Approaches to the calculation of losses in power networks Study and test of different approximate methods to the calculation of losses João Tiago Abelho dos Santos Calheiros Andrade Department of Electrical and Computer Engineering Instituto Superior Técnico - University of Lisbon, Portugal Email: joao.andrade@tecnico.ulisboa.pt Abstract In general, this dissertation appears in the ambit of the analysis of power networks, as well as the attempt of finding an alternative method for calculating the transmission losses after the occurrence of disturbances. Once the definition of the network operation topology can mean a large number of possible configurations, it becomes imperative to find an alternative way of knowing what the best solutions without involving the calculation of power flows to all cases. This way, in a first stage, it is treated the concept of adjoint network and the loss sensitivities are calculated for any network parameters, based on the Tellegen s Theorem and on the concept of adjoint networks. Posteriorly, it is developed a precise formula, continuing the study of the loss evaluation. Finally, we study the idea of get reliable results for the calculation of losses, reducing the computational effort and the network s observation space, based on an approximate model. All calculations and results obtained were performed using MATLAB. Index Terms Power networks, loss evaluation, adjoint networks, Tellegen s theorem, sensitivity analysis I. INTRODUCTION The electric power is an energy essentially produced in thermoelectric and hydroelectric centrals, wind, solar and nuclear systems, and it is one of the energies more used by the humanity. In Portugal, the main producers of electric power on regular mode are EDP Production, Iberdrola, REN Trading and ELECGÁS. There is also some production on special mode. Related to the transportation, REN is the exclusive public service licensee of the RNT (Transportation National Network), on very high and high voltage, connecting the producers to the consumption centers and covering all the continental territory. The distribution of electricity is divided by medium and high voltage and low voltage, being operated almost exclusively by EDP Distribution. On the distribution market of low voltage, there are some exceptions where the distribution of electric energy is assigned to small operators. Forecasts point to an increase in the consumption of electric power in the future. This way, it is important that there are studies which facilitate the analysis and raise the reliability and efficiency of the power systems. There are several criteria of quality in the power systems. A leading indicator of the efficiency of a power network is energy losses occurring throughout its structure. Many times, the study of the energy losses is based on the analysis of the difference between the purchased energy and the invoiced energy by the distribution company. The main issue with this analysis is that it is not possible to know the location of the losses and what are the network parameters responsible for them. Once the optimization plays an important role in planning, management and operation of the power systems, it is in this context that it is inserted the study developed on this dissertation, analysis of approximated models for energy losses calculation which allow to get accurate results, in a short time and with a reduced computational effort. A. Objectives The electric power produced in the generation centers, normally located far away from the consumption centers, is transported through transmission lines which feed subtransmission substations, located closer to the urban centers. From this point, the electric power distribution system is responsible to bring the electricity to all system consumers, wherever they are. Obviously, once power networks feed many different types of consumers and all of them are much distanced, it will exist some technological problems. The reconfiguration of distribution networks got an important role on planning the energy systems, where is needed to define the operation topology of the network. One of the goals of this study is to study different configurations of a power network and analyse the energy losses evolution to try to minimize them. There are other ways to reduce the losses, however, the reconfigurations are the solution more viable economically speaking. The reconfiguration processes consist in taking off and insert branches on the network, process that in the study corresponds to disturbances.
