Modeling of uncertainty in distribution network reconfiguration using ...

Modeling of Uncertainty in Distribution Network Reconfiguration Using Gaussian Quadrature Based Approximation Method Mojtaba Sepehry, Student Member IEEE, Mohammad Heidari-Kapourchali, Student Member IEEE and Visvakumar Aravinthan, Senior Member IEEE Department of Electrical Engineering and Computer Science Wichita State University Wichita, KS, 67260 [email protected] Abstract—Distribution feeder reconfiguration is a cost-effective scheme to improve the system operating condition. Making decisions on the status of switches along the feeder with high penetration of renewable resources requires a tractable and accurate input data analysis. This paper presents a method using Gaussian quadrature to approximate probability distribution function of input variables with a probability mass function. The method is applied to Weibull wind speed and Beta solar irradiance probability distribution functions. These approximated input variables are fed into an energy loss minimization problem to find the best configuration of 33-bus Baran test system in a typical summer day. Group search optimizer, a swarm intelligence optimization method, is applied to solve the problem. In order to analyze the accuracy and efficiency of the proposed method, Monte Carlo simulation and two other approximation methods- bracket midpoint and bracket mean- are used for comparison. Results show the superiority of the method in computational time and its adequate precision. Index Terms—discretization; reconfiguration; uncertainty; Gaussian quadrature

I.

INTRODUCTION

Historically, tradeoff between accuracy and computational burden of a time-step simulation of a large power system model has spawned a great deal of discussion [1]. High penetration of renewable resources with their intermittency and volatility along with active load participation gives rise to the need of accurate and time efficient decision making tools. Quality of resulting decisions relies on quality of information used in the assessments and how that information is processed [2]. Decisions in such an uncertain environment are influenced by either known statistical data or known probability distribution function (PDF) of input variables [3]. Practical limits on the size of most probabilistic models require that probability distributions be approximated by a few representative values and associated probabilities. However, a This work was supported in part by the Power Systems Engineering Research Center (PSerc). Project No: T-53

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criterion for the accuracy of a discrete approximation is to preserve as many as moments of the original distribution as possible [4]. In the power system, depending on the availability of historical data, most of the generation and demand behaviors are modeled using PDFs by which decisions are made. In smart grid era, decision making even comes down to minutely or hourly basis thereby extracted information out of PDFs needs to be precise and tractable. One of the frequent decisions being made in power distribution system is to find the best topology of the grid to meet the operating requirements. Changing the status of normally open switches and normally closed switches is called distribution feeder reconfiguration. This is a well-investigated topic in the literature of past decades [5, 6] and [7] . Traditionally, optimal power distribution feeder reconfiguration was selected and kept fixed throughout a year or a season [8]. However, high penetration of renewable resources and remote controlled switches have enabled and made it justifiable to change the status of switches in shorter periods of time. Value of daily or hourly reconfiguration has been discussed in [9] and [10]. Authors in [9] have evaluated the impact of hourly reconfiguration with variable loads but without renewable resources and ended up with the conclusion that hourly reconfiguration is not effective. However, in [10], adding renewable resources showed the worthiness of the hourly reconfiguration. Since the idea of short term reconfiguration is still in its infancy [10], more accurate estimation of the state of renewable generation is highly required. In this work, known renewable generation PDFs are chosen to represent the uncertainty of renewable generations. Proposed approach of [4] is presented as a tool to extract the most valuable information of PDFs while it keeps the decision making tractable. This approach is based on Gaussian Quadrature method of numerical integration which provides an approximation that preserves the prescribed moments of the original PDF. The output is fed into a reconfiguration optimization problem that minimizes expected daily energy

loss of 33-bus Baran test system. Group search optimizer (GSO) is applied to the problem to find the optimal solution. Monte Carlo simulation and two other approximation methods are run to validate the precision and efficiency of the proposed approach.

seasonal peak load. Each seasonal peak demand is represented as a percentage of the peak load of the system. III.

