CC BY
7Journal of Siberian Federal University. Engineering & Technologies 1 (2012 5) 39-56
УДК 621.316.11
Stochastic Simulation of Covariance Matrix and Power Load Curves in Electric Distribution Networks
Ivan V. Shulgina, Aleksey A. Gerasimenkoa* and Zhou Su Quanb*
a Siberian Federal University, 79 Svobodny, Krasnoyarsk, 660041 Russia b Harbin Institute of Technology China 150001, Harbin, 92 West Dazhi Street,
Nan Gang District 1
Received 6.02.2012, received in revised form 13.02.2012, accepted 20.02.2012
An algorithm of stochastic simulation of covariance matrix and nodal power load curves is developed for electric distribution networks based on factor analysis. Statistical stability of factor power load model is confirmed. Application of this model is able to identify a general regularity of nodal power changing, and to simplify the analysis of multivariate operating conditions in operational problems of electric distribution networks and their optimization.
Keywords: stochastic simulation, electric power load, covariance matrix, electric distribution network, principal component analysis, energy losses, operating condition optimization.
Introduction
The adoption of automated meter reading (AMR) systems in the industry makes it possible to store statistical data about power transmission and consumption. Based on the above-mentioned systems and modern mathematical methods, it is possible to solve a series of problems: multifactor simulation, prediction and standardization of energy consumption and some integral characteristics of power systems; production activity analysis and optimization of power system functioning; diagnostics of electrical equipment in electric power supply systems etc. [1-3].
Power supply continuity and safety of electric power supply depend on a stability of a whole chain: "electric power generation - transmission - distribution". Electric distribution networks, which are the master link in that chain, are the most problematic ones and outlay elements influence not only the electricity tariffs, but also the economic efficiency. About half of the current power sector's basic assets are related to electric distribution networks, and most of the energy is lost just in these networks. However, the role of the electric distribution networks is still often underestimated on the background of global construction problems. Thus, dangerous and far reaching consequences, both economic and social, may arise [4].
* Corresponding author E-mail address: gerasimenkoaa@yandex.ru
1 © Siberian Federal University. All rights reserved
Recently, taking into account the new computer technologies and the development of modern control measuring systems, the models of power consumption have been mainly developed by means of stochastic methods of component analysis, which include the principal component method [1, 5-12]. The models of power consumption or, in other words, the models of power load curves are able to decrease the volume of initial information needed for problem solving, and to simplify analysis of multivariate operating conditions in electric distribution systems. The problems include determination of power integral characteristics (energy losses, ranges of changing the operating condition parameters in the electric nodes and between power systems etc.), and reactive power compensation, both of which are very important when it comes to the complex optimization of power system and energy saving.
Deterministic methods, statistical simulation of power operating conditions, and determination of integral characteristics in power engineering have been advocated by different authors for some years [1, 5-8, 10-12]. The related research studies faced some difficulties: large dimension of the nodal power covariance matrix, large volume of information about power loads and operating condition parameters, complicated processing of initial information as well as underdevelopment of measuring systems, computers and programming. Therefore, the application area of stochastic analysis methods was limited. More recently, taking into account the adoption of SCADA and AMR systems, the above-mentioned disadvantages have been gradually disappearing, and the development of stochastic methods, partially the methods based on the principles of factor analysis, is more promising [13].
Conceptual Description of Principal Component Method
A component analysis as a method was developed by Pearson1; he proposed a method of databank compression which allocates a maximal variance. This method was also developed by Hotelling2.
The factor analysis is a multivariate analysis which researches an internal structure of the covariance or correlation matrices. It is applied for the statistical research of a system of random variables which have a correlation by means of stable random or nonrandom factors [9].
The principal component method is based on simple and ordinary conceptions, which depend on the covariance matrix analysis and the matrix linear transformation.
The modeling, characterizing the behavior of a random variable, is implemented by different ways of regression and factor analysis. In regression analysis the factors and model structure are entered a priori; in factor analysis we assume that, factors exist while their exact number and the model structure are only determined during the process of problem solving.
The principal component method is a dismemberment of a covariance matrix on the orthogonal vectors (components) or directions corresponding to the number of variables. These vectors correspond to the eigenvalues and the eigenvectors of the matrix. We agree that by a characteristic value we mean the set of eigenvalues and eigenvectors of the matrix.
Based on this method, the characteristic values are formed in descending order, which is important since only few components have to be used for the description of the initial data. The vectors are pairwise orthogonal ones, and their components are uncorrelated. A few components can reflect most
1 Pearson, K. On lines and planes of closest fit to systems of points in space. - Phil. Mag. - 1901. #6, p. 559-572.
2 Hotelling, H. Analysis of complex of statistical variables into principal components. - Jep, #24, 1933. - p. 417-441, 498-520.
of the sum variance of initial variables; however, all components are required for accurate reproduction of correlations between variable s.
