- Email: [email protected]
An affine arithmetic-based algorithm for radial distribution system power flow with uncertainties
Electrical Power and Energy Systems 58 (2014) 242–245
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Short Communication
An affine arithmetic-based algorithm for radial distribution system power flow with uncertainties Wei Gu a,⇑, Lizi Luo a, Tao Ding b, Xiaoli Meng c, Wanxing Sheng c a
School of Electrical Engineering, Southeast University, Nanjing 210096, China Department of Electrical Engineering, Tsinghua University, Beijing 100084, China c China Electric Power Research Institute, Beijing 100192, China b
a r t i c l e
i n f o
Article history: Received 15 June 2013 Received in revised form 30 December 2013 Accepted 16 January 2014
Keywords: Affine arithmetic Interval arithmetic Radial distribution system power flow Uncertainties
a b s t r a c t This letter presents an algorithm for radial distribution system power flow in the presence of uncertainties. To reduce the overestimation of bounds yielded by correlation of variables in interval arithmetic (IA), affine arithmetic (AA) is applied in this study to carry out tests of distribution system power flow. Compared with the algorithm based on IA, the proposed algorithm narrows the gap between the upper and lower bounds of the power flow solution. IEEE 33-bus and 69-bus test systems are used to demonstrate the effectiveness of the proposed algorithm. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction With an increasing number of renewable power generation systems connected to distribution systems, power injections are increasingly difficult to model, which makes it difficult for traditional deterministic methods to calculate the power flow. To deal with the uncertainties in power systems, many modeling approaches have been proposed in Refs. [1–5]. Moreover, popular power flow algorithms, such as Newton-Raphson and Fast Decoupled algorithms, generally fail to converge when analyzing the distribution system for its radial structure and high R/X ratio [6]. Therefore, the backward/forward sweep algorithm is used to solve this problem. In Ref. [7], an IA-based backward/forward sweep algorithm is developed to calculate the power flow of radial distribution systems, which uses ranges restricted by upper and lower bounds to express the uncertainties. However, the ranges estimated by IA tend to be too large, especially in complicated expressions or long iterative computations, because they ignore the correlation of different variables [8]. An AA-based algorithm is proposed to reduce this overestimation of bounds. This algorithm uses the affine form instead of the interval form to describe the uncertainties of power injections, thus accounting for correlations between different variables.
2. Affine arithmetic-based algorithm for radial distribution system power flow 2.1. Concepts of affine arithmetic In affine arithmetic, a quantity x is represented by an expression of the form
^x ¼ x0 þ x1 e1 þ þ xn en
ð1Þ
which is an affine expression of noise symbols ei with real coefficients xi. Each noise symbol ei is a symbolic real variable whose value is unknown except that it is restricted to the interval [1, +1] and is independent from other noise symbols. The coefficient x0 is called the central value of the affine form of ^x. The coefficients x1, . . ., xn are the partial deviations associated with the noise symbols e1, . . ., en in ^x. The number n of noise symbols depends on the affine form. Different affine forms use a different number of noise symbols, some of which may be shared with other affine forms. Affine forms provide interval bounds for the corresponding quantities: If a quantity x is represented with the affine form ^ x as above, then x 2 [x0 rx, x0 + rx]. Here rx = |x1| + . . . + |xn| is called the total deviation of ^ x [9]. 2.2. Calculation rules of affine arithmetic applied in complex field
⇑ Corresponding author. Tel.: +86 13814005169; fax: +86 25 87796196. E-mail address: [email protected] (W. Gu). http://dx.doi.org/10.1016/j.ijepes.2014.01.025 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
Two types of uncertainty sources exist in the power injections: active and reactive power. Because reactive power is represented
243
W. Gu et al. / Electrical Power and Energy Systems 58 (2014) 242–245
by an imaginary number, complex affine forms are used for the analysis of power flow with these uncertainties. Consider two complex quantities x and y represented by the complex affine forms
^x ¼ x0 þ x1 e1 þ þ x2n e2n ^ ¼ y0 þ y1 e1 þ þ y2n e2n y
^x þ y ^ ¼ ðx0 þ y0 Þ þ ðx1 þ y1 Þe1 þ þ ðx2n þ y2n Þe2n
ð3Þ
^x y ^ ¼ ðx0 y0 Þ þ ðx1 y1 Þe1 þ þ ðx2n y2n Þe2n
ð4Þ
^x y ^¼
i¼1
!