APPROXIMATION OF A PDF USING GAUSSIAN QUADRATURE METHOD

In this section the approximation of a PDF using Gaussian quadrature method [4] is presented. The common approach to Uncertainty in renewable resources generation is modeled approximate a PDF of an uncertain variable is to divide the using PDFs. Weibull and Beta are two widely accepted range of possible values for the random variable or the range distributions to model wind speed and solar irradiance, of cumulative probabilities into intervals. Each interval is respectively. In this work, demand is modeled as hourly represented by a probability-value pair. The value is selected constant load using IEEE RTS data. as the median or the mean of the interval. The probability shows the chance of the random variable to have a value in the A. Wind speed model and wind turbine output range of the interval. The Weibull wind velocity PDF is represented as In [4] it is proved that approximation of a PDF using these   v  k  k ( k 1) approaches will always underestimate all of even raw and (1) f weibull (v)  k v exp      central moments of the original distribution. If the intervals c   c   Where, k and c are shape and scale parameters of Weibull are represented by their means, the approximation will have distribution, respectively. v (m/s) is wind speed. Given the the same mean as the original distribution but it will wind velocity, power generated by a wind turbine is then underestimate or overestimate all the other odd raw or central moments. These discrepancies cause the deterioration of the represented by a step-wise function as given below: quality of the input variables. To show why these 0  v  vci discrepancies happen we assume the range of possible values 0  for a random variable is divided Into N intervals (R) and they  P v  vci are represented by their mean values. The selected vci  v  vr  w, r v v (2) probabilities and values used to represent these intervals are r  ci Pw   calculated using (5) and (6). P vr  v  vco w , r (5) pi    x dx for i  1, 2, , N  II.

RENEWABLE RESOURCES AND LOAD MODELING

0 vco  v  Where, vci, vr and vco are cut-in wind speed, rated speed

and cut-out speed respectively. Pw,r is the rated output of the wind turbine. B. Solar irradiance model and PV module output The Beta distribution is used to model the randomness of solar irradiance as given below:  1

(   )  s    f beta ( s)  ( )(  )  smax 

 s  1   smax  

 1

  

xi  R x i

x dx

for i  1, 2, , N

pi

N

 g ( x)   g ( x) x dx   R g ( x) x dx 

i 1

N

  pi R g ( x) i 1

(4)

sr  s

Where, Sr and Ps,r are rated irradiance and output of PV module, respectively. C. Load model Demand for each hour of the day is assumed to follow the IEEE RTS system [11]. This test system provides hourly peak load as a percentage of the daily peak load. The daily peak load is also represented as a percentage of its associated

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(7)

i

x dx pi

i

Based on Jensen’s inequality, if g(x) is a strictly convex function then,

x dx  g 

R g x  p i

i

x   R x p dx   g  xi  i  

(8)

i

Combining (7) and (8) results in: N

s  sr

(6)

Where,{x} is the PDF of the random variable. The average of any function g(x) of the random variable is:

(3)

Where, α and β are shape parameters of beta distribution. s and smax (w/m2) are actual and maximum solar irradiance, respectively. Given the solar irradiance, power generated by a PV module is calculated by:

  s  Ps , r  Ps    sr   Ps , r

Ri

 g ( x)   pi g ( xi )

(9)

i 1

Equation (9) shows that using the common approximation methods, the average of a strictly convex function of a random variable is always greater than the average obtained by its approximate mass probability function. The opposite of this statement is true for a strictly concave function. That is, the average of a strictly concave function of a random variable is always less than the average obtained by its approximate mass probability function. When n is a positive and even number, x  a n is a convex function for any constant a. Therefore,

N

 x 2    pi xi2 , i 1

N

( x  x ) 2    pi ( xi  x ) 2

(10)

i 1

Thus, the second raw moments and variance of the discrete approximation underestimate the corresponding moments of the original distribution. The same statement holds true for all the other even raw and central moments. When n is a positive and odd number, x n is convex for positive values of x and concave for negative values of x. So, the second and higherorder odd raw or central moments of the discrete approximation underestimate or overestimate that of original distribution. As mentioned before based on the approximation method used, the first moment may be equal or different from the original one. Gaussian quadrature method of numerical integration can be used to form a more accurate discrete approximation that its moments match the prescribed raw moments of the original distribution. The Gaussian quadrature method is an approximation of a definite integral of the product of a function g(x) and a weighting function ω(x) which is stated as the weighted sum of the values of g(x) at specified points within the domain of integration as follows.

