Combined Cumulant and Gaussian Mixture ... - IEEE Xplore

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 2, JUNE 2016

71

Combined Cumulant and Gaussian Mixture Approximation for Correlated Probabilistic Load Flow Studies: A New Approach B Rajanarayan Prusty, Student Member, IEEE and Debashisha Jena, Senior Member, IEEE

Abstract—In this paper, a probabilistic load flow analysis technique that combines the cumulant method and Gaussian mixture approximation method is proposed. This technique overcomes the incapability of the existing series expansion methods to approximate multimodal probability distributions. A mix of Gaussian, non-Gaussian, and discrete type probability distributions for input bus powers is considered. Probability distributions of multimodal bus voltages and line power flows pertaining to these inputs are precisely obtained without using any series expansion method. At the same time, multiple input correlations are considered. Performance of the proposed method is demonstrated in IEEE 14 and 57 bus test systems. Results are compared with cumulant and Gram Charlier expansion, cumulant and Cornish Fisher expansion, dependent discrete convolution, and Monte Carlo simulation. Effects of different correlation cases on distribution of bus voltages and line power flows are also studied. Index Terms—Correlation, cumulant, Gaussian mixture approximation, photovoltaic generation, probabilistic load flow.

N OMENCLATURE AM APM CCCFM CCGCM CCGMA CDF CFM CM DDC DLF EM FOR GCM GMA HM MCS

Analytical method. Approximate method. Combined cumulant and Cornish Fisher method. Combined cumulant and Gram Charlier method. Combined cumulant and Gaussian mixture approximation. Cumulative distribution function. Cornish Fisher method. Cumulant method. Dependent discrete convolution. Deterministic load flow. Expectation maximization. Forced outage rate. Gram Charlier method. Gaussian mixture approximation. Hybrid method. Monte Carlo simulation.

Manuscript received October 23, 2015; revised March 15, 2016; accepted May 9, 2016. Date of publication June 30, 2016; date of current version May 17, 2016. B Rajanarayan Prusty and D. Jena (corresponding author, e-mail: bapu4002@gmail. com) are with the Department of Electrical and Electronics Engineering, National Institute of Technology Karnataka, Surathkal, India575025. DOI: 10.17775/CSEEJPES.2016.00024

PDF PLF PMCC PMF PV

Probability density function. Probabilistic load flow. Pearson product moment correlation coefficient. Probability mass function. Photovoltaic. I. I NTRODUCTION

L

OAD flow study is a vital decision-making tool used for evaluating the performance of present and future power systems [1]. DLF completely ignores input uncertainty. Insertion of renewable energy sources such as PV generation at the transmission level increases the level of uncertainty in the power system. In conventional generation, uncertainty arises from FOR [2], whereas in PV generation it arises from inaccurate prediction of solar irradiance due to climatic changes [3]. Load demand uncertainty on the other hand, arises from forecasting and measurement errors. PLF incorporates effect of these uncertainties by treating each input as a random variable. Various methods used for PLF computation include MCS, AMs, APMs, and HMs. MCS requires numerous simulations to compute accurate results [4]. However, because of its accuracy, MCS is used as the reference for comparison and validation of AMs, APMs, and HMs. In AMs, input random variables are defined as either PMFs or PDFs. The desired random variables (bus voltages and line power flows) are represented as either PDFs or CDFs [5]. APMs have the ability to permit nonlinear analysis by using statistical properties of desired random variables. APM is computationally less efficient than CM. HM combines more than one of the AMs, and has gained additional interest since it suppresses the shortcomings of individual methods [2]. For large systems, CM is acceptable since computational time is less. Expansion methods such as GCM and CFM are used to approximate the shape of the distribution [6]. GCM is only applicable for unimodal distributions [7], whereas CFM is unable to accurately reflect skewness and kurtosis parameters in the distribution [8]. Both GCM and CFM fail to approximate multimodal distributions. DDC and GMA are two promising AMs in this context, but they still have some limitations. DDC requires discrete approximation of all continuous distributions [9], and in order to accurately incorporate input correlation, DDC also requires a smaller value of input sequence interval. Smaller sequence intervals increase the overall computational

