Probabilistic load flow calculation by using probability density evolution method

Electrical Power and Energy Systems 99 (2018) 447–453

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Probabilistic load flow calculation by using probability density evolution method Hui Zhanga, Yazhou Xub, a b

T

⁎

School of Information and Control Engineering, Xi’an University of Architecture and Technology, Shaanxi 710055, China School of Civil Engineering, Xi’an University of Architecture and Technology, Shaanxi 710055, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Probabilistic load flow Probability density evolution method Monte Carlo TVD difference scheme

This paper presents a novel framework methodology based on the probability density evolution method (PDEM) for solving the probabilistic load flow (PLF) problem. By leveraging a constructed visual stochastic process, the joint probability density evolution equation of a system statement and random inputs is derived based on the principle of preservation of probability. The probability density function of the system statement can then be numerically solved by means of a TVD-based difference scheme. The proposed method is validated through case studies in which the active power and reactive power consumptions of buses are assumed to obey a normal distribution and Weibull distribution, respectively. The cumulative probability functions of the voltage magnitudes of buses and active power branches are computed using the PDEM with 100 samples. The mean, standard deviation, skewness, and kurtosis are also examined. The comparison to Monte Carlo of 10,000 simulations demonstrates the accuracy and efficiency of the proposed approach and verifies its suitability to solve the PLF problem.

1. Introduction Load flow calculation is a fundamental issue for state estimation and system planning in the electric power industry. In addition to random input loads, the integration of wind energy and photovoltaic power has introduced uncertainty into electric network systems. Therefore, probabilistic load flow (PLF) analysis must be utilized to handle variation in state variables, such as bus voltages and line flows. Historically, load flow analysis considering randomness was first proposed by Borkowska [1] in 1974. Many mathematical methods handling uncertainty have since been introduced to the PLF problem, including fuzzy theory [2,3], set theory[4], interval methods [5], and probability theory. Additionally, unscented transformation [6], Gaussian mixture models [7], and univariate dimension reduction methods [8] have also been adopted to perform PLF analysis. Among these methods, probability theory-based methods are the most popular. Initially, the convolution technique [9] dominated PLF analysis by linearizing load flow equations. Thereafter, owing to the essential limitations of linearization and rapid development of computational capacity, the convolution technique was replaced by other methods that are capable of handling nonlinearity in load flow equations. Among these methods, Monte Carlo simulation [10,11] is particularly notable for its generality. Because the Monte Carlo method

⁎

requires a large number of simulations to ensure accuracy, it is often utilized to verify the accuracy of other methods instead of being used directly in engineering practice. To reduce the time cost of Monte Carlo simulation with simple or direct sampling, various modified sampling strategies (e.g., importance sampling, stratified sampling, and quasiMonte Carlo methods [11–13]) have been proposed. Additionally, other approximate methods have also been developed to balance calculation efficiency and accuracy. Combinations of cumulants and Gram-Charlier expansions [14] or Cornish-Fisher expansions [15] have been employed to calculate the PLF. Furthermore, a series of point estimation methods [16–18] have also been utilized to analyze PLF. Recently, considering the essential randomness of photovoltaic systems and using an index of photovoltaic penetrations, Ruiz-Rodriguez et al. [19] investigated the impact of the size of a single-phase photovoltaic system on voltage unbalance in a secondary radial distribution network using a point estimation method. Hernández et al. [20] assessed the impact of uncertainty in electric vehicles and photovoltaic generation on radial distribution systems. They proposed a general analytical technique based on the Cornish-Fisher expansion and a finite mixture distribution to handle the non-stationarity of loads. It is noteworthy that in the absence of a probabilistic distribution of random input variables, the point estimation method is a more attractive option. In recent years, based on the principle of preservation of

Corresponding author at: School of Civil Engineering, Xi’an University of Architecture and Technology, No. 13, Yanta Road, Xi’an 710055, China E-mail address: [email protected] (Y. Xu).

https://doi.org/10.1016/j.ijepes.2018.01.043 Received 2 September 2017; Received in revised form 8 January 2018; Accepted 28 January 2018 0142-0615/ © 2018 Elsevier Ltd. All rights reserved.