The principal component method is used for total simulation of initial randf m variables. However, we do not need to pui forward the hypotheses about variaf les because ahe variables do not even have to be random variables. In practice the observations of random variables are samples from some population.
In order to decrease the complexity of the statistical calculations we can replace an «-dimensional random variable by k<n linear functions from the initial variables. The simulation is called a reconstruction of function using a lindar predictor3, whicf is implemented by means of eigenvectors of the covariance matrix [12].
Consider ohe mulnivariate eandom variable X which is an «-dimension sample data
X =
... x
For the analysis of other random variables depending on X, it is necessary to determine the mathematical expectations, for instance, samp°e mean or average values MX, MX2, ..., MXn, and variations (changing) of initial random variables in the neighborhood of their average values AX, AX2, ..., AX„. A characteristir of a nandom vaeiable variation in the neighborhood of their average values is ;3i variance. It may be that the linear comb ination of initial random variables has the maximal variance (c ertainly, we should compare only the normalized linxab combination of random variables because any ranrlnm variable ean be multiplied by a large number, so any large variance can be obtained).
The linear transformation of initial variabies is implemented by meanr of uncorrrlated and normalized linear variables v.
Linear Combination of Random Variables with Maximal Variance
Consider ihie possiblx linrbr eomninaiionr of randam variables X t
G = X xu = [Xv. .X k ]x[tJ1. . . uk ].
where
X =
taking inao accouhi a iimination which hulfUls the faltowing condition oi noamalization
m
YU =1, i = i x-,k
j=i
(i)
Xjj Xk1 U11 Uk1
x12 Xk 2 U12 Uk 2
xh. II X ki U1 = U11 ; uk = Uk1
_ x1»_ _ Xkn _ U1m _
Predictor is a superior system ofvariables
x,, x
X
11 -"-21
X
X
X
nm
The variance of the linear combination (1) is determined by the following formula based on [7,
12]
с G = DG =
U11 U12 U21 U
u2
= ит X K(X) X и =
x K(X) x
X 0.0 ... 0.0" 0.0 X2 ... 0.0
0.0 0.0 ...
(3)
, Xk;
where K(X)=K - covariance matrix of initial random variables Xb X2, k - rank of matrix K(X); t - i ndex of tlie transpose of a matrix; m - total number of changing of the random variable X.
We assume that the matrix's elements k(XjXj) are the estimations calculated from samples: xih x, ..., i,m and xdl, x;2, ..., xsm. Tlius, the selection of a random factor having a maximal variance is found througha minimum of function (3) satisfying condition (2). The optimization problem is solved by a method of Lagrange multipliers. Introduce an auxiliary objective function, which is Lagrange function
Ф =a2 G + k X (ц*-1), j=1
(4)
where A - the Lagrange multiplier.
An absrlute minimum uf the function (4) corresponds to the с onditional miniimtm of function (3) subject to condition (2). Tfa function including all var(afles is differentiatfd; the minimum condition is oUtained as
дФ
ди.
= 2Ё ju -2AX = 0 1 = U,...,*; фТо^ = 1.
(5)
j=1
j=1
The solutions of sj^stem (5)) are all eormalizedl eigenvectors of the matrix K(X). Every solution determines an extreme pornt or spocific ppxpiin"^ of tloe function. The coordinates of the eigenvpctor cirresponding to the maximal eigenmalue X1 coroesponf to the globel minimum.
In tacOor analyst the components of the vector G are new ranckim variables which are tlee linear combinatieu af initial X or centered АХ random vciiri^bl^s.
The eiggnvaluet X ainc( eigenvectors o of power covariance mateix have useful properties wCocO are applied in component analycit. The eigenvalues ere feal values ;^n<e ehe ieiL|;5^^'^eic;t;t}ils can be ahosen as pespendicular to ^aicli other The eigenvectoos define en undergoing pure tention or compcer sion directio n of the li near transCormatiof corresponding to the matrix K(X). These vectors are also named the principal components of the matrix [12], and the eigenvalue X in a coefficient of the tuansformation. The variance of the i-th principef component is equae to the eigenvalue X of matrix K(Xf
It is known that the eigenvalues X and eigenvectors f of matrixes safisey the following expression
v., и
21
и
22
и
.. и
km
km
K x u = u x X . (6)
By multiply iniii; both sides of the expression (6) on the laft of the matrix u-1, we will arrive at
1 = u x Kx u.
(7)
Expxesaio n (7) is considerably simplified when the original matrix K is defined as positive, which is tfecase fortfe powea covariance matrix [7]. Thus, all eigenvecfora can be mhde orthfnormal ones, i. e. satiefy ing she expressions
Ui T x Uj = 0 when iit j; u=T x = 1 when i = j .