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time. GMA introduced in [10] takes a great deal of computational time to accomplish the convolution between inputs. Unlike DDC, however, it is free from discretization. This paper adopts CCGMA to establish multimodal distributions where the convolution operations are replaced by cumulant calculations. This technique saves time in achieving convolution operations and further, effectively incorporates multiple input correlations. A mix of Gaussian, non-Gaussian, and discrete distributions for inputs is considered. Multiple input correlation cases are also deliberated. The effectiveness of CCGMA is appraised by comparing its results with CCGCM, CCCFM, DDC, and MCS. A linearized model compatible with CCGMA is developed in Section II. Section III provides a description of CCGMA. Different steps to generate input random samples for MCS are discussed in Section IV. Input uncertainties are modeled in Section V. In Section VI CCGMA is verified in two transmission systems. Finally, the concluding remarks are presented in Section VII. II. P ROPOSED P ROBABILISTIC L OAD F LOW M ODEL For a power system consisting of n number of buses, the basic linearized load flow model [6] is given as,

Further simplification of (5) yields, 2n−m−2 X

xi =

aij yjG + yjNG + yjD + xi0

(7)

bij yjG + yjNG + yjD + zi0

(8)

j=1 2n−m−2 X

zi =

j=1

where yjG , yjNG and yjD represent Gaussian, non-Gaussian and discrete components of yj respectively. The range limits of i for x and z in (2) to (8) are [1, 2n − m − 2] and [1, 2l] respectively. III. CCGMA FOR PLF A. Theoretical Background Probability distribution of a non-Gaussian bus power yjNG , can be approximated by a gjth order Gaussian sum [10] given as, f

yjng

=

gj X

wjk fNk (µjk ,σ2 ) jk

yjng ,

k=1

¯ = K1 ∆y, ¯ ∆z¯ = K2 ∆y¯ ∆x

(1)

¯ ∆y¯ and ∆z¯ are uncertainty component vectors of where ∆x, ¯ (vector of bus voltage magnitudes and angles), y¯ (vector of x real and reactive bus power injections), and z¯ (vector of real and reactive line power flows) respectively, and K1 and K2 are sensitivity matrices. In expanded form, any ith element of the relationships in (1) are expressed as, ∆xi =

2n−m−2 X

aij ∆yj , ∆zi =

2n−m−2 X

j=1

bij ∆yj

(2)

xi = x0i + ∆xi = x0i +

2n−m−2 X

f

yjd

=

ldj X

wjk fNk (djk ,0) yjd

(3)

(9)

,

ldj X

wjk = 1

(10)

k=1

k=1

aij yj − yj0

wjk = 1

k=1

where wjk is the weight factor, fNk (µjk ,σ2 ) yjng is Gaussian jk 2 PDF of the k th component of yjNG , µjk and σjk are mean and th variance of the k component. 2 Parameters wjk , µjk and σjk pertaining to each component for a specific gj are obtained using the EM algorithm [11]. A discrete bus power yjD is considered as a mixture of Gaussian components given as,

j=1

where aij and bij (elements of K1 and K2 ) are known as sensitivity coefficients, m denotes the total number of P |V | buses in the system. ¯ and z¯ using (2) is given as, An element of the vectors x

gj X

where djk and wjk are the discrete values and corresponding probabilities of yjD respectively, ldj is the length of the discrete sequence. First two cumulants are adequate to describe each Gaussian component of (9) and (10). Cumulant based representation of (9) and (10) is given as,

j=1

zi = zi0 + ∆zi = zi0 +

2n−m−2 X

0

bij yj − yj

(4)

j=1

where x0 , y 0 and z 0 are the expected values of x, y and z respectively. Simplification of (3) and (4) yields, xi =