Electrical Power and Energy Systems 99 (2018) 447–453

H. Zhang, Y. Xu

density evolution equation pΦΘ (ϕ,θ,τ ) of Φ(Θ,τ ) and random parameters Θ are derived based on the principle of preservation of probability. Consequently, the PDF of Y can be obtained by integrating the joint PDF pΦΘ (ϕ,θ,τ ) at τ = 1 over the defined domain of the random parameters Θ. In other words, one can obtain the PDF of Y by using the following relationship:

probability, the probability density evolution method (PDEM) was been developed by Li and Chen [21,22]. Initially, it was adopted to predict structural responses in the presence of random excitations [23] or uncertain parameters [24]. Thus far, the PDEM has been successfully applied to seismic response and reliability analysis [25], static random buckling analysis [26], dynamic buckling analysis incorporating structural imperfections [27], fatigue reliability evaluation [28], and several other fields [29]. This paper proposes a novel methodology based on the PDEM to compute PLF. The remainder of this paper is organized as follows. The problem statement and formulas for PLF based on the PDEM and a numerical scheme are introduced in Section 2. Next, case studies with different coefficients of variation for normal inputs and Weibull inputs are presented to validate the proposed method in Section 3. Finally, various conclusions are drawn in Section 4.

pY (y ) =

2.1. Formulation This section presents a general framework for the PLF problem based on PDEM theory. Based on the formal solution for load flow analysis from probability density evolution theory, the joint probability density evolution equations of a system statement and random inputs are derived. Power flow analysis is performed to determine the steady operating state of a power system. Load demands and power generation are typically given, and the voltages (magnitudes and angles) of buses and the active and inactive power through lines can be obtained by solving the governing equations of the power system. The nonlinear load flow equations [13,30] for a power system can be expressed as:

̇ )·ndSΦ dθ −(pΦΘ ϕdτ

(1)

pΦΘ (ϕ,θ,τ + dτ ) = pΦΘ (ϕ,θ,τ ) +

∂pΦΘ (ϕ,θ,τ ) ∂τ

∂τ

dτ

(7)

dϕdθdτ

(8)

Because no new random factors are considered, according to the principle of preservation of probability [19,29], the incremental probability in Ω must be equal to the probability flowing into Ω through ∂Ω:

(2)

∫Ω

∂pΦΘ (ϕ,θ,τ ) ∂τ

Φ × ΩΘ

dϕdθdτ = −

∫∂Ω

Φ × ΩΘ

̇ )·ndSΦ dθ (pΦΘ ϕdτ

(9)

By using the Gaussian integration theorem, the right side of Eq. (9) can be converted into an integration over {ΩΦ × ΩΘ} and we have:

−

∫∂Ω

Φ × ΩΘ

̇ )·ndSΦ dθ = − (pΦΘ ϕdτ

∫Ω

∂ (pΦΘ Y (Θ)) Φ × ΩΘ

∂ϕ

dϕdθdτ

(10)

By substituting Eq. (10) into Eq. (9) and noting that Y (Θ) is independent over ϕ , one obtains:

(3)

∫Ω ×Ω

Here, a virtual time τ is introduced to form a stochastic process Φ(Θ,τ ) . One can see that the random variable of interest can be obtained using Y (Θ) = Φ(Θ,τ = 1) . Additionally, the deviation of Φ with respect to τ can be obtained as follows:

∂Φ Φ̇ (τ ) = = Y (Θ) ∂τ

∂pΦΘ (ϕ,θ,τ )

Therefore, the incremental probability in Ω during dτ is:

where X denotes the set of determine input parameters (e.g., system admittance), Θ is the set of random parameters (e.g., active or inactive injections of buses), and G is a transformation from considered factors to the statement variables of interest based on Eq. (1), which typically needs to be solved using numerical methods [31]. Because load flow calculation is essentially modeled as a steadystate problem, in order to employ the time-dependent PDEM to perform PLF analysis, a virtual stochastic process [25] must first be constructed as follows:

Φ(Θ,τ ) = Y (Θ)·τ = G (X,Θ)·τ

(6)

where ϕ ̇ is the deviation of ϕ with respect to τ . Let ∂ΩΦ be the boundary of ΩΦ , where SΦ denotes the area element of ∂ΩΦ , n is the norm vector of ∂ΩΦ , and dθ denotes the volume element of ΩΘ. Additionally, the minus symbol indicates that the probability flows into Ω. Next, the joint PDF pΦΘ (ϕ,θ,τ ) at τ + dτ can be expressed by first order expansion as:

where x denotes the state vector of nodal voltages and angles, y represents the vector of real and reactive power injections, z is the vector of real and reactive line flows, and g (·) and h (·) are nonlinear power injection functions and line flow functions, respectively. When load demand is uncertain, both the input and output of the power system need to be treated as random variables. Let Y be the state variable of interest. Then, its formal solution can be written as:

Y (Θ) = G (X,Θ)

(5)

The strategy for the derivation of the joint PDF pΦΘ (ϕ,θ,τ ) is formally similar to the conservation equations in computational fluid dynamics. In this study, the statement variable of a power system is modelled as a virtual stochastic process, where its actual statement corresponds to a particular instant with the virtual time being equal to 1. Because there are neither additional new random factors nor disappearing random factors in the virtual evolution process under consideration, {Φ(Θ,τ ),Θ} defined at {ΩΦ × ΩΘ} is a preserved stochastic system, where ΩΦ is the defined domain of the virtual stochastic process and ΩΘ is the defined domain of the random parameters under consideration. Now, let Ω be an arbitrary subdomain in {ΩΦ × ΩΘ} , where ∂Ω is the boundary of Ω. We inspect the change in probability in Ω and the probability flowing into Ω through ∂Ω during the virtual time interval dτ . First, the probability increment through ∂Ω during dτ is denoted:

2. Problem statement

y = g (x) z = h (x)

∫Θ pΦΘ (ϕ,θ,τ = 1) dθ

D

Θ

⎧ ∂pΦΘ (ϕ,θ,τ ) + Y (Θ) ∂pΦΘ (ϕ,θ,τ ) ⎫ dϕdθdτ = 0 ⎨ ⎬ ∂τ ∂ϕ ⎩ ⎭

(11)

Considering the arbitrary nature of Ω, the joint probability density evolution equation pΦΘ (ϕ,θ,τ ) is derived as:

∂pΦΘ (ϕ,θ,τ ) ∂τ

(4)

+ Y (Θ)

∂pΦΘ (ϕ,θ,τ ) ∂ϕ

=0

(12)

The corresponding initial condition can be written as:

For time-dependent loads, there is no need to construct the virtual stochastic process. One can directly solve the transient governing equations to obtain the probability density function (PDF) of interest by using the PDEM. With the help of the stochastic process Φ(Θ,τ ) , the PDF pY (y ) of Y can be determined by using the PDEM. First, the joint probability

pΦΘ (ϕ,θ,τ )|τ = 0 = δ (ϕ) pΘ (θ)

(13)

where δ (ϕ) is Dirac’s delta function. The initial condition implies that the probability distribution of random parameters was determined at the initial state. 448

Electrical Power and Energy Systems 99 (2018) 447–453

H. Zhang, Y. Xu 1

1

pj(k + 1)= pj(k ) − 2 (δY −|δY |)Δpj(+k )1 − 2 (δY + |δY |)Δpj(−k )1 2

− where

2

1 (|δY |−δ 2Y 2) ⎛ψj + 1 Δpj(+k )1 −ψj − 1 Δpj(−k )1 ⎞ 2 2 2 2 2

⎝

pj(k )

(15)

⎠

is the discrete joint PDF value at the k-th grid of τ and j-th

grid of ϕ , where Δpj(+k )1 = pj(+k )1 −pj(k ) and Δpj(−k )1 = pj(k ) −pj(−k )1. Additionally, 2

2

δ = Δτ /Δϕ denotes the grid ratio of τ to ϕ . The flux limiter is suggested as follows:

ψj + 1 ⎛r + 1 ,r − 1 ⎞ = H (−f ) ψ0 ⎛r + 1 ⎞ + H (f ) ψ0 ⎛r − 1 ⎞ 2 ⎝ j+ 2 j+ 2 ⎠ ⎝ j+ 2 ⎠ ⎝ j+ 2 ⎠ ⎜

r+

j+ 1 2

=

⎟

pj(+k )2 −pj(+k )1 pj(+k )1 −pj(k )