2n X y0 þ yi ei
!
ð5Þ
i¼1
where e2n+1 is a new noise symbol that is created during the computation. For any complex number z = a + jb, the function f() is defined as f(z) = |a| + j|b|.
" # 2n 2n X X ^x 1 ¼ x0 C þ ðC xi x0 yi Þei þ D x0 e2nþ1 þ f ðxi Þ ^ AB y i¼1 i¼1 " # 2n X 1 f yi þ f ðDÞ e2nþ2 AB i¼1
ðk1Þ
Þ
ð8Þ
where Si is the power injection at node i expressed by affine forms ðk1Þ in Step 2 and U i is the calculated voltage at node i during the (k 1)th iteration. Step 5: The details of this step are the same as the traditional backward/forward sweep except that the complex arithmetic has been replaced by complex affine arithmetic. At iteration k, ðkÞ the nodal affine form voltage U i is obtained by this step. ðkÞ Step 6: At the end of iteration k, the distance between U i and ðk1Þ Ui , henceforth denoted by di, needs to be calculated for all nodes i. Consider an affine form voltage expressed as
ð9Þ
"
n n X X ½U; U ¼ ½u0 rU ; u0 þ r U ¼ u0 f ðui Þ; u0 þ f ðui Þ i¼1
ð6Þ
2.3. Steps to radial distribution system power flow analysis The basic power flow analysis method used in this study is the backward/forward sweep power flow algorithm. However, to represent the uncertainties of the active and reactive power, power injections of each node have been treated as affine forms rather than fixed numbers, and consequently, the complex arithmetic has been replaced by complex affine arithmetic. Concrete steps are as follows: Step 1: Number the nodes of the distribution system and define the node count as N. Step 2: Transform the given power injections into affine forms with 2N noise symbols, with each node corresponding to two noise symbols. For example, assuming the power injection of node m is P + jQ and has a ±k% tolerance, it can be represented by
ð7Þ
The irrelevant noise symbols e1, . . ., e2m-2, e2m+1, . . ., e2N cannot be found in the formula because the coefficients associated with them just equal to zero.
# ð10Þ
i¼1
^ and f() has the same definition as where rU is the total deviation of U formulation (5)–(6). After transforming all the affine form voltages into interval forms according to (10), di is calculated by
ðkÞ ðk1Þ ðkÞ ðk1Þ di ¼ max U i U i ; U i U i
where e2n+1, e2n+2 are new noise symbols that are created during the P computation. These new variables are defined as A ¼ y0 2n i¼1 f ðyi Þ, pffiffiffiffi pffiffiffiffi P2n AB 1 AB B ¼ y0 þ i¼1 f ðyi Þ, C ¼ BþAþ2 AB y0 and D ¼ BþA2 , and the 2AB 2AB function f() has the same definition as in multiplication.
^Sm ¼ P þ jQ þ ðP e2m1 þ jQ e2m Þ k%:
ðkÞ
Ii ¼ ðSi =U i
its corresponding interval form with upper and lower bounds can be represented by
i¼1
i¼1
ðkÞ
Step 4: At iteration k, the nodal current injection Ii at node i can be calculated by
^ ¼ u0 þ u1 e1 þ þ un en U
" # " # 2n 2n 2n X X X ¼ x0 y 0 þ ðx0 yi þ y0 xi Þei þ f ðxi Þ f ðyi Þ e2nþ1 i¼1
With the preceding work above, the following steps are implemented for the iterative solution of the system.