2n−m−2 X

aij yj + xi0 , zi =

2n−m−2 X

j=1

bij yj + zi0

(5)

j=1

where xi0 =

x0i

−

2n−m−2 X j=1

aij yj0 ,

zi0 =

zi0

−

2n−m−2 X j=1

bij yj0

(6)

gj X

ldj X µjk djk D , C = w N . jk k 2 yj σjk 0 k=1 k=1 (11) Let the total number of Gaussian components required to approximate a desired random variable in (7) and (8) be Nr . Value of Nr can be calculated as the product of number of Gaussian components pertaining to all non-Gaussian bus powers and total number of impulses present in the discrete bus powers. Now distribution of a desired random variable can be established from the weighted sum of distributions of Gaussian components obtained in Nr evaluations. In each evaluation, equivalent probability weight is obtained as the product of probability weights of all the Gaussian components in that evaluation. CyjNG =

wjk Nk

PRUSTY et al.: COMBINED CUMULANT AND GAUSSIAN MIXTURE APPROXIMATION FOR CORRELATED PROBABILISTIC LOAD FLOW STUDIES: A NEW APPROACH 73

B. Cumulants of Linear Combination of Correlated Inputs PMCC matrix is independent of change of positive scaling. In case of negative scaling, it alters. Since, aij and bij are either positive or negative; hence, the linear combination of nr correlated inputs of (7) and (8) with PMCC matrix (ρX )nr×nr is generalized as Y = ±X1 ± X2 ± · · · ± Xnr .

(12)

In simplified form (12) is rewritten as, 0 Y = X10 + X20 + · · · + Xnr .

(13)

First two moments of Xi0 in (14) are obtained using (15). Xi0 = ±Xi , i = 1, 2, · · · , nr

(14)

µ0i

(15)

=

±µi , σi0

= σi

where (µi , µ0i ) and (σi , σi0 ) are mean and standard deviation values of (Xi , Xi0 ), respectively. An element ρXi0 Xj0 of the new PMCC matrix (ρX 0 )nr×nr is modified using ! µj µi ρXi Xj . (16) ρXi0 Xj0 = µ0i µ0j A compact form of (13) can be written as Y = Wnr−2 + th 0 where Wnr−2 is evaluated in (nr − 2) step. At any Xnr th i step as shown in Fig. 1, Wi−1 uses parameters of W0 th to Wi−2 . For example, (nr − 1) step uses parameters of W0 , W1 , · · · , Wnr−2 .

In (21) and (22), ρX10 X20 is PMCC between X10 and X20 . For 0 Wi given by (23), parameters σWi and ρWi Xi+2 are obtained using (24) and (25), respectively. Note, W0 = X10 . 0 Wi = Wi−1 + Xi+1 , 1 ≤ i ≤ nr − 2 q 2 2 0 0 σWi−1 σXi+1 + σX σWi = σWi−1 + 2ρWi−1 Xi+1 0

i+1

0 ρWi Xi+2 =

0 0 0 0 σXi+1 ρWi−1 Xi+2 σWi−1 + ρXi+1 Xi+2

σWi

(23) (24) (25)

A flowchart for the implementation of CCGMA to establish distribution of a desired random variable is provided in Fig. 2. IV. G ENERATION OF R ANDOM S AMPLES FOR MCS In PLF using MCS, random samples pertaining to each input random variable is required to be generated. A. Correlated Random Samples Pertaining to Continuous Parametric Distributions For the generation of correlated random samples pertaining to continuous random variables, Nataf transformation and Gauss-Hermite quadrature based algorithm is used [13]. B. Random Samples Pertaining to a Discrete Distribution 1) Define vectors d and p of length ld for discrete values of random variable and associated probability values respectively. 2) Develop uniform random generation vector u. 3) Obtain cumulative sum vector c for elements of p. 4) For i = 1, 2, · · · , ld a) Create vector lk that stores linear indices of the logical expression (u > c (i)) & (u ≤ c (i + 1)). b) Develop random generation vector rg which satisfies the condition rg (lk) = i. V. U NCERTAINTY C HARACTERIZATION A. Probabilistic Modeling of Conventional Generation

Fig. 1.

Linear combination of correlated input random variables.

In step 1, W1 = X10 + X20 .