It is easy to verify that the i nverse matrix j is equal to the conjugate one, and expression (7) can be rewritten as (3)
1 = uT x K x u ,
(8)
where u - an orthooormal matrix whose columns coneiet of eigenvectors uj, iT2,..., Condition (2) is satisfied for every column oh the ortOonsrmal matrix.
Inverse expression between nhe otigonal matrix K and the matrix k
K = [Dj Dt ...C=]x).x[iJ1 U2 ...UJT = ,
^11 ^ 211 U12 ^22
A 0 . 0. . .0 . 0 0.0 At ...0.0
0.0 0.0... A
^11 ^11 ■
(9)
Thie method allows us tto select the arthogonal Oactoms among tte i. e.
snetisticallyrtndepondent componenSs, which provide a linearity oi the mhthod and additive efficiency.
The above-mentioned properties of the eigenvectors show that the full totality of them is equivalent to the original ppobpbiliseic model corresponding to the vector X. Moreovem, both sets oi variables X and G define tha same vector space.
However, anon the sct of vectors G, it is hufficient to select a small number of M principal components (facdors) explaining mosl of the helatio nships between all components of the initial vector of random variables X. The main factors rre not directly observed, bul they characterize the change in the original variabks. Thtnefooe, wh con get the tark of obiaining a lineat peechctor of dimenskmM (M <k). Foat eaeh O the bnct pmr^dJliLc^oi' i.s tine fiastMeigenvecfost of K, rorretponding to the maximal eigenvalues. As a eesult of studying tOe internal stniciure oe the matrix Kh, the selection of main factors is made in such order thct at the begrnning the firat oe shem makes the gneatesi gontatbution to the -variance of SPe variables, than ihe recond one is ihe largest contribution to the varianco oh the -variables remaining ahtef taking Into account the main factor, eic. Ultimately, rhesg vectors conatitute a set of liineaLh-iiy indapendrnt brsis vectors, orianied in suth a way than each of nhiem makes the maximum conjribution to the vaeiance of the original variaMes X.
On the practira. sMe, the factor mode1 makes it possfble to adequately estimate the covariance stature between thee relatively large number of observed variables by means of a smaller number of common factots. Evhluetion or the factor structure is carried by the reqirired number of factors expfaining the correlations between variables and load factors in these variables. Component analysis
- L3 -
is most useful when all variables x,- are measured in the same units. If not, the method is much more difficult Io validate [9].
Factoaial or component methods of statistical analyeis are used in the computation of operational enengy losses S13, 14]. as wetl as for shot-Meem darecastingand optimization [15, 16]. When solving the problem of factorial simulation of electricloads via a stochattic approach, the information about 1he characteristics of the random verirbSe ii approximate^ determined by a partial sample from the general population. In tha factor simulation of power loads [5-7, 10, 11] the curves of active and reactive nodal powers ace consideted as a training sample having 2«-dimension. In operational computations the scope of the method is limited to the modeling of daily power load curves of an unobservable netwock.
The mottelmg of electric power loads on the balis of factor ansHysis allowt us to:
- find hidden regularities, which are determined by many internal and external causes of load changing;
- carayout tte compression of informatio n by descriting all curves ley means tf the common factors or principal components, whose number is much smaller than the number of initial curves;
e identity Sit statisSical dependence between the pawei load curves and the main factors;
-predict the random component curves based on the regression equation constructed on the basis of factor analysis;
- simplify the methods for determining the integral characteristics of power systems.
Determination Methods of Principal Components
The problem of deferminalioa of the principal componenhe is a classical probim of determining the eharnateriseic values from a matrix of random vsrifbdes, as which the nodal power
loeds are considered. The determination oS eigenvalutr and eigenvectors od matrices in linear algebra tt catted theh>rsC> lem of characteristic values, and it is s complicated task which is implemented in seveaal sSatistrca. aoftwsre ¡applications. The value C is called the elgenvatue of Ka, if there it a nonzero vector (eigenvrclor of" K) satrsaymg the equation
(K-Ax E) xu = 0, (10)
where E - unity malrix; 0 - nu;l vector.
The eystem(lO) is a tomogeneous systemof1 linnt equatiom because the free members of its equatinsit eee zeros. Id hrrs nontrsvial solutione if the determinsnt af atsisitr |K -Ax E| equals zero, i. e.
+ p.r-1 + P2X"~2 +... + pn_+x + pn = 0, (11)
where /?15.. - coefficients of the characfeeistir jp^l^jr^miail.
The methods for determining the eigenvalues and eigenvectors canbe divided into two groups [11]: the first group) includes iterative methods which often use a similarity transformation and solve a linear system of equations (10); the second group includes the direct methods that calculate the characteristic polynomial (11). The problems (10) and (11) have different conditionality, as the roots of polynomials (er) are often Wghly sensitive ro erroms which are inevitafly arising in the calculation of polynomial coeffirients. That was the main reason of the almosh complete exclurion of direct methods.