ð2Þ
where x0 and y0 are complex numbers and represent the central value of each complex affine form. When i is odd, xi and yi are real coefficients that represent the uncertainties derived from the active power injections. When i is even, xi and yi are imaginary coefficients that represent the uncertainties derived from the reactive power injections. Based on the calculation rules of affine arithmetic applied in real number field described in Ref. [9], addition, subtraction, multiplication and division of these two complex affine forms can be derived as follows:
2n X xi ei x0 þ
Step 3: Obtain the given voltage at the root node and set the initial voltages to all the other nodes as 1 p.u.
ð11Þ
If max(di), i = 1, 2, . . ., N, is less than the specified voltage error tolerance limit, the power flow analysis has converged. Otherwise the algorithm goes back to Step 4 to proceed with next iteration. 3. Case studies To demonstrate the effectiveness of the proposed algorithm, it is implemented on IEEE 33-bus and 69-bus test systems. In this letter, the voltage error tolerance limit for convergence of the iterative process is 104 p.u. First, the proposed algorithm is used to analyze the 33-bus system with an assumed ±20% tolerance on the given power injection of each bus. The power flow solutions obtained by different algorithms are compared in Figs. 1 and 2, with Fig. 1 showing the bus voltage magnitude bounds and Fig. 2 depicting the bus voltage angle bounds. Algorithm A is the algorithm proposed by this letter, algorithm B represents the interval arithmetic-based algorithm [7] and algorithm C is the Monte Carlo method with 10,000 trials. Obviously, the ranges of solutions obtained by both the AA-based and IA-based algorithms completely contain the bounds obtained by the Monte Carlo method. This overlap demonstrates that both of the first two algorithms can give proper approximations of the power flow solution bounds. Notice also that the solution ranges of the AA-based algorithm are narrower than that of IA-based algorithm, which proves that the AA-based algorithm is able to reduce the overestimation of bounds compared with the IA-based algorithm, due to its accounting for correlations between different variables in the distribution system. Next, both the 33-bus and the 69-bus test systems with uncertainty tolerances from ±10% to ±50% are analyzed. The maximum errors of bus voltage magnitudes and angles under different
244
W. Gu et al. / Electrical Power and Energy Systems 58 (2014) 242–245
Fig. 1. Bus voltage magnitude bounds in the IEEE 33-bus test system.
Fig. 2. Bus voltage angle bounds in the IEEE 33-bus test system. Table 1 Error analyses and iteration numbers under different uncertainty tolerances. Uncertainty tolerance (%)
10 20 30 40 50
Test system
IEEE-33 IEEE-69 IEEE-33 IEEE-69 IEEE-33 IEEE-69 IEEE-33 IEEE-69 IEEE-33 IEEE-69
Maximum error of bus voltage (magnitudes, p.u.)
Maximum error of bus voltage (angles, deg)
Iteration number
AA
IA
AA
IA
AA
IA
0.0096 0.0028 0.0195 0.0057 0.0296 0.0086 0.0400 0.0115 0.0506 0.0144
0.0130 0.0031 0.0272 0.0062 0.0427 0.0095 0.0601 0.0127 0.0800 0.0161
0.4822 0.1422 0.9826 0.2863 1.5027 0.4323 2.0436 0.5801 2.6069 0.7299
0.7591 0.1627 1.5741 0.3269 2.4611 0.4938 3.4505 0.6644 4.5870 0.8383
4 3 4 3 5 3 5 3 5 3
6 3 7 4 7 4 8 4 9 4
uncertainty tolerances are shown in Table 1, where the error specifically refers to the bound’s deviation from the deterministic power flow solution. It can be easily observed that the bounds increasingly deviate from deterministic power flow solutions as the uncertainty tolerances increase. Moreover, the bounds obtained by the AA-based algorithm consistently deviate less than that of the IA-based algorithm when the other conditions are the same. In addition, the iteration numbers needed for different scenarios are provided in the last two columns of Table 1. It is obviously that the AA-based algorithm always has less iteration numbers than the IA-based algorithm. 4. Conclusion Taking the uncertainties of power injections into consideration, an algorithm for radial distribution system power flow
based on affine arithmetic is proposed and tested in this letter. Compared with the interval arithmetic-based algorithm, the proposed algorithm accounts for correlations between different variables and thus obtains power flow solutions with narrower ranges. Acknowledgments This work was supported by the National High Technology Research and Development Program of China (863 Program Grant 2012AA050210), the National Science Foundation of China (51277027), the Natural Science Foundation of Jiangsu Province of China (SBK201122387), the Fund Program of Southeast University for Excellent Youth Teachers, the Southeast University Key Science Research Fund, the State Grid Corporation of China (dz71-13-036).