(17)

Cumulants of W1 are obtained using (18) or (19) [12]. CW1 ,k = A(k)CX10 ,k + CX20 ,k , σX20 ≥ σX10 CW1 ,k = C

X10 ,k

+ A(k)C k

X20 ,k

, σ

X10

A(k) = (1 + ρ) − ρk σX20 ρ = ρX10 X20 , σX20 ≥ σX10 σX10 σX10 ρ = ρX10 X20 , σX10 ≥ σX20 σX20

≥σ

X20

(18)

Uncertainty associated with real power generation of a conventional generator of rated output power P is described by a discrete distribution [5]. In this paper, PMF is assumed to follow Bernoulli distribution and is expressed as, ( F OR, PG = 0 f (PG ) = (26) 1 − F OR, PG = P where PG is the real power generated by the generator.

(19) (20) (21) (22)

where CX10 ,k , CX20 ,k and CW1 ,k are k th order cumulants of X10 , X20 and W1 , respectively.

B. Probabilistic Modeling of PV Generation Real power generation of PV units is modeled as per [6] with slight modification. It is strongly a function of irradiance and cell operating temperature and expressed as PPV = ηg rA (1 − KT ∆TF )

(27)

where irradiance r and forecasting error of PV cell temperature ∆TF are random variables, KT is the temperature coefficient,

74


ηg is PV generation efficiency and A is the total area of the PV module. Introducing two new random variables R and T as functions of r and ∆TF respectively, (27) is rewritten as PPV = RT

(28)

where, R = ηg rA, T = 1 − KT ∆TF . PDF of R is described by beta distribution [14] and the shape parameters a and b are obtained using µR (1−µR ) µR (1−µR ) −1 , b = (1−µR ) −1 a = µR 2 2 σR σR (29) where, µR and σR are mean and standard deviation of R, σR is µR times the coefficient of variation of R. Coefficient of variation of R is assumed as 30%. PDF of T is assumed to follow Gaussian distribution [6]. The linearized version of (28) is expressed as PPV = (µR + ∆R) (µT + ∆T ) = µR µT + µR ∆T + µT ∆R + ∆R∆T

(30)

where µT is mean value of T , ∆R and ∆T are uncertainty components of R and T , respectively. Since random variation of T is small, ∆R∆T in (30) can be neglected, and hence reduces to PPV = µR µT + µR ∆T + µT ∆R

(31)

Since, µT = 1 [6], (31) is rearranged as, PPV = µR ∆T + µR + ∆R = µR (T − 1) + R

(32)

Introducing T 0 = T − 1, (32) is rewritten as PPV = µR T 0 + R.

VI. C ASE S TUDIES AND C OMPARISON OF R ESULTS CCGMA is verified on modified IEEE 14 and 57 bus test systems. The verification includes 1) performance assessment of CCGMA compared to four different methods including MCS and 2) examining capability of CCGMA to handle multiple input correlations. Solution accuracy and computational efficiency are the two performance criteria considered. Input data for the systems are adopted from [16]. Solar parks are installed at the buses with higher load demand except those where either a conventional generator or synchronous condenser is connected. The technical details of the solar parks are provided in Table II. The details of discrete power generations for the test systems are given in Table III. PV penetration level is decided based on local bus load demand. The value of gj is considered as 2 while applying CCGMA. PMCC between Gaussian components pertaining to two different nonGaussian bus powers is assumed same as that among the non-Gaussian bus powers [17]. The following convention is adopted to designate a desired random variable: |Vi | and δi are bus voltage magnitude and bus voltage angle at ith bus, respectively. PLi−j and QLi−j are real and reactive power flows in the line connecting bus-i and bus-j, respectively. Absolute percentage error in estimating standard deviation (σ) for any desired random variable X is calculated using, σX,MCS − σX,AM × 100 (34) eσ = σX,MCS where σX,MCS and σX,AM are standard deviations of X obtained using MCS and AM, respectively. Programming codes are developed in MATLAB 7.10 and implemented on a system with 3.4 GHz Intel (R) core i7 processor and 8 GB of RAM. TABLE II T ECHNICAL D ETAILS OF S OLAR PARKS

(33)

Buses connected to PV systems are modeled as P Q type and reactive power capability of PV systems is assumed negligible, i.e., QPV = 0 [15].