The direct application of covariance matrix in various algorithms is greatly complicated by its dimension. In ordvr to compensate the above-mentioned Oisadvantage, the modeling of covariance matrix is implemented my using modern computer software interactive systems, such as MATLAB, MATCAD, C++, ANSYS, I7ORTRAN, etc.
The main coiterion for the normalization of the eigenvoctors in MATLAB consists of
uT X u = E. (12)
Small changes in matrix elomenti, such as rounding earors, can cause large changes in the characteristic values. The power covariance matsix is a square matrix that is easier to use for matrix transformations in comparison with other matrices.
Stochastic Model of Covariance Matrix and Eleceric Power Loads
Most oe ehe research ti-8] conducted in thir field was aimed at the modeling of power loads and its application for a daily time interval. This is due to tire; peculiarities of the energy business and information support of power utilities during the development of this technique. Structural changes that have occurred in the management of an integrated power grid led to the need of periodic computations between separate power business entities. Today, the main period of the financial settlement is one month. The modeling of power oonsumptfon on a montlily time interval was proposed for the first time in [10, 11]. At the moment the technique and its posribk application are implemented insufficiently and the refore further rese arch and elaboration is nee ded.
The simulation of the power covariance matrix is based on several properties of eigenvalues and eigenvectors of a matrix which is expanded by 2n eigenvalues and eigenvectors; the first few Mcharacteristic values (M<<2n) aecueately reflect the rental variance of the enitial power load curves [5-7, 10-12].
Statistical analy sir of power operating conditions uses information about variances of power loads a■2Pp,(T2Qj and cross-covarianse functinna Appf), kpPQj) , k2QQP) between random variables of different power nodes
=-7¿Q» -M-)2 ; v'Qt =d¿(Qlm -MQi)2 , i = in;
d m=l d m=l
k-P,Qj ) = QÍ-Pm - MP, )(Qjm - MQj ), i, j = m.
d m=1 ' (13)
1 d _
k(P>pj') = -; ÍM -MP)(Pjm - MP j), i,j = 1, n, i * j;
d m=1
1 d _
k(QiQj2 = QY(Qim-MQiXQjm -MQj), i, j = 1, n , i * j,
2 m=1
where i, j - indexes of nodes; m - index onevery time intenvaO ^or ^Itie; 7" period; n - the number of nodes of power Uistributton sastemt with know is or timalateol power eoad c;urves.
'^he elements (13) ssf a jscnwe^ covariance matoix characterize the dugrre of irregularity of power load curves, which remains approximately coostant over a long period. The degree may be determined on the basis o° daily measurement! performed on diffeoeot days in power sy stem sectors. T his is an impo rtant advantage cof nhe utatisiical method.
The variances and cross-covariance functions of power loads form the square covariance matrix K= K(P,Q) as folOows
K(P,Q) =
Kn Kif
K K
21 22,
KPA)......KPPj
KPP) (t2P2......KP.PJ
f(P.Qi) QPQ)......k(P<2n)
kPQk) kQP.Q2 )......k(PQ )
k(PnPx) k(PnP2)......a2
KQA) KQW)......HQfj
k{QA) k(Q.w.)......KQpp
k(P,Qi) k(PnQ2)......k(PQ)
kQQQ 2)......k(QlQn)
KQO a Q......kQQn)
k (Qn P.) k (Qn P.)......k (Qn P.) .
k(QnQi) k{QnQ2)......a2Qn
(14)
The simulatioia of the coaariance matrix is implemented coreesponding with expression (9), which may also be written in the usual form
KO = ^Tyl, x ( x u(T.
(15)
Every eigenvalue of covariance matrix corresponds to a general load diagram of power (GLD) , which it a lineac combination of 2n initial nodal power load curves centered at expectation MPt, MQ,
rin=[A/>...AJP AOi ..AeJxkMT T2 T3 ...Tln\i = \,...,n, (16)
wise re [u 2n ]T - the transp ose d matrix of eigenve ctr rs obeained Xxom the covariance matrix of statistical sample forinitial power loaOs, whice has ln*ln dimension;
APl , AQj - ttlie deviations crngered at oJli active and eeactive power in the node # 1 for
a certain time period T
m=1, 2,... Ad.
The obtaineed GLDs nanbn connidraed as ntw independent centered randomvariables with zero mathemetical expectations. The GLDr xave a property of erthogonatihy, i. is. crassrcotciiaace Ji^nc;^ions lUSC-Sm), k(rrf are to :zie;i:o. These new ranVom variables (faceotsf ore a L>uiLt;Ei-1^1e coordmate eystem flor aurcrurcrlue simulation on initial random variables P,-, Qq therefore, ussng M of shem
rk r [[...f,]^,,, *rif.. ,iVs< <<2n, (17)
which sonnesponds to Ihe maximal eigenvalues of power covaridnce mataix, allow us to eimulate initial load changing wtth r eufficient occueacy for a certain hme i.nt;i^^i3Ll T
Si «ONEfMp...] M2r..MgJ+[AP1....APn AQ^.AQn] =
(lid)
= ONEmrNMp.:MPn MQv.MQn]+Tk x[uk]T = p.Pn QAQ], k = 1,...,n, - 46 -
"4p„ " Ahi
Apn Aqn
AQ = APn Q 11 Mn
Ml. . ^qqi»-
where ONEm1 - a column vector consisting of units, and which have m = 1, 2,..., d rows.
