EXTENSION OF A PROBABILISTIC LOAD FLOW CALCULATION FOR THE CONSIDERATION OF INTERDEPENDENCIES Johannes Schwippe, Olav Krause, Christian Rehtanz Institute of Power Systems and Power Economics, TU-Dortmund University, Dortmund, Germany
[email protected] Abstract – In planning and operation of power systems uncertainties introduced by stochastic behaviour of loads and generators as well as the inherent error of predictions for future scenarios have to be taken into account. Information about the probability of violation of operational constraints is a useful parameter, especially for network planning. Because of the large amount of possible operational states that have to be evaluated when considering the stochastic nature of certain loads and generators, traditional load flow calculation is not a promising basis for deployment on models of large networks. This is the domain of probabilistic load flow calculation. In extension to existing convolution based probabilistic load flow methods an approach for the modelling and consideration of correlations between the input parameters is presented in this paper. Aspects like computational burden and achievable precision are discussed in detail.
Keywords: probabilistic load flow, convolution, considering dependences
̅ ̅
1 NOMENCLATURE Number of nodes Reference node Number of stochastic groups Current caused by shunt elements Current of the reference node Line admittance matrix, complex valued, nodal admittance matrix, complex valued, Shunt admittance at node j Probabilistic density function
2 INTRODUCTION In the process of liberalisation a paradigm change is becoming apparent in power system planning. The integrated utilities tended to avoid strictly congestions in their networks, even when this required significant higher investment costs. The networks were designed to cope even with seldom and extreme operational states. With the liberalisation proceeding, the trade-off between investment and congestion costs is gaining in importance. Furthermore causes the liberalisation and integration of renewable energies uncertainties of the fed in or consumed power of today or future power systems. The predicted future load conditions or installed wind power capacities have to be taken into account during the planning process of future power systems. Furthermore the long construction and eco-
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nomic life time of transmissions lines of several years and decades have to be regarded. Information about the probability of violation of operational constraints is crucial for the planning process. Especially for this purpose the techniques of probabilistic load flow (PLF) were developed. Up to now, they almost exclusively focused on the voltage-band and thermal limits of lines and transformers. The constraint of active power balance, which introduces probabilistic interdependences among the PLF input parameters, has to be integrated in the PLF calculation (PLFC). The PLFC is presented first in [1] uses in linear network model and in [2] uses a Taylor series extension of the non-linear load flow equations. Correlations between the nodal powers are not considered in these publications. The power fed in of wind turbines and the power consumed by domestic loads of small regions follow a common mean trend, as they are synchronized by weather conditions or repeating customs of the population [3]. Due to this, they form certain stochastic groups with a significant correlation among them. The correlations are determined exemplary for the fed-in wind energy in this paper. A linear relation between the nodal loads has been assumed in [6] uses a DC load-flow and in [7] uses a Monte-Carlo-Simulation. In contrast [8] uses a nonlinear model for the correlations even though the nonlinear behaviour is minor. The PLF is realized as well with a Monte-Carlo-Simulation which does not consider all possible nodal power combinations. In contrast [9] uses a convolution based method with a Taylor series extension. A Gram-Charlier-Expansion, which describes the propagation of the input quantities through the network equation based on a Taylor-series-extension of the load flow equations uses [10] additional with a probabilistic wind farm model. By these methods the nonlinear correlations are not considered and limitations of the probabilistic density function (PDF) of the nodal powers exist. This Paper classifies the different power injections of regenerative energies into groups. The characteristics of each group can be described with a “mean” probability density function and the nodal powers are mapped using the installed capacity to weight the influence of each group at the respective nodes. Deviations of specific nodes to the mean distribution can be considered with additional differential PDFs. Further correlations especially in transmission networks are involved through the necessary active power balance. The most PLFC neglect these correlations and
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the reference node has to ensure the active power balance. Through this, the estimated line currents in the surrounding area of the reference node have high deviation to the reality. This problem is described in [6] and is solved with Monte-Carlo-Simulation (MCS) using linear power equations by neglecting network losses. In contrast this paper describes PLF-solution with an enhanced convolution method considering of linear dependencies and the active power balance using a nonlinear power flow formulation. The accuracy is documented with a simple test network.