N

N

i 1

k 0

 pi xi ( xi )  0   Ck x k 1

(16)

The same procedure is repeated N times to yield the following set of linear equations:

 x 0  C0     x N 1  C N 1    x N    x C0     x N  C N 1    x N 1    2 N 1 N 2   x  C0     x  C N 1    x     N 1  x  C0     x 2 N  2  C N 1    x 2 N 1   

(17)

Solving (17), the coefficients of the polynomial are obtained. Then, roots of this polynomial are found to get each xi . Finally, each pi is calculated using the first N equations of (13). Because we used a linear combination of the equations of (13) to form (17), the last N equations of (13) holds true for (11) the obtained p s. i

N

b

Now, the second to (N+1)th equations are taken. The first one is multiplied by C0, the second one with C1 and so forth. Then, we add them together which results in:

a g ( x) ( x) dx   wi g ( xi ) i 1

As an example, the presented method is used to form an approximate probability mass function of a Weibull probability distribution representing the uncertainty of the wind speed in an area for a specific hour. The shape and scale parameters of the PDF are K=2 and C=6. Six probability-value pairs are used for the discretization of the PDF. The results are shown in Fig. 1 and presented in Table I. This table also (12) shows the output of an 850 KW VESTAS-V52 wind turbine.

If in (11) we replace function g(x) with raw moments of the random variable x, ω(x) with PDF {x} and weights wi with probabilities pi, the following equation is obtained which is used to match the raw moments of the approximate distribution with prescribed moments of the original distribution. N



 x k    x k {x} dx   pi xik 

for k  0, 1, 2,

i 1

Using (12), The (2N-1) defining equations [15] to find N probability-value pairs to discretize a PDF is as follows:

 p1  p2    pN   p 2 x2    p N x N  p1 x1   2  p2 x22    p N x N2  p1 x1     2 N 1  p2 x22 N 1    p N x N2 N 1   p1 x1

  x0   1   x   x2 

(13)

  x 2 N 1 

Equation (13) shows that N probability-pair values can match the first (2N-1) raw moments of the original distribution. The well-known method to solve (13) is to first defining the following polynomial: N

 ( x)  ( x  x1 )( x  x2 ) ( x  xN )   Ck x k

(14)

k 0

Equation (14) shows that CN  1 and  ( xi )  0 for i  1, 2, , N . The first N equations are taken. The first one is multiplied by C0, the second one with C1 and so forth.

Then, we add them together which results in: N

N

i 1

k 0

 pi ( xi )  0   Ck  x k 

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(15)

Fig. 1. PDF of wind speed and its approximation TABLE I. PROBABILITY-VALUE PAIRS OF THE APPROXIMATION Wind Speed (m/s) Probability Output (KW) 1.0495 0.0960 0 3.2616 0.3592 0 6.2779 0.3891 148.94 9.8930 0.1412 385.31 14.0997 0.0143 660.36 19.2210 0.0002 850

IV.

PROBLEM FORMULATION

In this section daily distribution network reconfiguration problem is presented. In each hour of a day, the PDF of wind speed and solar irradiance is approximated using the proposed

approach to produce different states of wind and solar generation and their associated probabilities. In this regard, the considered assumptions are as follows. Wind speed in different locations of a distribution network is assumed to be 100 % correlated. This is also the case for solar irradiance. Moreover, wind speed and solar irradiance states are considered to be independent. Subsequently, using the produced states, different scenarios of wind and solar generation combination and their probabilities are calculated which are then used as the input of the reconfiguration problem. Distribution reconfiguration problem objective function and constraints for power loss minimization can be written as

Min

nb 24

 Rb I b 1 t 1

Pnt  Q  t n

2

(18)

t b

VntVmt Ynm cosnm   nt   mt  ,

n  N

(19)



n  N

(20)

mN

V V

mN

t n

Y sin  nm    

t m nm

t n

which is the better configuration in the reconfiguration problem. Individual with the minimum fitness value gives the configuration which satisfies decision makers.

t m

,

Fig. 2. Paths of five scroungers moving toward the producer

VI.