W. Gu et al. / Electrical Power and Energy Systems 58 (2014) 242–245
References [1] Pereira LES, Da Costa VM, Rosa ALS. Interval arithmetic in current injection power flow analysis. Int J Electr Power Energy Syst 2012;43(1):1106–13. [2] Vaccaro A, Canizares CA, Villacci D. An affine arithmetic-based methodology for reliable power flow analysis in the presence of data uncertainty. IEEE Trans Power Syst 2010;25(2):624–33. [3] Mohammadi M, Shayegani A, Adaminejad H. A new approach of point estimate method for probabilistic load flow. Int J Electr Power Energy Syst 2013;51: 54–60. [4] Heleno M, Sumaili J, Meirinhos J, et al. A linearized approach to the symmetric fuzzy power flow for the application to real systems. Int J Electr Power Energy Syst 2014;54:610–8.
245
[5] Marinopoulos AG, Alexiadis MC, Dokopoulos PS. Energy losses in a distribution line with distributed generation based on stochastic power flow. Electr Power Syst Res 2011;81(10):1986–94. [6] Singh S, Ghose T. Improved radial load flow method. Int J Electr Power Energy Syst 2013;44(1):721–7. [7] Das B. Radial distribution system power flow using interval arithmetic. Int J Electr Power Energy Syst 2002;24(10):827–36. [8] Vaccaro A, Canizares CA, Bhattacharya K. A range arithmetic-based optimization model for power flow analysis under interval uncertainty. IEEE Trans Power Syst 2013;28(2):1179–86. [9] De Figueiredo LH, Stolfi J. Affine arithmetic: concepts and applications. Num Algor 2004;37(1–4):147–58.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Short Communication
An affine arithmetic-based algorithm for radial distribution system power flow with uncertainties Wei Gu a,⇑, Lizi Luo a, Tao Ding b, Xiaoli Meng c, Wanxing Sheng c a
School of Electrical Engineering, Southeast University, Nanjing 210096, China Department of Electrical Engineering, Tsinghua University, Beijing 100084, China c China Electric Power Research Institute, Beijing 100192, China b
a r t i c l e
i n f o
Article history: Received 15 June 2013 Received in revised form 30 December 2013 Accepted 16 January 2014
Keywords: Affine arithmetic Interval arithmetic Radial distribution system power flow Uncertainties
a b s t r a c t This letter presents an algorithm for radial distribution system power flow in the presence of uncertainties. To reduce the overestimation of bounds yielded by correlation of variables in interval arithmetic (IA), affine arithmetic (AA) is applied in this study to carry out tests of distribution system power flow. Compared with the algorithm based on IA, the proposed algorithm narrows the gap between the upper and lower bounds of the power flow solution. IEEE 33-bus and 69-bus test systems are used to demonstrate the effectiveness of the proposed algorithm. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction With an increasing number of renewable power generation systems connected to distribution systems, power injections are increasingly difficult to model, which makes it difficult for traditional deterministic methods to calculate the power flow. To deal with the uncertainties in power systems, many modeling approaches have been proposed in Refs. [1–5]. Moreover, popular power flow algorithms, such as Newton-Raphson and Fast Decoupled algorithms, generally fail to converge when analyzing the distribution system for its radial structure and high R/X ratio [6]. Therefore, the backward/forward sweep algorithm is used to solve this problem. In Ref. [7], an IA-based backward/forward sweep algorithm is developed to calculate the power flow of radial distribution systems, which uses ranges restricted by upper and lower bounds to express the uncertainties. However, the ranges estimated by IA tend to be too large, especially in complicated expressions or long iterative computations, because they ignore the correlation of different variables [8]. An AA-based algorithm is proposed to reduce this overestimation of bounds. This algorithm uses the affine form instead of the interval form to describe the uncertainties of power injections, thus accounting for correlations between different variables.
