution, there exists a
Copulafunction C(·,·) satisfying:
H x; yð Þ ¼ C F xð Þ;G yð Þð Þ ð22ÞThe density function of the
distribution function H(·,·)
can be derived by the density function C(·,·) of the
Reduction and transformationto m independent, standard
normal variables
End
start
Collocations of sampling points
Found polynomial chaosexpansion equation
Deduction ofcoefficients of y
Calculation of theexpected value of output
response
Calculation the windfarm correlation
Fig. 1 Flow chart of the analysis method
Fig. 2 The IEEE 24-bus system
Table 1 Parameters of wind farms
Wind farms 1 2
Fans 40 20
Rated capacity (MW) 0.6 1.5
Cut in wind speed (m/s) 4 3
Cut out wind speed (m/s) 22 24
Rated wind speed (m/s) 14 15
Ma et al. Protection and Control of Modern Power Systems (2017)
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Copula function and the edge distribution function F(·)and G(·)
as:
h x; yð Þ ¼ c F xð Þ;G yð Þð Þf xð Þg yð Þ ð23Þ
where c u; vð Þ ¼∂C u;vð Þ�
∂u∂v , u = F(x), v =G(y); f(·) and
g(·) are the density functions of F(·) and G(·), respectively.In
this paper, wind power output sequences of the twowind power plants
are x and y, and their distributionfunctions are F(x) and G(y),
respectively. u = F(x), v =G(y). H is the copula function of F (x)
and G (y).In this paper, Frank Copula function [19–22] is con-
sidered as the connection function of joint
probabilitydistribution of wind farms. The respective
distributionfunction and density function of Frank Copula can
beexpressed as:
CF u; v; βð Þ ¼ − 1βln 1þ e
−βu−1� �
e−βv−1� �
e−β−1
� ð24Þ
cF u; v; βð Þ ¼−β e−β−1
� �e−β uþvð Þ
e−β−1ð Þ þ e−βu−1ð Þ e−βv−1ð Þ½ �2 ð25Þ
where β is the relative parameter and β ≠ 0. If β > 0,
ran-dom variables u and v have positive correlation. If β→ 0,random
variables u and v tend to be independent. β
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κ ¼ P xi−xj� �
yi−yj� �
> 0n o
−P xi−xj� �
yi−yj� �
< 0n o ð26Þ
as the Kendall rank correlation coefficient, and к∈[−1,1],i ≠ j.
P indicates probability of occurrence. If к > 0, ran-dom
variables X and Y have positive correlation; if к < 0,the random
variables have negative correlation. If к = 0,the correlation
between random variables X and Y can-not be determined.Random
variable P1 and P2 is defined as the output
rates of the two wind farms, respectively. (p11, p12,…,p1n) and
(p21, p22,…, p2n) are the respective sample spaceof random variable
P1 and P2, n is the sample size, whichestablishes a one-to-one
relationship with p1i and p2i.The relation between the Kendall rank
related coeffi-
cient к and the related parameters β of Frank Copulafunction can
be expressed as:
κ ¼ 1þ 4βDk βð Þ−1½ � ð27Þ
where Dk βð Þ ¼ kβkZ β0
tk
et−1dt, k = 1.
2.4 Wind power uncertainty analysisThe relationships between
active power Pe that is sup-plied by the wind generation source and
wind speed vare expressed as [14, 20]:
Pe ¼cþ dv; asvin≤v≤vrP0; asvr≤v≤vout0; others
8<: ð28Þ
where vin and vout are respective cut-in wind speed andcut-out
wind speeds, vr is the rated wind speed, Pe is theactive power
generated by the wind farm, and P0 is therated active power. c and
d are constants.In this paper, the wind speed is assumed to follows
a
Weibull distribution and is shown by variation η ofstandard
normal distribution as:
v tð Þ ¼ − ln 12þ 1
2gerg
ηffiffiffi2
p� �� � � �1
t
ð29Þ
where gerg is the Gaussian error function.The analysis process
with SRSM is expressed in the
next section.
Fig. 6 Cumulative density distribution of voltage instability
for correlation κ = 0.257
Table 2 Average error indices of IEEE 24-node system in
differentcorrelation
Correlation coefficient Average error
0.162 0.039
0.257 0.041
~
~
~
~~
~ ~ ~
24 25
51
11 2612
2728
13
6
29
34
171819
151420
21
22 30 31 33
873
23
1
9
2
4 5
52
Fig. 7 The single-line diagram of the 36-node system
Ma et al. Protection and Control of Modern Power Systems (2017)
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3 Method3.1 Voltage uncertainty analysis with SRSMAs for actual
systems that contains wind power, whosewind speed follows Weibull
distribution rather than thenormal distribution, it should convert
the wind speed asthe standard normal distribution to analyze the
impact ofwind power uncertainty on voltage stability based onSRSM.
Some researchers also apply SRSM to analyze un-certainty of power
system dynamic simulation [23]. Theflow chat for the calculation
and analysis of the uncertaintyof wind power on stability using
SRSM is shown in Fig. 1.
4 Results4.1 Case studiesIn the test cases, dispersed wind
generation is consi-dered in the IEEE 24-bus system shown in Fig. 2
andtwo wind farms are added into the system at node 1 and7,
respectively. In all cases presented below, comparisonsare made to
the Monte Carlo (MC) numerical approach.
4.2 Case 1In this test case, the outputs of the two wind farms
are in-dependent from each other. In this paper, MC simulated6000
times to verify the accuracy and efficiency of SRSM.