3
PRESENTED METHOD
3.1 Basic method The method presented in this paper bases on the nodal admittance matrix and another linear map which maps the nodal voltages to the line currents. This map is called line-admittance-matrix and is shown for a simple network (Figure 3) in (1). Each row represents one line and the matrix contains the positive line admittance at the corresponding row to the starting and the negative one to the ending node. ̅
(
̅ ̅
̅ ̅
̅
̅ ̅
(1)
)
The second matrix, the nodal admittance matrix, maps the nodal voltages to the nodal currents and is only regular with consideration of the line capacitances. The values of the lines capacitances are very small whereby the calculation of the inverse is numerically ambitious. One possibility for the calculation of the inverse of singular matrixes is the pseudo inverse by using the singular value decomposition (SVD) [11]. A direct map from the nodal currents to the line currents arises by the multiplication of the line admittance matrix with the inverted nodal admittance matrix (2). ⏟
(2)
̃
Through dependencies within electrical networks the current of the reference (r) node depends on the sum of all other nodes (3). The current through the line capacitances, if they are considered, depends on the absolute voltage level. Due to this the current of the reference node has an additional part in dependence to the absolute voltage level (4). ∑
̅
With
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(3)
(4)
∑
These dependencies are considered in the matrix (5). The first row represents the influence of the currents that is caused by the absolute nodal voltage level. The other rows describe the influence of all nodal currents without the reference node. ( ̃
̃
̃
̃
̃ )
(5)
The current of the line j can be expressed as a weighted sum of all nodal currents without the reference node and a part of the absolute voltage level. ∑(
)
(6)
The elements of the used row of the matrix describes the influence of each node to line current. Distant nodes have a less influence than nodes of the surrounding area of the line. Usually the input quantities of load flow calculations are nodal powers and not nodal currents. An estimation of the nodal voltage angle is possible with a linear map like the DC load flow calculation (7) to map the nodal powers to the space of the nodal currents. Due that the matrix is singular the pseudo inverse is used. (7)
3.2 Estimation with expected values If the nodal voltages are estimated with the expected values of the nodal powers and adopt a statistical independency between the nodal powers the convolution can be used to calculate the PDF of the line currents. At first the PDF of the nodal powers are turned with the estimated voltage angle and argument and elongated or compressed by the absolute value of the corresponding element of the matrix (8). Hereby the represents the part of the node at the line current n. (
)
(
)
(8)
In a second step, the full is calculated with the convolution of the modified of the nodal powers (9). For this, the modified PDF will be converted to uniform sampling points. The accuracy of this PLFC decreases, if the distance between the sampling points is too large but with a smaller distance the computation burden increases significantly.
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After the convolution the resulted PDF is shifted with the current of the line capacitances, which is estimated with expected nodal voltages (9). The reference node is here donated as 1.
This method is designated as enhanced convolution below. The number of necessary calculations of the enhanced convolution is constant to the classic convolution except the additional linear operations to estimate the voltage angles for each nodal power combination. (13)
( (
)
(
)
| ̅ | | ̅ |
∫
̅
( )
∑ (14)
∑(
(10)
3.3 Extension for the consideration of the corresponding voltage profile A further increase of the accuracy is possible with the consideration of the estimated voltage profile for each nodal power combination and not only for the expected values. In this case the convolution is described as integral of the product of the probabilities of the nodal powers. The upper and lower limits of each integral represent the nodal power limits at each node and the address of the last PDF (m) is chosen so that the combination of the nodal powers corresponds to the line current (11), (12). )
∫∏
With
This method only shows a good accuracy for narrowed PDF of the nodal powers [1], [5]. Deviations to a MCS with full-AC-load flow arise through the simplification of the voltage angle profile estimation and by neglecting the absolute voltage. The accuracy can be increased with considering the absolute voltage level with a linear model [5].