[uk ]T - the transposed matrix of prime k eigenvectors Uk corresponding to the first maximal eigenvalues Sik of the power covariaoce matrix K(P,Q) (14);
MQ1 - a mathematical expectation of reactive power curve in the node #1 for the accounting period T.
Pt - pos sible variation of active power in i node for the accounting period T.
The simulation of the power loads allows us So track; rhe variation oa load parameters. It should be noted that initial powea load curves are fulty ssmulaVed on formula (18) using all GLDs. The eigenvalues k of the initsal covoriance matrix K(PrQ) are variancer of GLDs (3). Hence, any power load turve can be represented as a linear combination of the GLDs which reflect the general regularities of power expectation changing for the initial collection of power nodes.
The following daily/monthly samples of the power load curves have been analyzed:
1) 36 simple weekday and weekend power curves [24 hrs] of a 10 kV electric distribution network (d=4); the unit of active power is kW; the unit of reactive power is kvar.
2) 18 atypical real daily active and reactive power curves of a 10 kV electric distribution network (d=24) [11]; the unit of active power is kW; the unit of reactive power is kvar.
3) 42 typical daily power curves for different industries (d=12) [12]; the unit of power is relative unit [r. u.].
4) 30 monthly active power load curves for 110-220 kV overhead lines for August-September in 2009 [11]. Number of intervals d of period T was reduced from 744 to 31 by calculating the average power for each day. The unit of active power is MW.
Computational results of eigenvalues and GLDs (17) from the above-mentioned samples of data #1-4 are presented in Table 1-3, and in Fig. 1-2.
The original power load curves characterize a different degree of irregularity, therefore every sample data #1-4 has its own principal factors linking the power load curves to the system of characteristic values. In all cases, the error of simulation of power load curves is, in a dozen times and sometimes more, less than the error of the covariance matrix simulation (9) taking into account
Table 1. The Six Maximal Eigenvalues Obtained from Sample Data #1-4 for Simulation of Covariance Matrix and Power Load Curves
Sample Data Contribution of principal components to the total variance of loads Eigenvalues of the original sample data in decreasing sequence
Xi X2 X3 X4 X5 X6
1) X 45806.73 25494.41 5039.49 4.64-10"12 - -
0, % 60.00 33.40 6.60 0.0 - -
2) X 17699.71 9289.16 2247.51 1136.81 595.20 502.52
0, % 53.28 27.96 6.77 3.42 1.79 1.51
3) X 1.06705 0.432602 0.104090 0.102429 0.0284292 0.0157218
0, % 60.10 24.36 5.86 5.77 1.60 0.885
4) X 1510.38 369.29 172.26 80.00 68.83 47.70
0, % 65.06 15.91 7.42 3.45 2.96 2.05
Fig. 1. Two Principal Daily GLDs Corresponding to the Maximal Variances for Sample Data #1 in the Same Units as the Initial Variables
-220
Fig. 2. Three Principal Daily GLDs Corresponding to the Maximal Variances Obtained from Sample Data #2 in the Same Units as the Initial Variables
Table 2. Six Daily Nonnormalized GLDs Corresponding to the Maximal Variances Obtained from Typical Daily Power Curves for Different Industries (Sample Data #3) [r. u.]
t, hrs A 9 2 ro r2 r- r0
0-2 -L58281 0.107(504 -0.185982 0.0554811 -0.003-0906 0.0530191
2-4 -L63002 0.111505 -0563562 0947225- 05274-11 0.0268960
4-6 -L00645 -0.0037509 0.339926 .0.521240 0.201222 -0430268
6-8 f.826063 f.369568 -0465288 -0,464285 -0.953642 0.0141667
8-10 0.965851 0 : 888989 0.169168 0.1H99e -0.0532470 -0^22-010^6
10-12 0.186523 0.654527 0.089099 -6.8899502 -0.052270e 0.265027
12-14 0.671546 0:5908fl 0.0328468 0 . 665991 0.1(599393 0.90600122
14-16 1.331077 -0.655575 -05269-56 ^^OH 0.503330 0.0769630
16-18 0: 832953 -t .12614 0.168290 0.0509246 0.0587576 0525265
18-20 0.687603 -0.651610 0.279512 0.0721762 -0.101280 50.107559
20-22 -0.0217858 -0.704435 0.00610245 0.325773 -0.0908384 -0.0676617
22-24 -1.26054 -0.t 2e be a rf.ia»C26 0.3ia45a -0.0 et0239 -0.0h40258
the same numb er oJf cherectee^tic values and GLDs. The recson foe this is that the elements of the power eevariance niaerix are relativity small velues compnoed wiith tins valuar oa the initial power ioad curves. The maximaO ^eeroir of simulation nf power ioad eurves daes nrS ooceed and tine; average one ir 8.a3% Oor sampte heSe #S takmg inte accourt only sie maximae chasoeSasiatic values.