⎜

, r−

j+ 1 2

=

⎟

⎜

⎟

(16a)

pj(k ) −pj(−k )1 pj(+k )1 −pj(k )

(16b)

ψ0 (r ) = max(0,min(2r ,1),min(r ,2))

(16c)

where H (·) is the unit step function and ψj − 1 obeys the same rule. 2

(4). Compute the cumulative distribution function CDF of interest PY (y ) by integrating pΦΘ (ϕ,θ,τ = 1) with respect to the random parameter set Θ and ϕ . (5). The mean, standard deviation, skewness, and kurtosis are computed using their definitions. Recently, an ensemble evolution numerical method was proposed for solving the generalized density evolution equation [32]. A more detailed description and discussion can be found in [23,33]. 3. Case studies In order to validate the accuracy and efficiency of the proposed approach for PLF analysis with coefficients of variation (CVs) of 5%, 10% and 15%, we first examined Case30. In Case30, we computed input variables obeying a normal distribution and Weibull distribution, and compared the results to Monte Carlo simulation with 10,000 samples. To further investigate the performance of the proposed computational procedure for large systems, the proposed method was also tested on systems having 300 and 1354 buses (i.e., Case300 and Case1354pegase). Because the results and trends of change for these cases were similar to those of Case30, to avoid redundancy, the predicted cumulative probability function and statistical parameters are presented only for Case30. However, a detailed computation time comparison is provided later.

Fig. 1. Flowchart of PLF computation by PDEM.

Consequently, the marginal distribution of interest pY (y ) can be obtained by integrating pΦΘ (ϕ,θ,τ ) with respect to θ :

pY (y ) =

∫Ω

Θ

pΦΘ (ϕ,θ,τ = 1) dθ

(14)

Because the PDF pY (y ) is given, the statistical moments or reliability of the state variable can be easily computed using their definitions. 2.2. Numerical method

3.1. Case30 study with normal distribution inputs

In most cases, one cannot obtain analytical solutions for the probability density evolution equation shown in Eq. (12). Typically, the finite difference method is employed to solve this problem. Chen and Li proposed a type of TVD scheme with satisfactory results [23]. For the sake of comprehension, a flowchart outlining the PDEM is presented in Fig. 1 and the corresponding numerical procedures are briefly summarized below.

The Case30 bus system consists of 30 load buses, 41 transmission lines, and 6 generator units. The active power consumed by each load bus was assumed to follow normal and Weibull distributions with means equal to the values provided in MATPOWER [34]. The CVs were specified as 5%, 10%, and 15%. The reactive power consumption of each load was determined in such a way that its power factor remained constant [35]. Load power factors can also be computed from the bus load data provided by MATPOWER. For the sake of simplicity and without loss of generality, random inputs were assumed to be uncorrelated. To remove correlation, an orthogonal transform [36] or POD technique can be employed to transform correlated random variables into independent random variables. Then, one can generate samples in the independent random domain, meaning correlated samples of interest can be simulated by summing the series based on independent random variables. Subsequently, the PDEM can be used to solve PLF problems. As mentioned above, the mean reactive power consumption of each

(1). A discrete point set Θ = {θ1,q,θ2,q···θs,q} is first obtained by partitioning the random parameter space Θ, where q is the total number of discrete points and s is the number of random parameters. (2). Solve Eq. (1) using the point set Θq to obtain samples of the state variables of interest. (3). Substitute the sample values into Eq. (12). Then, the numerical solution of pΦΘ (ϕ,θ,τ ) could be calculated using the aforementioned TVD scheme, which is written as:

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H. Zhang, Y. Xu

(a) CV=5%

(a) CV=5%

(b) CV=10%

(b) CV=10%

(c) CV=15%

(c) CV=15%

Fig. 3. CDF of PF of branches 6–8 with normal inputs.

Fig. 2. CDF of VM of bus 3 with normal inputs.