2. Affine arithmetic-based algorithm for radial distribution system power flow 2.1. Concepts of affine arithmetic In affine arithmetic, a quantity x is represented by an expression of the form
^x ¼ x0 þ x1 e1 þ þ xn en
ð1Þ
which is an affine expression of noise symbols ei with real coefficients xi. Each noise symbol ei is a symbolic real variable whose value is unknown except that it is restricted to the interval [1, +1] and is independent from other noise symbols. The coefficient x0 is called the central value of the affine form of ^x. The coefficients x1, . . ., xn are the partial deviations associated with the noise symbols e1, . . ., en in ^x. The number n of noise symbols depends on the affine form. Different affine forms use a different number of noise symbols, some of which may be shared with other affine forms. Affine forms provide interval bounds for the corresponding quantities: If a quantity x is represented with the affine form ^ x as above, then x 2 [x0 rx, x0 + rx]. Here rx = |x1| + . . . + |xn| is called the total deviation of ^ x [9]. 2.2. Calculation rules of affine arithmetic applied in complex field
⇑ Corresponding author. Tel.: +86 13814005169; fax: +86 25 87796196. E-mail address: [email protected] (W. Gu). http://dx.doi.org/10.1016/j.ijepes.2014.01.025 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
Two types of uncertainty sources exist in the power injections: active and reactive power. Because reactive power is represented
243
W. Gu et al. / Electrical Power and Energy Systems 58 (2014) 242–245
by an imaginary number, complex affine forms are used for the analysis of power flow with these uncertainties. Consider two complex quantities x and y represented by the complex affine forms
^x ¼ x0 þ x1 e1 þ þ x2n e2n ^ ¼ y0 þ y1 e1 þ þ y2n e2n y
^x þ y ^ ¼ ðx0 þ y0 Þ þ ðx1 þ y1 Þe1 þ þ ðx2n þ y2n Þe2n
ð3Þ
^x y ^ ¼ ðx0 y0 Þ þ ðx1 y1 Þe1 þ þ ðx2n y2n Þe2n
ð4Þ
^x y ^¼
i¼1
!
2n X y0 þ yi ei
!
ð5Þ
i¼1
where e2n+1 is a new noise symbol that is created during the computation. For any complex number z = a + jb, the function f() is defined as f(z) = |a| + j|b|.
" # 2n 2n X X ^x 1 ¼ x0 C þ ðC xi x0 yi Þei þ D x0 e2nþ1 þ f ðxi Þ ^ AB y i¼1 i¼1 " # 2n X 1 f yi þ f ðDÞ e2nþ2 AB i¼1
ðk1Þ
Þ
ð8Þ
where Si is the power injection at node i expressed by affine forms ðk1Þ in Step 2 and U i is the calculated voltage at node i during the (k 1)th iteration. Step 5: The details of this step are the same as the traditional backward/forward sweep except that the complex arithmetic has been replaced by complex affine arithmetic. At iteration k, ðkÞ the nodal affine form voltage U i is obtained by this step. ðkÞ Step 6: At the end of iteration k, the distance between U i and ðk1Þ Ui , henceforth denoted by di, needs to be calculated for all nodes i. Consider an affine form voltage expressed as
ð9Þ
"
n n X X ½U; U ¼ ½u0 rU ; u0 þ r U ¼ u0 f ðui Þ; u0 þ f ðui Þ i¼1
ð6Þ
2.3. Steps to radial distribution system power flow analysis The basic power flow analysis method used in this study is the backward/forward sweep power flow algorithm. However, to represent the uncertainties of the active and reactive power, power injections of each node have been treated as affine forms rather than fixed numbers, and consequently, the complex arithmetic has been replaced by complex affine arithmetic. Concrete steps are as follows: Step 1: Number the nodes of the distribution system and define the node count as N. Step 2: Transform the given power injections into affine forms with 2N noise symbols, with each node corresponding to two noise symbols. For example, assuming the power injection of node m is P + jQ and has a ±k% tolerance, it can be represented by
ð7Þ
The irrelevant noise symbols e1, . . ., e2m-2, e2m+1, . . ., e2N cannot be found in the formula because the coefficients associated with them just equal to zero.