The obtained GLDa can also be used to determine the total normalized or weight average GLDs which are used dor the oimulaiion ot unknown powes lorh curves»
By the means ah MATLAB syarem using fill GLDs ond choractecistic values for the sample data, the power covariance matrix (9) and power load curves (18) are simulated with high accuracy. The simulation on the basis of expression (18) is primarily designed for modern automated meter systems and it uses operating; condition informasioee from theica The lock of the alrove-menrioaed syetems in most distfibution sjsSens non lair reptaced by ahe powt) soed simulation [7, US anad so ta usee iha advantages oa facioe rimuJlatioa
Stability ef Fcclor Power Load Model
Factor simulation of power loads has a practical application only if the estimations of factor values (GLDs), which are obtained for different random processes of load changing, are statistically similar ones, i. e. have statistical stability. A research study of the statistical stability of the factor model, which was based on real data on power load curves for different power utilities with large statistical volume of information, showed the presence of a collective and dynamic stability for daily, weekly and monthly power load curves [4, 5, 7, 10, 11].
Statistical stability describes the possibility of using GLDs derived from one learning sample for the simulation of powers which were not included in the learning sample. Dynamic stability describes the comparison of different time realizations of the factor model for constant power utilities; collective stability is the comparison of GLDs belonging to different power utilities.
Table 3. Six Monthly GLDs Corresponding to the Maximal Variances Obtained from Sample Data #4 [MW]
d A r2 r- r A A
1 -61.0910 -19.4394 -^.H-l- 0.98T755 -47. 54 51 2.85404
2 -49.9621 -14.9897 0.604059 «e^-iij-ioic);:^ 404088
3 -56.1344 -11.5645 10.9850 l», 1 -498926 41.4969
4 -44.3974 -8.20962 1048674 5.94788 0.69444-6 47.9750
5 -96.6974 2.44254 49.8471 8.92353 2.65649
6 7.69748 .16.7;)'81 8.35295 iL.^lO^lO
7 55-). 05525-03 42.8409 0.199554 4.75245 447875 D14595
8 90. 1995 42.7496 41.0442 16121149 -10.4229 97.4151916
9 54.5808 14.4486 -22.9847 -2)68463 6) 0;2;;304-
10 —.7969 -33.5047 4.19782 2 7 12644 14.75286
11 42.7563 -28.2635 -14.5850 4.16292 2.03908 -4.62368
12 2X.2400 0.475527 -4.00582 43.7653 46.8796 3.542819
13 -82.9028 02.X289 8.68478 -27.8n98 W. 37166 -0.580486
14 -89.8760 32.9655 -14.5903 2.03812 25047500
15 -64.0213 8.74729 19.1788 4.. 56532 4.64756
16 24.1308 -4.0 3705 40.5377 6.70718 8.72019 0.639262
17 7.18X90 -15.8859 8.20769 -349646 7^9(306:5 ^^■18714!1
18 6.47624 -14.2350 3.81054 -949059
19 08. 5 007 ■4.964X9 -L90678 -11.5262 51081 5 98 -2.05565
20 13.6620 -8:17086 -4.04740 42.0936 0.49460 -0.301653
21 9.60770 -13.5638 -3.17290 -11.0283 3.86244 2.55857
22 8.20646 -17.2128 1.59427 .4.97636 3.m-0 27 .nOM
23 3.85626 1.29999 17:8203 8515507 -1.(57-9141 3.97740
24 11.9X44 -70.1443 8.961n2 4.82252 59292 6.58522
25 14.4479 45.7646 14.6704 5.18767 -o^:2íз:- 5.0.1 7. ■5191^4-
26 04.5485 41. M — 10.0805 4.09407 274395 8.39X03
27 lX.7817 19.1X72 24.4280 9.96452 4.36413 -0.910460
28 14.4501 26.3028 30.7772 14.1072 -2.136420
29 07.8795 0.40787 15.9746 1.122875 6.85280 090314
30 05.8722 5.89048 9.71 574 4 . 84532 5.26982 5.52796
31 6.55574 -19.8384 0.394553 0.291731 -6.93900 16.5671
The research studies of daily power load curves obtained by the AMR system in more than 100 points of head line sections in 6-110 kV distribution networks for 13 days identified a strong statistical relationship between the first GLDs, i. e. proximity of variances of the power load curves and close correlation dependence [11]. This allows us to conclude that the factor model of the covariance matrix and power load curves have statistical stability, and it is possible to apply it for the simulation of power consumption irregularity for the posterior or previous analogous time periods.