Typically, the calculation of the losses can be done with a power flow, considering methods as Newton-Raphson, Gauss- Seidel or Decoupling. Other solution is the classic method of Bs. With these hypotheses, it is possible to calculate the exact value of the losses. The problem here is to calculate the losses for a high number of possible configurations, once the time and computational effort would be very high. This way, this study suggests analysing the performance of different approaches to the energy losses calculation on power networks, in terms of accuracy of the results, depending of the computational effort and the network s observation space. II. LOSSES EVALUATION SENSITIVITIES, TELLEGEN S THEOREM AND ADJOINT NETWORKS The concept of network addressed on this dissertation and the Tellegen s Theorem are a theoretical base for calculating losses in power networks. Tellegen s Theorem expresses a relation between magnitudes of the branches of two adjoint networks, i.e. networks with the same graph (1). The evaluation of losses starts with a study of the sensitivities, once they play an essential role on planning and reconfiguring distribution networks. On this section is established the relation between incremental variations of dependent variable L ( ) and a series of incremental variations on independent variables of the network (,,,, ). A. Tellegen s Theorem Considering a power network N where is the voltage across the branch k ϵ K (set of indexes for all network branches), and also considering a network (adjoint network of the network N), where is the current which crosses the branch k, the Tellegen s Theorem says that: ( ) In case of networks which suffer incremental variations the Tellegen s Theorem suggests: ( ) 1. The representation of the reference node of the power network N is done with an independent voltage source on the adjoint network equation 2.4 2. The representation of the lines on the adjoint network is done with the same admittance of the power network s line, i.e. the branch k of the passive network corresponds to an admittance on the symbolic representation of the adjoint network equation 2.5 3. The representation of the branches which correspond to PQ nodes (load branch) is performed on the adjoint network by a dependent current source equation 2.6 4. The representation of the branches which correspond to PV nodes (generators) is performed by a dependent voltage source equation 2.7 Fig. 2.1 Symbolic representation of the adjoint network s elements (1) Independent voltage source (2) admittance (3) Dependent current source (4) Dependent voltage source C. Sensitivity Formulas The adjoint values present on sensitivity formulas are deduced from the following system: k ϵ (2.4) k ϵ (2.5) k ϵ (2.6) k ϵ (2.7) Considering the different partitions of the set K Once the losses are a real quantity and the equation above is a complex equation, only the real part of the equation is used: B. Adjoint Network ( ) The symbolic representation of the adjoint network is topologically equivalent to the original power network, presenting both the same graph structure. The adjoint network is defined based on the following conditions: the following formulas [1] are the basis for the computation of the loss sensitivities of the system relatively to any parameter: k ϵ (2.8) { k ϵ (2.9) k ϵ (2.10) { k ϵ (2.11) (1) Graph is a set of points (vertices) interconnected by lines (edges)
{ k ϵ (2.12) { k ϵ (2.13) { k ϵ (2.14) The loss sensitivity formulas related with e, k ϵ derive from the previous formulas which are related with parameters of the network branches ( Network): D. Tellegen s Sum { { k ϵ (2.15) k ϵ (2.16) The first order variation on the power can be obtained by summing the contributions of all the branches, correspondent to the Tellegen s sum (contributions specified on the dissertation). Taken into account that this first order variation corresponds, on the reference node, to: Fig. 2.2 Representation of the example power system. The nodes are numbered in bold, while the branches numeration does not have any formatting. Node 0 reference. Node 2 PV. Nodes 1, 2 e 3 PQ Based on section B. Adjoint Network, and following the symbolic representation of adjoint networks, one obtains for the power system of the figure 2.2 the following adjoint representation: (2.17) one obtains: {( ) (2.18) Fig. 2.3 Correspondent adjoint network to the power system of figure 2.2. The nodes and branches numeration follows the same logic of the previous figure The next step implies the calculation of the voltage values for each node and the correspondent voltage values of the adjoint network. Only with these values is possible to calculate the loss sensitivities based on the section C. formulas. This way, on the table below are represented the voltage values for the network and adjoint network nodes. The partial derivatives, represented on the section C. Sensitivity Formulas, are easily obtained taking into account the following differential k 0 1 1 1 0.9966 - j0.0828 1.0390 + j0.0863 2 1.0621 + j0.0213 1.0487 - j0.0370 3 0.9338 - j0.1366 1.0088 + j0.1680 4 0.9935 - j0.0431 1.0166 + j0.0481 E. Example and results ( ) Considering the network of the figure 2.2 (2.19) Tab. 2.1 Voltage values of the power system and correspondent voltage values of the adjoint network Note that to get the voltage values for the passive branches, one does or, for the branch which connects the nodes i and j. For example, to get one does. Doing one gets the value of.