RESULTS

Baran 33-bus system [14] with 16 switches including 5 (21) Vmin  Vnt  Vmax, n  N normally open ties is chosen in this paper to evaluate the (22) proposed approach. Fig. 3 shows the initial configuration of Ibt  I max, b , b  N the 33-bus system. Location of switches and renewable th Where, Rb is the resistance of b line. I bt is current of of resources are also shown in Fig. 3. Two VESTAS-V52 wind bth line at time t. nb is number of branches. Pnt and Qnt are turbines with capacity of 850 KW have been located in buses 6 and 12. Two solar arrays with capacity of 600 KW have also th net injected active and reactive power at the n node at time t, been added to buses 16 and 31. All the other required data for respectively. Vnt and  nt are magnitude and angle of the this test system are available in [14]. Hourly generation of th voltage at the n node at time t, respectively. Ynm and  nm wind and solar, representing by Weibull and Beta probability distributions, is approximated by a probability mass function th are magnitude and angle of the admittance between n and as explained in section III . For each PDF, six probabilitymth nodes, respectively. N is set of nodes of the system. Vmin value pairs have been extracted to form the approximate and Vmax are minimum and maximum allowable bus voltages, probability mass function. Combination of generation scenarios produced by this method is fed to group search th respectively. I max, b is the current carrying capacity of b line. optimizer as inputs. Then, GSO finds the optimal Equations (19) and (20) are power flow equations. GSO is configuration that minimizes the expected energy loss of the applied to optimize the above objective function. Optimal system. Table II shows daily losses for three cases. In this table loss value for optimal configuration is compared with solution gives the best combination of switches. losses of initial configuration with and without renewable V. GROUP SEARCH OPTIMIZER resources. Open switches for the optimal configuration are Group search optimizer [13] is based on animal searching given in the last row of the table. Significant loss reduction up behavior. Living creatures search for better resources like food to 28.7 percent is achieved after reconfiguration. Fig. 4 shows and shelter to survive. In optimization problems we try to find hourly losses in a typical summer day for the three cases. the best possible solution or at least find a near optimum one. Computational efficiency and precision of inference from It means there is a similar strategy in both of the above data and models are two main factors in making data-driven behaviors. decisions. Hence, the results of the proposed method are Due to this analogy some operators of group search can be compared with Monte Carlo simulation and bracket midpoint adopted in optimization problem. The population of the GSO [12] and bracket mean [16] in terms of both the accuracy and algorithm is called a group. An individual with the best fitness time complexity as tabulated in tables III and IV. The Monte value is considered as the producer. Then selected producer Carlo simulation is the most accurate approach to model uncertainty. However, its accuracy is highly dependent on the searches for a better and richer resource. 80 % of group members are called scroungers. Scroungers number of samples. As the number of samples increases more search around the producer to change their current position to accurate results at the expense of higher simulation time are a position with a better fitness. Fig. 2 shows how scroungers achieved. As shown in table III, all three Monte Carlo simulations have been run for 100 times to make sure that the move toward a producer positioned in a minimum point. The standard deviation lies in an acceptable range. In table IV, the rest of the group is called rangers. results of proposed method is compared with [12] and [16] for Each GSO group member is representative of the Location different number of probability-value pairs (NP). of normally open switches. GSO searches for better resources

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I.

CONCLUSION

In this paper a reconfiguration problem with the aim of loss minimization was formulated. Time varying nature of renewable resources were considered as known Weibull and Beta distribution probabilities for wind speed and solar irradiance, respectively. Gaussian Quadrature method was used to derive an approximation of PDFs. This approximation preserves the prescribed moments of the original PDF which translates into higher quality input fed to the optimization problem. Accuracy of the results was validated using Monte Carlo simulation and two other approximation methods. This comparison shows the effectiveness of the proposed approach in decision making problems. REFERENCES . Fig. 3. Initial 33-bus network with DGs

Fig. 4. Power loss during a typical day TABLE II. SYSTEM POWER LOSS BEFORE AND AFTER RECONFIGURATION 3.5596 Intial config (MWh) 2.6460 Initial config.with DG (MWh) 1.8865 Optimal config. with DG (MWh) 6-7, 8-9, 13-14, 31-32, 8-14 Open Switches TABLE III. MONTE CARLO SIMULATION (MCS) BASED POWER FLOW RESULTS FOR THE OPTIMAL CONFIGURATION Method Samples Runs Mean (MWh) Std (KWh) Time (S) 50 100 1.8907 7.190 26 MCS 100 100 1.8895 5.495 52 1000 100 1.8895 1.775 523 Proposed 6 1 1.8865 0.106 TABLE IV. POWER FLOW RESULTS USING DIFFERENT APPROXIMATING METHODS FOR THE OPTIMAL CONFIGURATION NP Proposed method [12] (MWh) [16] (MWh) Power Loss Error Power Loss Error Power Loss Error (MWh) (kWh) (MWh) (kWh) (MWh) (kWh) 2 1.9008 5.8 2.0412 146.2 1.8762 18.8 4 1.8876 7.4 1.9150 20.0 1.8870 8.0 6 1.8865 8.5 1.8688 26.2 1.8856 9.4 8 1.8891 5.9 1.8512 43.8 1.8869 8.1

Table IV shows that the proposed method provides better accuracy than [12] and [16] with respect to the high accuracy value provided by MCS. In large networks with higher penetration of DGs the superiority of the proposed method in providing more accurate approximation will be more revealed.

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