ABOUT CONFIGURATION UNDER UNCERTAINTY OF A POWER DISTRIBUTION NETWORK Alexis Aubry ∗,1 Marie-Laure Espinouse ∗ Mireille Jacomino ∗ Bertrand Raison ∗∗ ∗ {firstname.name}@lag.ensieg.inpg.fr LAG (Laboratoire d’Automatique de Grenoble) ENSIEG-B.P 46 38402 St-Martin d’H`eres CEDEX France ∗∗ {firstname.name}@leg.ensieg.inpg.fr LEG (Laboratoire d’Electrotechnique de Grenoble) ENSIEG-B.P 46 38402 St-Martin d’H`eres CEDEX France

Abstract: The problem which is presented here concerns the configuration of a power distribution network. On the basis of an existing meshed network, configuring this network consists in building a forest which makes it possible to feed all the customers while checking electrotechnical constraints. However, some perturbations can occur and deeply damage the performances of the configuration: particularly having not served customers. This paper aims to describe the perturbations to which this system is subjected c and for which robust configurations should be built.Copyright °2005 IFAC Keywords: Power distribution network, scheduling under uncertainty, configuration and reconfiguration.

1. INTRODUCTION Load Line + switch

In this paper the configuration of a power distribution network is considered. This network is composed of several power sources, electrical lines with their switch and customers with loads as depicted in figure 1.

Customer Power source

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Fig. 1. A power distribution network

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Corresponding author

The network of figure 1 consists of 2 sources, 17 customers and 26 portions of electrical lines

with their switch. The set of the power sources must serve the set of customers with supporting their load and verifying some electrotechnical constraints (like disjunction of the sources in functioning). Configuring the network means to choose what power source will serve what customer with opening or closing the appropriate switches. So the set of the switches positions represents the configuration of the network. However, the network is naturally subject to perturbations like the uncertainty on customers demand or the loss of an electrical line. Thus the goal of this paper is to present this new problem in the domain of optimization under uncertainty and to carry out a reflection about the robustness and the flexibility of a power distribution network. In the next part, the static problem is presented with its modeling by an undirected graph. In section 3, the framework of the study is given: a resolution process of the scheduling problems in uncertain context is particularly presented with the definition of terminologies used when uncertain context must be taken into account. In the fourth section, some guidelines are given to adapt the concepts developed for scheduling under uncertainty to solve the problem of the configuration under uncertainty of a power distribution network: i.e. how the principles presented in the third section can be used to answer to the addressed problem?

of the network. The set of vertices T k are junction vertices. They represent intersection points between at least 3 switches. For instance, vertex T 1 of figure 2 is representing the intersection point between switches 1, 2 and 3 of figure 1. These particular vertices behave as customers with no load, which are not to be necessarily served. The structure of the network is represented by edges between the vertices of V . Each edge Ek ∈ E with Ek = (Vi , Vj ) for Vi , Vj ∈ V 2 represents an electrical line with one switch. The existence or non-existence of the edge (Vi , Vj ) is defined by the binary adjacency matrix A whose size is (n + m + l)2 . If Aij = 1 then the edge (Vi , Vj ) exists and does not exist else if. The demand in current of each customer is represented by the real n-size vector Wc . The configuration of the power distribution network can be modeled by a (n + m + l)2 binary matrix Q. Qij = 1 if the switch represented by the edge (Vi , Vj ) is closed and Qij = 0 if the switch represented by the edge (Vi , Vj ) is opened or if the edge (Vi , Vj ) does not exist (i.e. Aij =0).

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2. PRESENTATION OF THE STATIC PROBLEM The power distribution network is composed of m power sources which have to feed n loads (residential, commercial or industrial customers). Each customer must be served by one and only one power source. To configure the network means to open or close the good switches to serve all the customers with satisfying some technological constraints. The network of figure 1 can be modeled by the undirected graph of figure 2.