Exemplifying: ( ) ( ) ( ) ( ) Having all the voltage values, of the power network and its correspondent adjoint network, it is possible to calculate the loss sensitivities for the different parameters of the system. On the following table are some sensitivity values for the energy system of the example. Type of branch k Reference 0 Passive Branches 8 10-0.0913 0.3578-0.0022 0.2014-0.0068 ( ) ( ) ( ) ( ) ( { ( ) { ( ) ) PQ Nodes 1 3 4 PV Node 2 0.0426-0.0000 0.0834 0.0213 0.0234 0.0040-1.0942 0.5313 Tab. 2.2 Sensitivity values of the branches and nodes of the energy system related with its different parameters III. LOSSES EVALUATION EXACT FORMULA The study developed on this section is based on the study of the previous section II. Losses Evaluation Sensitivities, Tellegen s Theorem and Adjoint Networks, however, with the difference that are not done any approximations in the Tellegen s equations, making possible the development of the exact formula for energy losses calculation. It is by comparing with this exact model that the errors will be considered and it will be possible take conclusions about the performance of the approximated models. Note that, to calculate the losses with the exact formula, it is necessary to do, first, the evaluation of the system s voltages when it is subject to disturbances. This way, are firstly presented the exact equations to the voltage evaluation. Those equations, similarly to the previous section II, use adjoint values which correspond to adjoint networks (real values) and (imaginary values). A. Voltage Variations Exact equations The voltage variations calculated with the following equations [5] will be used on the exact formula for the energy losses calculation. Real part of the voltage variations on system nodes: { (3.1) Imaginary part of the voltage variations on system nodes: ( ) B. Exact Formula (3.2) ( ) ( ) ( ) ( { ( ) { ( ) Considering again the partitions of the set K ) (3.3) and being the adjoint values defined for the following system: (3.4) k ϵ (3.5)
k ϵ (3.6) k ϵ (3.7) Considering also the higher order terms, the Tellegen s Theorem suggests that the exact formula [5] is wherein ( ) (3.8) ( ) ( ) ( ) ( ) {( ) (3.9) The highlight in the first example: With the exact formula, contour lines are obtained instead of straight lines, since the losses can evolve nonlinearly with the variation of a parameter. The energy losses calculation by sensitivities is approximated of the exact values, at least considering the losses evolution. For the base operating point ( e ), the sensitivities straight line is tangent to the contour line resultant of the exact formula, being the losses value at this point 0.0475. When the active power of the node is 0.2 and the reactive power is 0.1, the losses value for this operating point when calculated by the sensitivities is 0.03, while the real value is 0.0345 (approximated error of 10%). For the operation point where the active and reactive power are, respectively, 0.65 and 0.15, the losses by the sensitivities are 0.07, while the real value is 0.078 (approximated error of 15%). In this example, the errors of the sensitivities model are considerable. ( { ( ) { ( ) ) C. Examples and results This section allows taking the first conclusions related to the accuracy of the sensitivities model for the energy losses calculation. For the network of the figure 2.2 were analyzed the losses, for some parameter spaces, with the exact formula and by the sensitivities. The results were overlapped with the idea of getting conclusions about the accuracy of sensitivities relatively to the exact formula. Fig. 3.2 Comparison of the results obtained relatively to the energy losses calculation by sensitivities and by the exact formula, in the space ( e ) The highlight in the second example: For some operating points, the results obtained with the sensitivities are very close to the exact values, once the straight lines are practically tangent to the contour lines of the exact formula. For lower active powers, the error will grow and the results start to have a worse accuracy. At the point where both active powers are 0.8, the losses calculated with the sensitivities are 0.0607 and the real value is 0.060 (approximated error of 1.2%). When the active power of the node 1 is 0.45 and the active power of the node 4 is 0.4, the losses value calculated by the sensitivities is 0.0364 and the real value is approximately 0.040 (approximated error of 9%). Despite all, in this example, the sensitivities model got a very reasonable accuracy. Fig. 3.1 Comparison of the results obtained relatively to the energy losses calculation by sensitivities and by the exact formula, in the space ( e )