(
∫
(9)
The PDF of the absolute line current is calculated with (10). | ̅ |
)
(11)
)
̅
3.4 Modelling of dependencies The characteristics of the fed in wind power and the domestic loads of one region have a similar behaviour and can be depicted with a general characteristic or PDF. Due to this, they form stochastic groups with a significant correlation among them. A Matrix which contains the installed capacities of each group maps the global powers of the stochastic groups to the nodes (15).
( )
(
)(
(15)
)
with: : installed capacities of the group m at node n Differences from the injected or consumed power at the nodes to the global PDF can be expressed with an additional ) of the deviations. The nodal active power is described as weighted sum of the power of each group and the power deviations. (16)
∫
∫∏
∑ ̅
̅ With ̅ ̅
̅
(12)
∑
Assuming statistical independency among the power differences of each group, their convolution to one PDF is possible (17). (
Due to dependences between nodal active powers and complex valued nodal voltages which are nonlinear because they have only a significant impact to the argument of the nodal voltages the classic convolution cannot be used anymore. An alternative solution is possible with the determination of the complex valued line current as function of the control variable of the integrals (13), (14).
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)
(
)
(17)
In contrast to the feed-in of active power, the reactive power consumption or injection depends on the absolute nodal voltage. Its description with a general PDF is not meaningful. Exceptions are domestic loads where the ratio of active and reactive power is assumed to be fix. Consequently the reactive power injection or consumption is modeled within the PDF of the deviations and
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individual possibilities of nodes can be considered individually. The PDF of the line current is calculated with (18), whereby every combination of the global powers is combined with every nodal deviation. The reference node is donate here as 1.
(
)
∫
∫∏
(
) (18)
∫
∫∏
( )
The nodal power and are calculated with (16) and the line current with (14). 3.5 Including the unit commitment The power flows especially in transmission networks are also determined by the unit commitment. The power plants provide the active power difference between the demand and the feed-in of the renewable energies and the network losses. A solution with the classic convolution is impossible, due to the additional dependencies and non-linearity. The unit commitment is considered with an additional probability (pPP) which represents the availability of the determined unit commitment (19).
gration of the linear estimation method for the network losses improves the accuracy. 4 ANALYSIS OF CORRELATIONS The correlation of the injected or consumed power of different renewable energies or domestic loads is analyzed exemplary for the wind energy. Time series of the feed-in wind power of the four german control areas in 2009 (values for each ¼ h, [11]) are analyzed with a cross correlation which estimates the time shift between different signals (Figure. 1). The time shift between the injected wind powers is only a few hours and having the greatest difference between the control area of 50Hertz and Amprion which also have the largest geographical distance to each other. A modeling of the correlation with a global and a PDF of the nodal deviations seems to be suitable. 1 R 0,8
0,6 0,4 ENBW - Transpower ENBW - Amprion Transpower - Amprion
0,2 0 -48
-36
-24
-12
ENBW - 50Hertz Transpower - 50Hertz 50Hertz - Amprion 0
12
24 t[h]
36
48
Figure 1: Cross-correlation between the injected wind power of the german control areas
(
)
∫
∫∏
∫
(
∫∏
)
( )
(19)
∑
Figure 2 shows exemplary the correlation of the fed in wind power between the sum of all four control areas and the area of Amprion, having a linear dependency. The probability of high deviations to the total fed in wind power is small. 1 PA[pu] 0,8