Parameters Bus no. No. of PV units Penetration level

14 Bus System 57 Bus System SP1 SP2 SP1 SP2 13 14 16 17 2 2 2 2 20% (1st unit), 25% (2nd unit)

C. Probabilistic Modeling of Load Demand Load demand is highly time dependent, comprising of both deterministic and random components. The random component arises from forecasting and measurement errors. The distribution of real component of aggregate bus load demand at any bus is modeled as per the three cases ascertained in Table I.

TABLE III D ETAILS OF D ISCRETE P OWER G ENERATIONS IEEE 14 Bus Bus 2 Unit no. 1 2 Capacity (p.u.) 0.22 0.22 FOR 0.10 0.08 Parameters

IEEE 57 Bus Bus 3 Bus 12 1 2 1 2 3 0.22 0.22 0.84 0.84 0.84 0.10 0.08 0.09 0.09 0.07

4 0.84 0.07

TABLE I P ROBABILISTIC D ESCRIPTION OF L OAD D EMAND U NCERTAINTY [5] Load Power Gaussian Discrete One point

Uncertainty Description Small variance (forecasting error)

Parameter Definition

Mean: Specified deterministic value Standard deviation: 7% of mean Series of discrete values along with Large variance their probability of occurrence Precisely forecasted Specified deterministic value value with probability of one

A. The IEEE 14 Bus System Real and reactive load powers at buses 5, 13, and 14 are assumed to follow Gaussian distribution. Loads at remaining buses except 9 (see Table IV) are assumed to follow one-point distribution. PMCC matrix assumed between random variables associated with solar parks is provided in Table V. Real and reactive components of load demands at buses 5, 13, and 14 are


Fig. 2.

Flowchart for establishing distribution of a desired random variable using CCGMA.


correlated as per PMCC matrix assumed in Table VI. Further, PMCC of 0.3 is assumed between random variable R of PV units in solar parks 1 and 2 with that of local real load power demands. The above correlation is considered as base case. Total number of input random variables is 18. TABLE IV D ETAILS OF D ISCRETE L OAD P OWER AT B US 9 (IEEE 14 B US ) Load Power Real Reactive Probability value

0.134 0.075 0.100

Capacity (p.u.) 0.196 0.302 0.348 0.110 0.170 0.196 0.150 0.300 0.250

1 Cumulative Probability

76

R2 0.7 0 1 0.3 0 0 0 0

T20 0 0.2 0.3 1 0 0 0 0

T30 0 0 0 0 0.3 1 0 0.2

R3 0 0 0 0 1 0.3 0.7 0

PD5 1 0.2 0.2 0.5 0 0

PD13 0.2 1 0.5 0 0.5 0.2

PD14 0.2 0.5 1 0 0.2 0.5

QD5 0.5 0 0 1 0.2 0.2

T40 0 0 0 0 0 0.2 0.3 1

R4 0 0 0 0 0.7 0 1 0.3

Cumulative Probability

0.6 0.4 0.2

QD13 0 0.5 0.2 0.2 1 0.5

QD14 0 0.2 0.5 0.2 0.5 1

0.96 0.94 0.92 0.9 0.88

1.035

0.32 0.34

0.2 0.25 0.3 Real Power Flow in Line 7–9 (p.u.)

0.35

0.8 0.6

CCGMA CCGCM CCCFM DDC MCS

1.02 1 0.98

0.4

0.025

0.2 0 −0.005

0

0.005 0.01 0.015 0.02 0.025 Reactive Power Flow in Line 13–14 (p.u.)