However, once the idea i sis to study the power flow equations in the context of the behaviour of the system to local disturbances, based on Tellegen s Theorem, the new power flow equations [2] are: { ( ) ( ) ( ) (4.3) { ( ) Fig. 3.3 Comparison of the results obtained relatively to the energy losses calculation by sensitivities and by the exact formula, in the space ( e ) The highlight in the third example: The variation of the reactance is the main factor to reduce the accuracy of the sensitivities model. If a disturbance reduce the resistance of the branch to 0.02pu and increase the reactance to 0.45pu, the exact losses are 0.045, while calculated by the sensitivities are 0.04. It is a considerable error (around 11%) and observing the evolution of the contour line tends to grow. In this example, the straight lines generated by the sensitivities analysis give the idea that the losses decrease when the reactance grows when, in fact, the losses increase. IV. APPROXIMATE MODEL TO THE EVALUATION OF VOLTAGES IN DISTRIBUTION NETWORKS In this section, the evaluation of losses will be addressed in an approximate way. One presents a model that it makes possible to obtain approximate voltage values which will be posteriorly used on the approximate model of losses. Addressing the topic power flow equations to distribution networks, it is known that the voltages are the unknown values and that the equations are based on the power conservation principle of each node. The new equations presented in this section are based on the concept of adjoint networks and on Tellegen s Theorem. Note that here only are considered networks with a reference node and the remaining nodes PQ distribution networks. A. Conventional Power Flow VS Equations based on Tellegen s Theorem The power flow equations [2] are: ( ( ) ( )) (4.1) ( ( ) ( )) (4.2) ( ) ( ) (4.4) where m corresponds to the index of a node, corresponds to the branches which suffer a disturbance in its admittance, for the nodes which a disturbance affects their complex power and for all nodes of the network. In these equations, the values of,, and are known, as well as the values and, which correspond to the adjoint quantities and which are computed together with all the other known values, for the considered base case. The main differences of the equations based on Tellegen s Theorem are that these equations take into account incremental quantities ( e ) and that they are more located. This means that are not necessary the same number of equations and nodes, being enough a set of equations which contemplates the nodes directly involved on the disturbances, or a set which also includes the neighbour nodes. One of the goals is precisely prove the more local nature of these equations, facing the conventional power flow equations. B. Local solution for the new equations A local solution model consists on the achievement of the system s answer, considering only a located region around the disturbance(s). Excluding all the terms which are not included in the defined region in the equations 4.3 and 4.4, are obtained the local solution equations [2] of the approximate model for voltages evaluation. { ( ) ( ) ( ) (4.5) { ( ) ( ) ( ) (4.6) Now, there is the partition, instead of the partition. This new partition contains all the nodes inside the defined region around the perturbation (only involved nodes short region or including neighbour nodes extended region. It is also possible to do the same approximation for the conventional equations.
C. Examples and results The first test taking into account the network presented on figure 4.1. different load levels at the network s nodes. Fig. 4.1 Distribution network with a disturbance (remove branch which connects the nodes 7 and 9) In this first test, the disturbance considered was the removal of one of the network s branches (branch 7 9), being the idea to compare the error on the calculation of the voltages after the disturbance, when calculated by doing an approximation by local solution for the two groups of equations considered before (conventional eqs. and based on Tellegen s Theorem eqs.). Fig. 4.3 Error values of the approximate model relatively to the exact calculation of the voltages for different load levels in the nodes How it is possible to observe in the figure 4.3, the calculation error of the node voltages, for the local solution of the equations based on Tellegen s Theorem, increases with the load level of the nodes relatively to the nominal values. However, the errors presented are always very close to zero (10-6 order). The next tests consider the network of the figure 4.4 and the disturbances involve the removal of a branch and the insertion of another one (remove branch 2 4 and insert branch 5 9). Fig. 4.2 Comparison of the error of a local solution for the two types of equations How it is possible to check on the figure above, the calculation error for the local solution of the equation based on Tellegen s Theorem is practically null (line with circles which overlap the axis), while the local solution of the conventional equations presents considerable errors on the voltages calculation to the nodes involved on the disturbance. It can be concluded that the new equations have a more located character then the conventional equations. The following test takes into account the same disturbance and has the goal of evaluate the calculation errors for the local solution of the equations based on Tellegen s Theorem for Fig. 4.4 Distribution network with disturbances (remove branch 2 4 and insert branch 5 9) The idea of the following tests is to verify if, considering a region for the local solution of the new equations that beyond the nodes directly involved in the disturbances also includes the neighbour nodes, the accuracy on the calculation of the node voltages after the disturbances increases. The first image (4.5) shows the values of the real part of the voltage, calculated taking into account only the nodes directly involved in the disturbances short region. How it is possible to observe there is an error between the approximate values and the real values (more visible on nodes 4 and 9).