# ð10Þ
i¼1
^ and f() has the same definition as where rU is the total deviation of U formulation (5)–(6). After transforming all the affine form voltages into interval forms according to (10), di is calculated by
ðkÞ ðk1Þ ðkÞ ðk1Þ di ¼ max U i U i ; U i U i
where e2n+1, e2n+2 are new noise symbols that are created during the P computation. These new variables are defined as A ¼ y0 2n i¼1 f ðyi Þ, pffiffiffiffi pffiffiffiffi P2n AB 1 AB B ¼ y0 þ i¼1 f ðyi Þ, C ¼ BþAþ2 AB y0 and D ¼ BþA2 , and the 2AB 2AB function f() has the same definition as in multiplication.
^Sm ¼ P þ jQ þ ðP e2m1 þ jQ e2m Þ k%:
ðkÞ
Ii ¼ ðSi =U i
its corresponding interval form with upper and lower bounds can be represented by
i¼1
i¼1
ðkÞ
Step 4: At iteration k, the nodal current injection Ii at node i can be calculated by
^ ¼ u0 þ u1 e1 þ þ un en U
" # " # 2n 2n 2n X X X ¼ x0 y 0 þ ðx0 yi þ y0 xi Þei þ f ðxi Þ f ðyi Þ e2nþ1 i¼1
With the preceding work above, the following steps are implemented for the iterative solution of the system.
ð2Þ
where x0 and y0 are complex numbers and represent the central value of each complex affine form. When i is odd, xi and yi are real coefficients that represent the uncertainties derived from the active power injections. When i is even, xi and yi are imaginary coefficients that represent the uncertainties derived from the reactive power injections. Based on the calculation rules of affine arithmetic applied in real number field described in Ref. [9], addition, subtraction, multiplication and division of these two complex affine forms can be derived as follows:
2n X xi ei x0 þ
Step 3: Obtain the given voltage at the root node and set the initial voltages to all the other nodes as 1 p.u.
ð11Þ
If max(di), i = 1, 2, . . ., N, is less than the specified voltage error tolerance limit, the power flow analysis has converged. Otherwise the algorithm goes back to Step 4 to proceed with next iteration. 3. Case studies To demonstrate the effectiveness of the proposed algorithm, it is implemented on IEEE 33-bus and 69-bus test systems. In this letter, the voltage error tolerance limit for convergence of the iterative process is 104 p.u. First, the proposed algorithm is used to analyze the 33-bus system with an assumed ±20% tolerance on the given power injection of each bus. The power flow solutions obtained by different algorithms are compared in Figs. 1 and 2, with Fig. 1 showing the bus voltage magnitude bounds and Fig. 2 depicting the bus voltage angle bounds. Algorithm A is the algorithm proposed by this letter, algorithm B represents the interval arithmetic-based algorithm [7] and algorithm C is the Monte Carlo method with 10,000 trials. Obviously, the ranges of solutions obtained by both the AA-based and IA-based algorithms completely contain the bounds obtained by the Monte Carlo method. This overlap demonstrates that both of the first two algorithms can give proper approximations of the power flow solution bounds. Notice also that the solution ranges of the AA-based algorithm are narrower than that of IA-based algorithm, which proves that the AA-based algorithm is able to reduce the overestimation of bounds compared with the IA-based algorithm, due to its accounting for correlations between different variables in the distribution system. Next, both the 33-bus and the 69-bus test systems with uncertainty tolerances from ±10% to ±50% are analyzed. The maximum errors of bus voltage magnitudes and angles under different
244
W. Gu et al. / Electrical Power and Energy Systems 58 (2014) 242–245
Fig. 1. Bus voltage magnitude bounds in the IEEE 33-bus test system.