The results of multiple computations indicate a rather large contribution of the three principal components (70-80%) to the total variance of the whole sample of original power load curves. The first
principal components of GLDs indicate the presence of common internal reasons for daily irregularity in power load curves and intersystem power flows [6].
The collective stability of the factor model allows us to suggest that GLDs reflect the main reasons for power changing without specific factors. Therefore, in order to determine the GLDs, we do not need to analyze the power load curves in all nodes; it is enough to take only some modeling subset of power load curves into account, for instance, combined diagrams of consumer groups. These are formed on the basis of check measurements of power consumption, which is carried out by an inspectorate. The computation results showed that sets of GLDs which were obtained on the basis of analysis of large samples of power load curves are quite close to each other [6].
Computations for different samples of daily and monthly power load curves have also confirmed the statistical stability of the factor model (see above). The contribution of the first principal component to the total variance of power loads is over 50%, while the significant contribution of the first three components was also confirmed (see Table 1).
Wide application of factor analysis for power load simulation offers a possibility to limit the volume of the statistical sample to not more than 100 elements [7, 8], which allows to manipulate not large covariance matrices. In this case, the requirements of representativeness are carried out, and the obtained GLDs are statistically stable diagrams.
A Number of Principal Components
A recent review ot the approaches to determine the sequired number M of eigtnvalues and eigenvectors for covariance matrix simulation identified the absenct of a consensus among specialists on lhe factor analysi si [11]. Ono opinio n is thbt this aumbe r it usually not higher Shan tour. However, in some rasta depending on the accuracy of tice aavariance matrix simulrtioo, it is required to take into accouot a grealer numbrr off characteristir values and GLDs. Researches of power covariance matrix and power lntd curves showed thai tise sequired number of charecteeistic values depends on sample data ptopesties and on she ireegularity of power load rurvee. Tie maximal aumber is equal to the first six maximal characteristic values in MATLAB system. Most likely, this number was chosen after extenrive analy sps oC covariaaee matricef and faator rimulation.
A simplr approach for the; detesmiaalion of a rrtsonabUe number of characteristic values is to estimaie lhe cavr^irfilll contribution ni the principal components' sequence fls f2,..., .fi to the total varirnc e. If thus summarized coascibution to the total variance is 75-90%, we should stop o n the value A=2, 2r a., M[5, 6]. The total percentage contribution (9 tact thevariance for fixed A/is caOculated by the foltowing eormula
M
Xk
O = 4=--100%; 75 < O < 90%, (19)
i=1
2n
where YfK - the rum of eigenvalunt oi origmal power covdrianca matrix, which is called spur of
i=1
matrix i7, 8]. &io a criterion for tite accuracy of the coaariance matrix s^mi^ul^t;i(3n nnd Che simulation of origmal active and re active power load curves. This criterion is suffietenS to carry out the computations of titc power integral charanteristics and power syiiem optimization.
Component analysis is used to simulate both random and deterministic dependencies similar to the regression method of approximating the functions dependent on the time. The most effective test of statistical hypothesis and the determination of the required number of principal components is the repeated application of component analysis for various samples of the same general population. If the statistical hypothesis about the existence of common dominant trends in all random variables X, is true, then the statistical characteristics, at least for the first principal component selected on the basis of different sample data, will be close to each other.
The statistical characteristics for principal components r not characterized by the properties of one general population differ substantially from sample to sample. Repeated factor simulation for different sample data is a universal method to select statistically stable principal components which characterize the properties of the general population.
The factor simulation of a set of random variables is a useful instrument of statistical analysis if the dimension of the model space of M initial random variables is a sufficiently small one. Such a situation is typical for the simulation of nodal power loads. The application of factor analysis methods allows us to simulate hundreds of power load curves by the means of 2-3 GLDs [7, 8].
For a reliable application of the method, it is necessary that the first eigenvalues of the K are significantly different from each other. The matrix K, which corresponds to the nodal power load curves, usually satisfies this condition [12].
The required number of GLDs for the simulation of original power load curves or required number of characteristic values for the simulation of power covariance matrix depends on the properties of the concrete set of research random variables. The research studies on different collections of power load curves confirmed the hypothesis about high quality simulation of power covariance matrix. This simulation is based on a small number of characteristic values reflecting the maximal part of the total variance for the whole general population; hereby a small number means 2+5 GLDs, the exact number depending on the properties of the sample data, desired accuracy and purposes of simulation.
Stochastic Simulation of Covariance Matrix and Power Load Curves' Algorithm
Referring to the above-mentioned points we can formulate a simulation algorithm for covariance matrix and power load curves:
1. The original power covariance matrix K(P,Q) is formulated on formulas (13), (14) based on retrospective analysis of active and reactive power curves in distribution networks for a certain time period T and control measuring data.