The line current is calculated with (14) whereby the nodal powers consist of a part of the power plants, the general groups and the nodal specific deviations. The computation burden is the main challenge in PLFC. A simple fundamental market model is chosen for this reason, which has no significant impact on the complexity of this method. The primary energy carrier (e.g. running water, lignite, hard coal, gas, etc.) and the marginal cost are chosen to group related power plants. Economic power plants are used primarily. If the power of one group is not completely required, the residual power is distributed uniformly on all power plants of this group. The availability of power plants is not considered in the actual implementation wherefore the probability (pPP) is one. This aspect is also a possible extension to the market model but increases the number of possible combinations significantly and thus the computation burden. Network losses are not considered and have to be covered of the reference node. An inte-
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0,6 0,4 0,2 0 0
0,25
0,5
P0,75 T[pu]
1
Figure 2: Correlation between the fed in wind power of the sum of all four control areas and the control area of amprion
5 TEST CASES The different variants of the presented method are verified with a simple test network, demonstrated in figure 3. The lines correspond to typical overhead lines
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(
, , ). The length of the lines 1-2, 2-3 and 3-4 is 80 km and of line 2-4 160 km. 1 2 3
5.2 With consideration of linear dependencies The scenarios with the consideration of correlations and the unit commitment are more complex. The installed capacities of the power plants, wind turbines and the distribution of the domestic loads to the nodes are shown in Table 1. Node 1 2 3 4
4 Figure 3: Simple test network
The accuracie of the different methods are verifed with a Monte-Carlo-Simulation with 50.000 experiments and a time series simulation of the nodal powers. Both uses a full AC-load flow method to take the effect of active and reactive power flows into account. 5.1 Without consideration of dependencies At first the variant with independent input quantities is analyzed (Section 3.2 and 3.3). The nodal powers at each node are represented with 2-dimensional normal distributions with an expected value of 0.5 pu for the active and 0.25 pu for the reactive power. The variance is 0.3 pu for all nodes. The number of sampling points of the PDF of the nodal powers is 21x13 per node. This causes to 20.34 million ) possible combinations. The accuracy of the convolution based method decreases for widespread PDF of the nodal powers because the voltage angles are only estimated for the expected values [1][5] and is not included in the comparison for this reason. Exemplary shows Figure 4 the PDF of the absolute line current of line 2-3 for the MCS and the enhanced convolution.
Domestic loads [MW] 600 200 1000 1200
Wind [MW] 200 1000 200 400
Power plants [MW (type, order)] 800 (gas,3) 1200 (nuclear,1) 1000 (hard cole,2)
Table 1: Installed capazities for the scenario consideration of correlations and the unit commitment
with
The used global PDF are one-dimensional because they only describe the active power distribution. The global PDF has 21 and the PDF of the nodal deviations ( ) 15x5 sampling points in this scenario. The number of combinations is 186.047 million ( ). The used PDF of domestic loads, the summed up wind power and the PDF of the deviations of the Amprion control area to the total wind power are shown in Fig. 5. 20%
domestic loads 16% p[%] 12% 8% 4% 0% 0 20%
0,25
0,5
0,75 P[pu]
1
total wind power
p[%] 16% p[%] 12%
1,20% enhanced convolution Monte-Carlo-Simulation
p[%] 0,80%
8%
4% 0% 0
0,25
0,5
0,75 P[pu] P[pu]
1
75%
deviation wind power p[%] 60%
0,40%
45% 30%
0,00% 0,00
0,50
1,00
1,50
2,00 I[pu]
2,50
Figure 4: PDF of the absolut line current of line 2-3 without correlations
It is visible that the enhanced convolution has deviations to the MCS for higher line currents. A possible cause is the neglecting of the reactive absolute voltage sensitivity.