Considering now the extended region, the lines are overlapped, what it means that the approximate values are practically equal to the exact values. Note that even for the node 9 there is not any difference in the values. Fig. 4.5 Comparison between approximate and exact models on the calculation of the real part of the voltages in the nodes directly disturbed short region Including in the observation region the neighbour nodes extended region one can observe clearly that the approximate values approach the exact values. The lines of the image below are almost overlapped, even for the nodes where was noted a higher difference of the values for the real part of the voltage. Fig. 4.8 Comparison between approximate and exact models on the calculation of the imaginary part of the voltages in the nodes directly disturbed and neighbour nodes extended region It can be concluded that, by extending the observation region, the accuracy of the results for the new equations of voltage calculation is increased. V. APPROXIMATE MODEL TO THE EVALUATION OF LOSSES IN DISTRIBUTION NETWORKS The main idea of this section is to present an approximate model for the losses calculation which encompassing the studies presented previously. It is also intended compare the different models presented, since the begin, relatively to the results accuracy. Fig. 4.6 Comparison between approximate and exact models on the calculation of the real part of the voltages in the nodes directly disturbed and neighbour nodes extended region The next images present the results of the tests performed for the imaginary part of the node voltages. By following the same logic, the image below shows that, even really close, the values for the node 9 have a difference. A. Approximate model of local solution In this section it will be analysed what kind of approximations will be done in the exact formula from the section III, in order to get an approximate model for the losses. 1) Concept This approximate model involves the use of the formula for the exact calculation of the losses, however, are not used all the voltage variations, but only some of them (some variations can be considered negligible). For those variations will not be used exact values but approximate values (calculated by the voltage model presented in the previous section). So, the voltage variations calculated will focus exclusively the more influent nodes (nodes connected by branches where occur the disturbances and/or nodes of their neighbourhood. 2) Procedure The next diagram presents the procedure followed by the approximate model for the calculation of the losses. Fig. 4.7 Comparison between approximate and exact models on the calculation of the imaginary part of the voltages in the nodes directly disturbed short region
The highlight in the first test: Higher losses on the two last cases, what means that a disturbance of the type has greater influence on the behavior of the network than a simple reconfiguration, as removing or inserting branches. For those cases which the disturbances affect the values of the load, the accuracy of the models decreases, being obtained results farther apart of the exact results. The approximate model developed on the section V which is based on the exact formula is much more accurate than the sensitivities model. Fig. 5.1 Procedure diagram of the approximate model for the calculation of the losses on distribution networks B. Models to compare On the next section will be checked the results obtained by the sensitivities studied in the section II, the results obtained by using the approximate model developed in the section V and also by using the exact formula presented in the section III. The proximity of the results of the models, relatively to the results obtained by the exact formula, will indicate the accuracy level of each of them on each tested situation. C. Examples and results The tests performed in this section put face to face the three models developed along the dissertation, for different operating conditions of a distribution network and upon occurrence of different disturbances on it. Also refer that, for the all tests, the approximate model for the evaluation of losses only considered the nodes involved in the disturbances, not taking into account the neighbour nodes. For the first test was used the distribution network of the figure 4.1 have being studied the value of the losses for five different cases, as remove branches ( ) and variation of the load on some nodes ( ). The following graphic presents the test results. Fig. 5.2 Losses of the figure 4.1 network, considering different disturbances, calculated by the exact formula, by the sensitivities and by the approximate model The second test performed has as main goal show the behaviour of the different models when the network is operating with different load levels at its nodes. In the network of the figure 4.1 was removed the branch which connects the nodes 1 and 6. The results are on the graphic below. Fig. 5.3 Losses in the network of figure 4.1 to different % of nominal load, calculated by the exact formula, by the sensitivities and by the approximate model The highlight in the second test: Higher accuracy for the approximate model comparatively to the sensitivities model. The sensitivities model can be very accurate when the load at the nodes is decreased relatively to its nominal values. For load values higher than the nominal ones, the error of the sensitivities model can achieve almost 40 %. The approximate model developed on the section V, in spite of being very accurate, also presents values with a higher error when the load at the nodes is above their nominal values. The third test as realized with the network of the figure 4.4, on which is removed the branch 2 4 and is inserted the branch 5 9. The loss results obtained were analysed by the approximate model of the section V and by the exact formula, for the nominal values of the load and varying also the load at some nodes. The sensitivities are not taken into account once, for not existent branches which will be insert, the sensitivities are not calculated.