2. 2n eigenvalues and 2n eigenvectors of the K(P,Q) are determined by means of any suitable software based on the principal component method. The characteristic values identify a degree of statistical relationship between random deviations of powers Pt, P, Q, Q. from their mathematical expectations MP, MP, MQ,, MQ.
3. A selection of first M maximal eigenvalues of the K(P,Q) kk < k and eigenvectors uk < u is arranged in descending order. The number of eigenvalues and GLDs needed for covariance matrix simulation is an acceptable one for practical computations of integral characteristics and system optimization if condition (19) is satisfied. In other words, we should stop on such value of M where the contribution of sum of the first diagonal elements of the covariance matrix to the total variance
of the general population of active and reactive powers is 75-90%. The recommended range of M is 2 < M < 5.
4. The simulation of the power covariance matrix K(P,Q) is carried out on formula (9) by the means of M characteristic values. Computations like that can be performed only once a month based on the analysis of a training sample of power load curves.
5. The GLDs are determined on formula (16) for the whole collection of the corresponding sample
data.
6. The first M maximal GLDs rk are selected on (17), which correspond to the M eigenvectors uk and eigenvalues lk of K(P,Q). The original active and reactive power curves are simulated on the expression (18) by the means of rk.
This method is effective when the condition M <<2n can be limited to take into account only the first few and rk. In addition, the GLDs rk obtained for the different random process realizations of power load changing must have statistical stability. The property of universality was confirmed by means of computations of the GLDs rk of statistically representative sample data for different power utilities. In all cases, 3-4 rk are usually sufficient for the reflection of up to 75-95 % of the total variance of original power loads (see tab. 1-3).
Thus, the original power load curves can be presented in the form of some characteristics: the mathematical expectations and the coefficients kk and uk. This is used for effective determination of integral characteristics and optimization algorithm in power distribution systems.
In order to obtain a more accurate model of covariance matrix and power load curves it is necessary to take into account a larger number of characteristic values and GLDs, and it is possible to use the expansion in a Fourier series or similar methods [8]. However, the component analysis differs from other statistical methods by offering a more economical and convenient way for system optimization and a good form for the presentation of information. One advantage of the component analysis is that it is determined in such way that the function package of simulation is not selected randomly, as for instance, in Fourier analysis, but on the basis of analyzing the principal regularity of power load changing. The indicated regularities obtained on the basis of the main factors, their number is less than in other simulation methods, determine the possibility of their application for small samples containing 4-6 points of a daily curve.
The methodology of the integral characteristic determination is developed on the basis of stochastic simulation of covariance matrix and power load curves particularly the load-dependent energy loss determination [13, 14] and optimization algorithms of operating conditions on reactive power in electric distribution networks [15, 16].
Conclusions
1. This paper proposes the factor model of power loads on daily and monthly time periods, which allows us to identify the general regularity of nodal power changing in electric distribution networks. The advantages and possibilities of the model application are described.
2. The computational results of the general load diagrams are derived for different sample data of original active and reactive power curves for daily and monthly time periods T. The statistical stability of factor power load model is confirmed for 6-110 kV electric power networks. The contribution of the first principal component to the total variance of power loads is more than 50%.
3. The simulation algorithm for covariance matrix and nodal power load curves is formulated by means of only a subset of the first main factors (2 < М < 5); it is possible to decrease the complexity of computations in comparison to traditional computation of multiple operating conditions and to simplify the determination of power integral characteristics.
4. The algorithms of energy loss determination and ranging of reactive power changing are developed on the basis of the proposed stochastic model of covariance matrix and power load curves [13-16]. This can be effectively used for solving the problems of energy saving and power system optimization.
References
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Стохастическое моделирование матрицы корреляционных моментов и графиков нагрузок мощностей узлов
распределительных электрических сетей
И.В. Шульгин3, А.А. Герасименко8, Су-Чуан Джоуб
a Сибирский федеральный университет Россия 660041, Красноярск, пр. Свободный, 79 бХарбинский политехнический университет
КНР, Харбин
Разработан алгоритм стохастического моделирования матрицы корреляционных моментов и графиков активной и реактивной мощностей узлов распределительных электрических сетей на основе факторного анализа. Обоснована статистическая устойчивость факторной модели электрических нагрузок. Применение данной модели позволяет выявить общие закономерности изменения мощностей нагрузочных узлов сети и упростить, сделать эффективными методы анализа и учёта многорежимности в задачах эксплуатации распределительных электрических сетей и оптимизации их режимов.
Ключевые слова: вероятностно-статистическое моделирование электрических нагрузок, матрица корреляционных моментов, распределительная электрическая сеть, метод главных компонент, потери электроэнергии, оптимизация режимов.