0.03

0.03

In all the cases CCGMA and DDC plots are closer to MCS plots. CCGCM and CCCFM plots are significantly biased from MCS plots in case of PL7−9 and QL13−14 . As expected, CCGCM and CCCFM are unable to approximate multimodal distributions. It can be seen from Fig. 4 and 5 that CCGCM leads total probability value more than unity, whereas CCCFM is unable to reflect skewness and kurtosis parameters accurately in the distributions. Performance of CCGMA is compared in Table VII. Average time is calculated as the ratio of total time and total number of desired random variables (62 in case of 14 bus system). Average time required in obtaining distribution of any desired random variable using all the AMs are less than MCS time. However, total time of DDC exceeds MCS time. Average time of CCGMA, CCGCM and CCCFM are nearly same. Further, average eσ in CCGMA is less as compared to CCGCM and DDC. Hence, CCGMA is accurate and efficient to approximate multimodal distributions. However, in the absence of discrete inputs CCGCM and CCCFM are the better options. TABLE VII P ERFORMANCE C OMPARISON OF CCGMA I N 14 B US S YSTEM

1.048

0

0.3

0.2

Fig. 5. Comparison of cumulative probability plots of reactive power flow in line 13–14 in 14 bus system.

1 CCGMA CCGCM CCCFM DDC MCS

0.98

1

CCGMA, CCGCM, CCCFM, DDC, and MCS are applied for PLF. Nr computes to 1600 while applying CCGMA. In DDC, sequence interval of the discrete sequences is set to 0.0001. First six cumulants of desired random variables are considered while applying CCGCM and CCCFM. MCS with 20000 samples is found to be sufficient to produce minimum variation of results. Cumulative probability plots of |V14 |, PL7−9 and QL13−14 using above methods are compared in Fig. 3–5 respectively.

0.8

0.4

Fig. 4. Comparison of cumulative probability plots of real power flow in line 7–9 in 14 bus system.

TABLE VI BASE C ASE L OAD D EMAND PMCC M ATRIX (IEEE 14 B US ) Random Variable PD5 PD13 PD14 QD5 QD13 QD14

1

0.15


T10 0.3 1 0 0.2 0 0 0 0

R1 1 0.3 0.7 0 0 0 0 0

0.6

1.02

0 0.373 0.210 0.200

TABLE V BASE C ASE PV G ENERATION PMCC M ATRIX (IEEE 14 B US ) RandomVariable R1 T10 R2 T20 R3 T30 R4 T40


0.8

1.04 1.045 1.05 Voltage Magnitude at Bus 14 (p.u.)

1.049 1.055

Fig. 3. Comparison of cumulative probability plots of voltage magnitude at bus 14 in 14 bus system.

Performance Criteria Total time (sec) Average time (sec) Average eσ (%)

CCGMA 8.15 0.13 2.58

CCGCM 5.72 0.09 2.64

CCCFM 6.88 0.11 2.50

DDC 37.69 0.61 2.60

MCS 20.06 –

In order to perceive the effect of multiple input correlations on distributions of desired random variables, three correlation conditions are ascertained in Table VIII. Load power factors


at buses 5, 13, and 14 are assumed constant. Effect of C3 correlation on probability density plots of |V13 | and δ13 is depicted in Fig. 6. Effect of C-1, C-2, and C-3 correlations on mean value (µ), standard deviation (σ), skewness (γ1 ), and kurtosis (γ2 ) is compared in Table IX. It can be observed that correlation does not have any effect on µ. However, its effect on σ, γ1 and γ2 is quite noticeable. A constant change in PMCC causes nearly constant change in σ, γ1 and γ2 .

are applied for PLF. Elements of K1 and K2 that represents coupling between (|Vi |, QLi−j ) and injected real bus powers as well as (δi , PLi−j ) and injected reactive bus powers are found to be less than 0.005 and hence are neglected. Nr computes to 5120 while applying CCGMA. Cumulative probability plots of δ16 and PL12−16 using above methods are compared in Fig. 7 and 8, respectively. TABLE X D ETAILS OF D ISCRETE L OAD P OWER AT B US 47 (IEEE 57 B US )