(VM) of buses and real power injected at the “from” bus end (PF), were obtained. Finally, the PDEM procedure was performed to compute the PLF. For the sake of brevity, the results for bus 3 are presented in Fig. 2(a)–(c), while branches 6–8 are presented in Fig. 3(a)–(c), respectively. The results for other buses were very similar. The CDF of Bus 3 was also compared to the results of the Monte Carlo method with 10,000 samples. To further quantitatively evaluate the proposed method, an index called average root mean square

load was taken as the default value from MATPOWER and its standard deviation was calculated with CVs of 0.05, 0.1, and 0.15. Then, for each CV, 100 samples for PDEM were generated using MATLAB. Accordingly, 100 samples of reactive power consumption for each load were computed based on the principle of maintaining a constant power factor. Then, MATPOWER code was employed to perform deterministic analysis in which samples of load flow, such as the voltage magnitude 450

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Table 1 ARMS (×100%) of voltage magnitude (VM) and real power (PF) for Case30. Probability distribution of random input variables

Gaussian Weibull

VM of bus 3

PF of branches 6–8

CV = 5%

CV = 10%

CV = 15%

CV = 5%

CV = 10%

CV = 15%

0.070 0.220

0.227 0.320

1.31 1.20

0.087 0.244

0.368 0.400

0.417 0.353

Table 2 Statistical parameters of Case30 (Normal inputs, CV 5%). Statistical parameters

VM of bus 3

Mean Std. Skewness Kurtosis

PF of branches 6–8

MC

PDEM

MC

PDEM

0.9831 0.0006 0.0046 3.0907

0.9832 0.0006 −0.0033 3.1070

24.8284 1.2896 −0.0236 3.0736

24.7310 1.3279 0.0669 3.0895

Table 3 Statistical parameters of Case30 (Normal inputs, CV 10%). Statistical parameters

VM of bus 3

Mean Std. Skewness Kurtosis

PF of branches 6–8

MC

PDEM

MC

PDEM

0.9831 0.0012 −0.0141 3.0924

0.9834 0.0010 0.0957 2.6930

24.8356 2.5791 −0.0218 3.0736

24.7598 2.2319 −0.0683 2.8303

(a) CV=5%

Table 4 Statistical parameters of Case30 (Normal inputs, CV 15%). Statistical parameters

VM of bus 3

Mean Std. Skewness Kurtosis

PF of branches 6–8

MC

PDEM

MC

PDEM

0.9831 0.0019 −0.0975 3.0135

0.9811 0.0020 −0.7364 3.3333

24.8330 3.9535 0.0509 3.0078

24.6633 4.1518 −0.0429 3.1147

(ARMS) is defined as [14]: N

∑ ARMS =

(b) CV=10%

(piMC −piPDEM )2

i=1

N

(17)

where piMC is the i-th point value on the cumulative probability distribution function computed by the Monte Carlo method with 10,000 simulations, piPDEM is the i-th point value on the cumulative probability distribution function calculated by the proposed PDEM, and N denotes the number of discrete points. The quantitative errors in the voltage magnitude of bus 3 are listed in Table 1. Fig. 3(a)–(c) display the CDFs of PF of branches 6–8 computed by the PDEM and Monte Carlo method. Quantitative errors are also presented in Table 1. One can see that for the normal case the PDEM achieves high efficiency with a small error of less than one-tenth of a percent, while the number of samples for the PDEM was only one percent of those for the Monte Carlo method. For different CVs, the predicted values for the mean, standard deviation, skewness, and kurtosis of voltage magnitude and active power are listed in Tables 2, 3, and 4.

(c) CV=15% Fig. 4. CDF of VM of bus 3 with Weibull inputs.