To enable a balanced comparison of the accuracy betweenSRSM and
MC, the same number of uncertainties areused for each moment model.
The parameters of the twowind farms are shown in Table 1. In study,
the mode ofload change is that increase the whole network load
si-multaneously, with the load of each node increased by thesame
proportion. In the load direction, the load compo-nent of the
partial load follows normal distribution.Figure 3 shows the
cumulative probability curve ob-
tained by SRSM and MC method for the IEEE 24-nodesystem, and the
voltage instability probability of the sys-tem are determined under
different load levels.In Fig. 4, the probability density of
spectral radius with
380 MW wind power generation is given. The calcula-tion results
also show that SRSM has high accuracy thanthe MC method, and it can
meet the needs for the prac-tical applications.
4.3 Case 2In this test case, two wind farms are correlated each
other.Figures 5 and 6 compare the cumulative probability
curveobtained by SRSM and MC for the IEEE 24-node system,and the
system voltage instability probability are deter-mined under
different load levels.In Figs. 5 and 6, MC was used to verify the
accuracy of
the proposed SRSM method in the paper, and is can beseen that
SRSM method has good accuracy. Under thecondition of different
correlation coefficients, the ave-rage error of voltage instability
analysis with SRSM isgiven in Table 2. Calculation results shown
that Kendallrank correlation coefficient method in analyzing
voltageinstability has good accuracy, and can satisfy the
engi-neering requirement.The results of numerical calculation show
that wind
farm correlation had a significant influence on systemvoltage
stability. According to Figs. 5 and 6, the greaterthe correlation
between the wind farms has, the morenoticeable influence on system
voltage stability. The
100 150 200 250 300 350 400 450 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Critical power(MW)
Vol
tage
uns
tabi
lity
prob
abili
ty
SRSMMC
Fig. 8 Cumulative density distribution of voltage instability
for correlation κ = 0.280
0 50 100 150 200 250 300 3500
0.2
0.4
0.6
0.8
1
Critical power(MW)
Vol
tage
uns
tabi
lity
prob
abili
ty
MCSRSM
Fig. 9 Cumulative density distribution of voltage instability
forcorrelation κ = 0.794
Ma et al. Protection and Control of Modern Power Systems (2017)
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simulation results of case 1 and case 2 shown that thesystem
voltage may reached an instability state is under-estimated without
considering the correlation, that isunderestimate the potential
risks. From the simulationresults, it also got that the proposed
method can reflectaccurately the system voltage stability as
analyze thevoltage stability uncertainty problems.
4.4 Case 3In the test case, dispersed wind generation is
consideredin the EPRI 36-node system shown in Fig. 7. Two
corre-lated wind farms whit the same parameters as shown inTable 1
are added to the system and are connected atnode 4 and 5,
respectively.In this case, MC is simulated 4000 times to verify
the
accuracy and efficiency of SRSM. The system referencepower is
100 MW and the 2-rank SRSM polynomial isused for calculation. In
the simulation, the load of eachnode is again increased by the same
proportion.For correlation coefficient κ = 0.280 and κ = 0.794,
the
cumulative density distributions of voltage instability
areexpressed in Figs. 8 and 9, respectively.As can be seen from the
Figs. 8 and 9, the larger the cor-
relation between the wind farms has, the lower the
voltageinstability critical power is, and the probability of
instabilityis greater under the same power condition. Under the
con-dition of strong correlation, it is also indicated that more
at-tention should be paid to the voltage instability problem.
5 ConclusionsThis paper presents a method that establishes a
dynamicsystem including node voltage to study power system
volt-ages stability incorporating wind farm uncertainty. Ratherthan
the eigenvalues of the Jacobi matrix, the criterion ofpower system
voltage stability is given by the spectral ra-dius of the composite
matrix. In the study process, thecorrelation of wind farms is
considered, such that the un-certainty of the wind farms and the
analysis method arecloser to actual systems. The proposed method
which usesSRSM to study the uncertainty can provide power
systemoperators with useful real-time estimation of the powersystem
voltage stability with wind power integration.Compared to the
traditional methods, e.g. the MonteCarlo method, the proposed one
is more efficient.The analysis and the simulation results also
shown
that the proposed method has a higher accuracy and hasa good
application prospect to actual system operationand stability
analysis. The effect of the correlation be-tween multiple
distributed energy source on system vol-tage stability will be
considered in future research.
AcknowledgementsThis work is supported by project of the Jiangsu
Province University NaturalScience Research Foundation
(14KJB470003).
Authors’ contributionsZM contributed to the study design and
analysis and drafted the manuscript;HC worked on aspects of the
study relating to wind farm correlation; YC wasinvolved in data
acquisition and revision of the manuscript. All authors haveread
and approved the final manuscript.
Competing interestsThe authors declare that they have no
competing interests.
Author details1College of Automation Engineering, Qingdao
University of Technology,Qingdao 266033, China. 2Jiangsu Electric
Power Company, Nanjing 210024,China. 3Jiangsu Normal University,
Xuzhou, China.
Received: 6 January 2017 Accepted: 9 May 2017
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Ma et al. Protection and Control of Modern Power Systems (2017)
2:20 Page 9 of 9
AbstractIntroductionDiscussionSystem model analysisStochastic
response surface method analysisCopula theory correlation
analysisCopula function definitionCorrelation analysis
Wind power uncertainty analysis
MethodVoltage uncertainty analysis with SRSM
ResultsCase studiesCase 1Case 2Case 3
ConclusionsAcknowledgementsAuthors’ contributionsCompeting
interestsAuthor detailsReferences