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15% 0% -0,50
-0,25
0,00
0,25 ΔP[pu]
0,50
Figure 5: Uses PDF of demestic loads, total Wind power and the deviation of the area amprion
The nodal reactive power is a uniform distribution between -0.4 and 0.4 pu. For the evaluation of the accuracy of the different variants, the absolute value of the
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line current of line 1-2 is exemplary presented without consideration of the unit commitment in Figure 6. 6% enhancend convolution Monte-Carlo-Simulation time-series
p[%] 5% 4% 3% 2% 1% 0% 0,00
0,25
0,50
0,75 I[pu]
1,00
Figure 6: PDF absolut line current of line 1-2 without unit commitment
The PDF of absolute line currents of the enhanced convolution method is similar to the PDF of the MCS and time series simulation. The MCS deviates for higher line currents to the other method, which is potentially caused by the used random generator. The results without consideration the unit commitment are shown in Fig 7. In addition to the MCS also a time series analyses is realized. The time series of the fed-in wind power of the four control areas are assigned with the installed capacities of Table 1 to the nodes. The domestic loads are simulated with the representative load profile [11]. 7% enhancend convolution Monte-Carlo-Simulation time-series
p[%] 6% 5% 4%
3% 2% 1%
0% 0,00
0,25
0,50
0,75 I[pu]
1,00
Figure 7: PDF absolut line current of line 1-2 with unit commitment
A comparison with the convolution method (expected values) is not realized because the accuracy of this method is, compared to a MCS for widespread PDF, worse. The deviations between the MCS and enhanced convolution are minor but major deviations to the time-Series are visible. Due to the good results without consideration of the unit commitment and the similarity to the MCS, the deviations may be caused by the decoupled modeling of
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the domestic loads to the fed wind power. Another reason is the lower number of sampling points of the global and nodal PDF’s in comparison to the times series, restricting the number of possible combinations. All in all the enhanced convolution method shows good results in comparision to the MCS but also to the used time series. A modelling of the wind power with linear depedencies seems to expedient. The impact of a higher number of sampling points to the accuracy have be investigated further. 6 ANALYSYS OF COMPUTATION BURDEN The computation burden is a main aspect for the applicability of PLFC. The classic convolution function uses high optimized methods wherefore a comparison only on the computation time is unqualified. This section describes the differences of the proposed methods and the impact on the number of operations. In contrast to the method which uses the classic convolution (expected values, section 3.2) the method enhanced convolution (section 3.4 and 3.5) executes for each nodal power combination two matrix vector multiplications (calculation of the estimated voltage angle and the complex valued line current). Each matrix vector multiplication requires (n-1)2 multiplications and (n2)2 additions (whereby n is the number of nodes). Thus, the enhanced convolution requires significantly more operations than the classic convolution but takes the corresponding voltage angle for each nodal power combination into account. This is strictly required for a high accuracy by widespread PDF. The MCS and the used time series analysis require much more operations for each considered combination because the used full-AC-load flow computation needs more matrix operations, even including matrix inversions. The used matrices by the enhanced convolution method depend only on the network topology. An inversion is only required in the beginning of the method. Due to the independence sequence of nodal power combinations a parallelization of the operation is possible and reduces the computation time significantly. The use of GPU-Computing which is specialized to matrixmatrix and matrix-vector operation reduces the computation time, too. 7 CONCLUSION This paper presents a new method to consider linear dependences in the probabilistic load flow calculation. An exact model between the nodal currents and line currents and a linear model between the nodal powers to the nodal currents is used. The dependences between the injected wind powers are analyzed with time series of the wind power of the fours control areas in Germany. A linear modeling of the dependences seems to be possible. Deviations to these linear modeling are considered with nodal specific PDF of the deviations. The dependencies between other group’s e.g Photovoltaic and domestic load require a