In the section V was then presented the approximate model for the energy losses calculation, which is based on the approximate calculation of the voltages from the section IV. The tests performed allowed to compare this model with the exact formula and the sensitivities model. This approximate model for the calculation of losses showed itself a very accurate alternative, once, for all the considered tests, since the occurrence of disturbances at the branches, to the occurrence of disturbances in the load values of some nodes, considering also the operation of the networks at load levels above and under the nominal values, always presented results with almost negligible errors, when compared with the results of the exact formula. The calculation of the energy losses by the study of the sensitivities revealed itself a considerable solution when the accuracy is not the most important at all and just to have an idea about the influence of some parameter. Fig. 5.4 Losses in the network of the figure 4.4 considering different disturbances, calculated by the exact formula and for the approximate model. The highlight in the second test: Despite the graphic, on all of the cases, suggest a difference between the values of the two models, by observing the scale of loss values, it is possible to check that the error is very small. When disturbances occur at the load of the nodes, the losses move away from the value calculated for the base case. Also for the same cases (case 2 and 3 on the figure 5.4), the approximate model presents values farther apart of the real values, although always very close. A. Synthesis VI. SYNTHESIS AND CONTINUITY OF THE STUDY In the context of network analysis was followed a study which was suggesting different approximate models for the calculation of energy losses which could be a reliable and accurate alternative to the conventional power flow, once, as it was said in the section Objectives, for networks with a high number of possible configurations, perform this operation to each possibility can require too much time and a high computational effort. The study made in the section II presented the Tellegen s Theorem and the concept of adjoint networks as a basis to model formulas which allow calculating the sensitivity to the occurrence of disturbances of each network s parameter. In the section III was studied the modelling of an exact formula based on the previously presented Tellegen s Theorem and were made some result comparisons between the results of this formula and the results obtained with the sensitivities. In the section IV was analysed the possibility of calculate the voltages at the nodes of a distribution network in an approximate way, taking into account the generation of an approximate model and with a local solution for the calculation of energy losses. B. Continuity of the study The energy efficiency of a power network is directly related with the losses while the energy is transported until each consumer. The distribution networks are, in general, very large networks. They are complex networks which connect a very high number of nodes and can be considered the most important power networks from the economic and dimension points of view. This way, there is a great interest in find an optimal solution for the operation of these networks, relatively to their configuration. The solutions presented in this study are based on Tellegen s Theorem and on the concept of adjoint networks. This concept was addressed for the first time in the seventies, with the intent of studying the sensitivities on small electronic networks. The studies conducted in this field led to the development of new power flow equations and enabled, consequently, the generation of methods to calculate the energy losses, by doing a faster and more efficient analysis of possible configurations for distribution networks. In this context would be interesting to continue the study about the application of the presented models for distribution networks with real dimensions, trying to find, for a high number of possible configurations, an optimal operating point. REFERENCES [1] C.M.S.C. Jesus and L.A.F.M. Ferreira, Loss sensitivity formulas by adjoint networks, Electric Power Systems Research, ELSEVIER, 23 de Junho 2010. [2] C.M.S.C. Jesus and L.A.F.M. Ferreira, Comparing Power Flow Equations on a Local Basis for Distribution Networks, Int. Conf. on Power Engineering, POWERENG, Setúbal Portugal, 12-14 de Abril 2007. [3] T.S. de Almeida e B. Costa Pinto, O sector eléctrico em Portugal continental contributo para discussão, Banco BPI, 2011. [4] J. P. Sucena Paiva, Redes de energia eléctrica uma análise sistémica, IST Press, 2ª Edição, Dezembro 2007. [5] C.M.S.C. Jesus, Aplicação do Teorema de Tellegen e de Redes Adjuntas ao Cálculo de Tensões e Perdas em Sistemas de Energia Eléctrica, Dissertação para obtenção do grau de Doutor, Instituto Superior Técnico, Dezembro 2004. [6] MathWorks - MATLAB Documentation Center [Online]. http://www.mathworks.com/help/matlab/