TABLE VIII D IFFERENT C ORRELATION C ONDITIONS C-2 Base case Base case 0, 0.5, 1

Load Power Real Reactive Probability value

C-3 0, 0.5, 1 Base case 0, 0.5, 1

Correlation

Random Variable |V13 |

C-1 δ13 |V13 | C-2 δ13 |V13 | C-3

Probability Density

δ13

400 300

µ

PMCC 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0

1.0548 1.0548 1.0548 −0.2320 −0.2320 −0.2320 1.0548 1.0548 1.0548 −0.2320 −0.2320 −0.2320 1.0548 1.0548 1.0548 −0.2320 −0.2320 −0.2320

0 0.5 1

40 30

200

20

100

10

1.05 1.055 1.06 Voltage Magnitude at Bus 13 (p.u.)

0

0.134 0.070 0.100

Capacity (p.u.) 0.196 0.308 0.349 0.090 0.120 0.150 0.150 0.300 0.250

0.373 0.110 0.200

1

TABLE IX E FFECT OF D IFFERENT C ORRELATION C ONDITIONS σ

γ1

γ2

0.00119 0.00120 0.00121 0.01404 0.01410 0.01416 0.00114 0.00120 0.00126 0.01384 0.01410 0.01436 0.00113 0.00120 0.00127 0.01380 0.01413 0.01444

0.2048 0.2006 0.1966 0.5528 0.5463 0.5399 0.2367 0.2019 0.1749 0.5774 0.5458 0.5169 0.2402 0.2003 0.1706 0.5818 0.5433 0.5089

−0.0393 −0.0297 −0.0334 −0.2346 −0.2306 −0.2266 −0.0430 −0.0387 −0.0298 −0.2484 −0.2304 −0.2143 −0.0447 −0.0334 −0.0260 −0.2512 −0.2289 −0.2094

14 12 10 −0.22 −0.21

−0.26 −0.24 −0.22 −0.2 Voltage Angle at Bus 13 (rad)

Fig. 6. Effect of C-3 correlation on probability density plots of voltage magnitude at bus 13 and voltage angle at bus 13.

B. The IEEE 57 Bus Test System Real and reactive load powers at buses 13, 14, 15, 16 and 17 are assumed to follow Gaussian distribution. Loads at remaining buses except 47 (see Table X) are assumed to follow one-point distribution. PMCC matrix for real and reactive load powers at buses 13, 16, and 17 are same as Table VI. Other correlations are as per IEEE 14 bus test system. Total number of input and desired random variables is 26 and 266 respectively. CCGMA, CCGCM, CCCFM, DDC and MCS


C-1 0, 0.5, 1 Base case Base case

0.8 0.6


0.4

1.02 1 0.98 −0.1

−0.08 −0.06

0.2 0 −0.25

−0.2 −0.15 −0.1 Voltage Angle at Bus 16 (rad)

−0.05

Fig. 7. Comparison of cumulative probability plots of voltage angle at bus 16 in 57 bus system.

1 Cumulative Probability

Correlation PV-PV PV-load Load-load

0.8 0.6 0.4


1.02 1 0.98 −0.3

−0.2

−0.1

0.2 0 −1

−0.8 −0.6 −0.4 −0.2 Real Power Flow in Line 12–16 (p.u.)

0

Fig. 8. Comparison of cumulative probability plots of real power flow in line 12–16 in 57 bus system.

From the comparison of plots in Fig. 7 and 8, similar conclusions as in the 14 bus system are obtained. CCGMA and DDC closely follow the reference whereas CCGCM and CCCFM are inaccurate in relation to approximate multimodal distributions. Performance of CCGMA for a 57 bus system is compared in Table XI. Though average eσ for CCGCM TABLE XI P ERFORMANCE C OMPARISON OF CCGMA IN 57 B US S YSTEM Performance Criteria CCGMA CCGCM CCCFM DDC MCS Total time (sec) 62.37 55.86 57.14 196.84 100.30 Average time (sec) 0.23 0.21 0.22 0.74 Average eσ (%) 4.45 4.43 3.78 4.48 –

78



and CCCFM are less but they are inefficient to approximate multimodal distributions. Simulation run time of CCGMA is less compared to DDC and MCS. It is also capable of approximating multimodal distribution of desired random variables. Hence CCGMA is accurate and efficient to establish multimodal distributions of desired random variables. The effect of C-3 correlation (see Table VIII) on cumulative probability plot of PL9−12 is depicted in Fig. 9. 1 0.8

0 0.5 1

0.6

0.4 0.84

0.4 0.2 0

0.99

0.86

0.3 −0.02

0

0.82 0.0532

0.0552

0.97 0.106

0.02 0.04 0.06 0.08 0.1 Real Power Flow in Line 9–12 (p.u.)