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different modeling because they are synchronized with the time of day. A possibility is more dimensional PDF whereby each dimension represents one group. The unit commitment which has a great impact to the powers flows in transmission networks is also considered in the PLF. Only a simply fundamental market model, based on the marginal cost, is used. The computation burden of the new method with the different variants is analyzed in detail and compared to the classic convolution. By using the fast fouries transformation the classic convolution can be calculated faster. A parallelization of the other variants is possible. Only by significantly increasing computation speed an application for realistic scenarios is possible. The enhanced convolution method with considering dependences has a good accuracy in comparison to a MCS and time-series-simulation but the computation burden is significant smaller. Possible future extensions are the parallelization and the implementing of GPU-computing. In a second step the development of a probabilistic market model to consider the availability and failure rate of power plants is possible. REFERENCES [1]Borkowska, B.; , "Probabilistic Load Flow," Power Apparatus and Systems, IEEE Transactions on , vol.PAS-93, no.3, pp.752-759, May 1974 [2]Allan, R.N.; Leite da Silva, A.M.; Burchett, R.C., "Evaluation Methods and Accuracy in Probabilistic Load Flow Solutions," power apparatus and systems, ieee transactions on , vol.PAS-100, no.5, pp.25392546, May 1981 [3]Widen, J.; , "Correlations Between Large-Scale Solar and Wind Power in a Future Scenario for Sweden," Sustainable Energy, IEEE Transactions on , vol.2, no.2, pp.177-184, April 2011 [4]Schwippe, J.; Krause, O.; Rehtanz, C.; , "Probabilistic Load Flow Calculation based on an enhanced convolution technique," PowerTech, 2009 IEEE Bucharest, vol., no., pp.1-6, June 28 2009-July 2 2009 [5]Schwippe, J.; Krause, O.; Rehtanz, C.; , "Extension of a probabilistic load flow calculation based on an enhanced convolution technique," Sustainable Alternative Energy (SAE), 2009 IEEE PES/IAS Conference on, vol., no., pp.1-6, 28-30 Sept. 2009 [6]Leite da Silva, A.M.; Arienti, V.L.; Allan, R.N.; , "Probabilistic Load Flow Considering Dependence Between Input Nodal Powers," Power Apparatus and Systems, IEEE Transactions on , vol.PAS-103, no.6, pp.1524-1530, June 1984 [7]Rodrigues, P.R.; Castro, R.; Villafafila-Robles, R.; Sumper, A.; , "Modeling the stochastic dependencies in a probabilistic load flow including wind generation," Sustainable Alternative Energy (SAE), 2009 IEEE PES/IAS Conference on , vol., no., pp.1-7, 28-30 Sept. 2009
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[8]Mori, H.; Wenjun Jiang; "A New Probabilistic Load Flow Method Using MCMC in Consideration of Nodal Load Correlation," Intelligent System Applications to Power Systems, 2009. ISAP '09. 15th International Conference on , vol., no., pp.1-6, 8-12 Nov. 2009 [9]Allan, R.N.; Al-Shakarchi, M.R.G.; , "Linear dependence between nodal powers in probabilistic a.c. load flow," Electrical Engineers, Proceedings of the Institution of , vol.124, no.6, pp.529-534, June 1977 [10]Lei Dong; Weidong Cheng; Hai Bao; Yihan Yang; , "Probabilistic Load Flow Analysis for Power System Containing Wind Farms," Power and Energy Engineering Conference (APPEEC), 2010 Asia-Pacific , vol., no., pp.1-4, 28-31 March 2010 [11]Time series of the injected windpower of the year 2009, amprion, www.amprion.de, 50 Hertz Transmission, http://www.50hertz-transmission.net, ENBW, http://www.enbw.com, Transpower, http://www.tennettso.de; [12]G. Strang, Introduction to Linear Algebra, 3rd Edition. Wellesley-Combridge Press 2003
BIOGRAPHIES Johannes Schwippe is research associate at the Institute of Electrical Power Systems and Power Economics, Technical University of Dortmund, Dortmund, Germany. He received his diploma degree in Electrical Engineering at the Technical University of Dortmund in 2008. His research interests are Power System Planning and Modeling.
Olav Krause is Lecturer at the School of Information Technology & Electrical Engineering at the University of Queensland, Australia. He received his diploma degree in Electrical Engineering in 2005 and his doctorate in 2009 at the TU Dortmund University, Germany. His research interests are strategies for the coordinated use of power systems and issues related to network stability. Christian Rehtanz received his diploma degree in Electrical Engineering in 1994 and his Ph.D. in 1997 at the Univ. of Dortmund, Germany. From 2000 he was with ABB Corporate Research, Switzerland and from 2003 Head of Technology for the global ABB business area Power Systems. From 2005 he was Director of ABB Corporate Research in China. From 2007 he is professor and head of chair for power systems and power economics at the Univ. of Dortmund. His research activities include technologies for network enhancement and congestion relief like stability assessment, wide-area monitoring, protection, and coordinated FACTS- and HVDCcontrol.
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