Fig. 2. Bus voltage angle bounds in the IEEE 33-bus test system. Table 1 Error analyses and iteration numbers under different uncertainty tolerances. Uncertainty tolerance (%)
10 20 30 40 50
Test system
IEEE-33 IEEE-69 IEEE-33 IEEE-69 IEEE-33 IEEE-69 IEEE-33 IEEE-69 IEEE-33 IEEE-69
Maximum error of bus voltage (magnitudes, p.u.)
Maximum error of bus voltage (angles, deg)
Iteration number
AA
IA
AA
IA
AA
IA
0.0096 0.0028 0.0195 0.0057 0.0296 0.0086 0.0400 0.0115 0.0506 0.0144
0.0130 0.0031 0.0272 0.0062 0.0427 0.0095 0.0601 0.0127 0.0800 0.0161
0.4822 0.1422 0.9826 0.2863 1.5027 0.4323 2.0436 0.5801 2.6069 0.7299
0.7591 0.1627 1.5741 0.3269 2.4611 0.4938 3.4505 0.6644 4.5870 0.8383
4 3 4 3 5 3 5 3 5 3
6 3 7 4 7 4 8 4 9 4
uncertainty tolerances are shown in Table 1, where the error specifically refers to the bound’s deviation from the deterministic power flow solution. It can be easily observed that the bounds increasingly deviate from deterministic power flow solutions as the uncertainty tolerances increase. Moreover, the bounds obtained by the AA-based algorithm consistently deviate less than that of the IA-based algorithm when the other conditions are the same. In addition, the iteration numbers needed for different scenarios are provided in the last two columns of Table 1. It is obviously that the AA-based algorithm always has less iteration numbers than the IA-based algorithm. 4. Conclusion Taking the uncertainties of power injections into consideration, an algorithm for radial distribution system power flow
based on affine arithmetic is proposed and tested in this letter. Compared with the interval arithmetic-based algorithm, the proposed algorithm accounts for correlations between different variables and thus obtains power flow solutions with narrower ranges. Acknowledgments This work was supported by the National High Technology Research and Development Program of China (863 Program Grant 2012AA050210), the National Science Foundation of China (51277027), the Natural Science Foundation of Jiangsu Province of China (SBK201122387), the Fund Program of Southeast University for Excellent Youth Teachers, the Southeast University Key Science Research Fund, the State Grid Corporation of China (dz71-13-036).
W. Gu et al. / Electrical Power and Energy Systems 58 (2014) 242–245
References [1] Pereira LES, Da Costa VM, Rosa ALS. Interval arithmetic in current injection power flow analysis. Int J Electr Power Energy Syst 2012;43(1):1106–13. [2] Vaccaro A, Canizares CA, Villacci D. An affine arithmetic-based methodology for reliable power flow analysis in the presence of data uncertainty. IEEE Trans Power Syst 2010;25(2):624–33. [3] Mohammadi M, Shayegani A, Adaminejad H. A new approach of point estimate method for probabilistic load flow. Int J Electr Power Energy Syst 2013;51: 54–60. [4] Heleno M, Sumaili J, Meirinhos J, et al. A linearized approach to the symmetric fuzzy power flow for the application to real systems. Int J Electr Power Energy Syst 2014;54:610–8.
245
[5] Marinopoulos AG, Alexiadis MC, Dokopoulos PS. Energy losses in a distribution line with distributed generation based on stochastic power flow. Electr Power Syst Res 2011;81(10):1986–94. [6] Singh S, Ghose T. Improved radial load flow method. Int J Electr Power Energy Syst 2013;44(1):721–7. [7] Das B. Radial distribution system power flow using interval arithmetic. Int J Electr Power Energy Syst 2002;24(10):827–36. [8] Vaccaro A, Canizares CA, Bhattacharya K. A range arithmetic-based optimization model for power flow analysis under interval uncertainty. IEEE Trans Power Syst 2013;28(2):1179–86. [9] De Figueiredo LH, Stolfi J. Affine arithmetic: concepts and applications. Num Algor 2004;37(1–4):147–58.