0.12

0.116 0.14

Fig. 9. Effect of C-3 correlation on cumulative probability plot of real power flow in line 9–12.

VII. C ONCLUSION A new PLF technique CCGMA is effectively implemented on two transmission systems with solar parks installed. Input uncertainties pertaining to conventional generations, PV generations, and system load demands are modeled as correlated input random variables. Multiple input correlation cases are effectively incorporated. Results are compared with CCGCM, CCCFM, DDC, and MCS. CCGMA accurately established the multimodal distributions of desired random variables, whereas CCGCM and CCCFM failed to obtain the distributions precisely. DDC is also found to be accurate, but computational time is comparatively high. Effect of different values of correlation on desired random variables is studied for both the systems. It is important to note that the correlation has a significant effect on second and higher order moments without affecting the mean value. Finally, CCGMA can also be applied to other power system uncertainty analysis problems such as reliability assessment. R EFERENCES [1] G. J. Anders, Probability Concepts in Electric Power Systems. New York: Wiley, 1989, pp. 9–10. [2] O. A. Oke, “Enhanced unscented transform method for probabilistic load flow studies,” Ph. D. dissertation, University of Nottingham, 2013. [3] S. Conti and S. Raiti, “Probability load flow using Monte Carlo techniques for distribution networks with photovoltaic generators,” Solar Energy, vol. 81, no. 12, pp. 1473–1481, Dec. 2007. [4] R. N. Allan and A. M. Leite da Silva, “Probabilistic load flow using multilinearizations,” IEE Proceedings – Generation, Transmission and Distribution, vol. 128, no. 5, pp. 280–287, Sep. 1981. [5] R. N. Allan, B. Borkowska, and C. H. Grigg, “Probabilistic analysis of power flows,” Proceedings of the Institution of Electrical Engineers, vol. 121, no. 12, pp. 1551–1556, Dec. 1974. [6] M. Fan, V. Vittal, G. T. Heydt, and R. Ayyanar, “Probabilistic power flow studies for transmission systems with photovoltaic generation using cumulants,” IEEE Transactions on Power Systems, vol. 27, no. 4, pp. 2251–2261, Nov. 2012. [7] L. A. Sanabria and T. S. Dillon, “Stochastic power flow using cumulants and von Mises functions,” International Journal of Electrical Power & Energy Systems, vol. 8, no. 1, pp. 47–60, Jan. 1986.

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B Rajanarayan Prusty (S’14) received the B.Tech. degree in electrical and electronics engineering and the M.Tech. degree in power systems from National Institute of Science and Technology (NIST), Berhampur, India, in 2007 and 2011, respectively. He is currently working towards the Ph.D. degree in the Department of Electrical and Electronics Engineering, National Institute of Technology Karnataka (NITK), Surathkal, India. His research interests include application of probabilistic methods to various power system analysis.

Debashisha Jena (M’10–SM’16) received his Bachelor of electrical engineering degree from University College of Engineering, Burla, India, in 1996, Master’s degree of Technology in electrical engineering in 2004, and Ph.D. degree in control system engineering from the Department of Electrical Engineering, National Institute of Technology, Rourkela, India, 2010. He was awarded a GSEP fellowship in 2008 from Canada for research in control and automation. Currently, he is an Assistant Professor in the Department of Electrical & Electronics Engineering in the National Institute of Technology Karnataka, Surathkal, Mangalore, India. His research interests include evolutionary computation, system identification, and neuro-evolutionary computation.