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Table 5 Statistical parameters of Case30 (Weibull inputs, CV 5%). Statistical parameters

Mean Std. Skewness Kurtosis

VM of bus 3

PF of branch 6–8

MC

PDEM

MC

PDEM

0.9831 0.0006 0.6265 3.7226

0.9831 0.0006 0.5751 3.5072

24.8215 1.2920 −0.8967 4.3324

24.7043 1.4061 −0.8851 3.7700

Table 6 Statistical parameters of Case30 (Weibull inputs, CV 10%). Statistical parameters

(a) CV=5%

Mean Std. Skewness Kurtosis

VM of bus 3

PF of branch 6–8

MC

PDEM

MC

PDEM

0.9831 0.0012 0.4305 3.3361

0.9831 0.0011 0.4236 3.7849

24.8092 2.6425 −0.6554 3.6917

24.9329 2.4111 −0.6506 3.1868

Table 7 Statistical parameters of Case30 (Weibull inputs, CV 15%). Statistical parameters

Mean Std. Skewness Kurtosis

VM of bus 3

PF of branch 6–8

MC

PDEM

MC

PDEM

0.9831 0.0018 0.3184 3.1195

0.9834 0.0020 0.0813 2.6522

24.8108 3.8783 −0.5163 3.2990

24.8726 4.0922 −0.3231 2.5903

Table 8 Computation time comparison of the PDEM to the Monte Carlo method (seconds).

(b) CV=10%

Case

PDEM

Monte Carlo

Case30 Case300 Case1354pegase

3.6 11.9 12.7

262.1 1503.5 2171.1

their relationship to the given mean and standard deviation. Then, 100 samples in a Weibull distribution were generated using “wblrnd” in MATLAB. For comparison, the means and CVs of the Weibull cases were the same as those of the normal distribution cases. The detailed scale and shape parameters are not listed for each bus for the sake of brevity. Finally, the PDEM procedures were performed again to calculate PLF. The CDFs of the voltage magnitude of bus 3 with Weibull inputs are presented in Fig. 4(a)–(c). A comparison to the Monte Carlo method with 10,000 samples is presented in the same figures. The CDFs of the PFs of branches 6–8 computed by the PDEM and Monte Carlo method are displayed in Fig. 5(a)–(c). Additionally, the quantitative errors for the Weibull cases are listed in Table 1. One can see that for the Weibull cases, the results of the PDEM have good agreement with those of the Monte Carlo method, but the quantitative error increases with an increased CV. For the cases of different CVs, the predicted values for the mean, standard deviation, skewness, and kurtosis of voltage magnitude and active power are listed in Tables 5, 6, and 7. To further analyze the performance of the proposed computational procedure for large systems, the proposed PDEM was also used to investigate Case300 and Case1354pegase. As mentioned previously, to avoid redundancy, only a detailed computation time comparison is presented in Table 8. It is easy to see that the PDEM is more efficient for large systems.

(c) CV=15% Fig. 5. CDF of PF of branches 6–8 with Weibull inputs.

3.2. Case study with Weibull distribution inputs Considering renewable energy sources such as wind farms, the Case30 system is analyzed with inputs following a Weibull distribution in this section. The mean reactive power consumption of each load was taken as the default value from MATPOWER and the standard deviation values were calculated with CVs of 0.05, 0.1, and 0.15. We used “fsolve” in MATLAB to evaluate scale and shape parameters based on 452

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4. Conclusions [11]

Based on the principle of preservation of probability, the governing equation of the joint PDF of PLF with random inputs was derived and a TVD difference scheme was implemented to obtain numerical solutions. Subsequently, the PDF of interest, namely PLF, could be obtained by integrating the joint PDF with respect to its random parameters. In order to validate the efficiency and accuracy of the proposed method, Monte Carlo simulations were performed for comparison in several case studies. With the assumption of normal and Weibull distributions for the power consumption of buses, cases with different system scales and CVs were studied. For validation, the CDFs of the voltage magnitudes of buses and active power at the “from” bus ends were obtained and compared to the Monte Carlo method with 10,000 samples. The mean, standard deviation, skewness, and kurtosis were also validated. ARMS was also employed to examine the proposed method. A computation time comparison demonstrated the efficiency of the PDEM. In conclusion, the predicted results from the PDEM have good agreement with those of the Monte Carlo method. It was found that error was largely dependent on CVs. A larger CV increases error. Additionally, with an increase in the CV, the errors for normal inputs were close to those of Weibull inputs.

[12]

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[15]

[16] [17] [18] [19]

[20]

[21] [22]

Acknowledgements

[23]

The support of the Natural Science Foundation of China (Grant No. 51578444) and Program for Innovative Research Team in University of Ministry of Education of China (Grant No. IRT_17R84) is greatly acknowledged.

[26]

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