2.1 Problem formalization The power distribution network is modeled by an undirected graph G=(V,E) with V = S ∪ C ∪ T . S = {Sj/j ∈ 1 . . . m}, C = {Ci/i ∈ 1 . . . n} and T = {T k/k ∈ 1 . . . l}. Each vertex Ci represents a customer whereas a vertex Sj represents a power source. For the example of the figure 2, the vertices C1 to C17 are representing the customers of the network just like on the figure 1. The vertices S1 and S2 are representing the 2 power sources

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Fig. 2. Network modeling of Figure 1

• Electrotechnical Constraints of the problem The electrotechnical constraints inherent in the problem can be defined as follows: (1) The sources must be disconnected in functioning: an admissible configuration does not contain any path connecting two sources. (2) There cannot be loop in the resulting configuration: an admissible configuration must be a spanning forest. (3) All the customers must be served one and only one time. (4) Voltage drop must be limited. The depth of each tree (contained in the admissible solution) is bounded. (5) Each source can provide only a limited quantity of power characterized by a capacity in current. (6) Each line can support a maximal current: the edges are constrained by a capacity of maximum flow.

2.2 Nature of the problem solutions

2.4 About the perturbations

Regardless any objective function, a solution of this problem is an oriented forest whose roots are the source nodes. This forest must cover all the customers nodes and respect the previous constraints. A solution for the example of figure 2 could be the forest of figure 3. For this configuration the switches represented by the edges (C16, T 3),(C6, C7),(C17, C9) are opened whereas all the others are closed.

Naturally a lot of perturbations can occur in the electrotechnical domain: risk to lose a line, uncertainties on the customers demand, risk of non correct functioning of a switch. . . Actually loads vary with time of day, of the week and of season. Moreover the profile of the load varies with the type of customer (residential, commercial, industrial). That is the reason why, the performance (like power losses) of a power distribution network configuration can deeply be degraded by the variation of the customers demand. Computing the configuration which offers the best guarantees (to be defined) against uncertainties on the customers demand is a necessity but also an interesting and challenging problem. Using the notations presented previously, the fact that the load is really fluctuating means that the data Wc is an uncertain data. Another critical perturbation can be the risk to lose a line. When the lines are buried, it frequently happens on a building site that a line is cut inopportunely by a machine. For the case of overhead lines, an ordinary road crash can cause the fall of an electrical pylon and so the break of the cable. Taking into account the uncertainties on the demand or taking into account the risk to lose a line will not imply the same performances to be guaranteed. The methods of resolution will be undoubtedly different.

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Fig. 3. An example of solution 2.3 About the problem in certain context Searching the configuration of a power distribution network in certain context is very widespread in the literature, especially in the field of electrical engineering. A lot of objective functions can be considered. However the most used objective function in literature is the power losses. In fact Ohm’s law show that a line of resistance r dissipates r × i2 watts of power if i is the current which circulates in the line. This problem is really critic because in France the total power losses are 13TWh (i.e. 2.5% of the totality of the produced power: these values can be found in (Quinquempoix et al., 2004)). Considering the power losses, the authors of (Ahuja et al., 1993) place it in the context of convex cost flow problems and present some methods to solve these types of problem. The majority of the methods solving the problem of configuration or reconfiguration of power distribution network are randomization techniques based methods like genetic algorithms ((T¨ urkay and Arta¸c, 2005), (Ramos et al., 2005), (Duan and Yu, 2003)), simulation annealing (Gotzig et al., 1997) or tabu search (Gotzig et al., 1997). Moreover the authors of (Duan and Yu, 2003) considered that power distribution system optimization problems are NP-hard problems. Up to now, the majority of work carried out on the configuration of a power distribution network did not consider the perturbations since the computation of the configuration but made a partial reconfiguration when a perturbation occurred. In the next section, the possible perturbations to take into account will be presented.

In the next section the principles and the methods of resolution for scheduling will be approached. And then the fourth section will explain how its principles can be useful in the addressed problem for the perturbations which have been just highlighted.

3. SCHEDULING UNDER UNCERTAINTY Work presented here is set in the framework defined in (Billaut et al., 2005). The authors particularly define a process of resolution for scheduling problems under uncertainty. This process of resolution includes three steps which are defined as follows: - step 0: Definition of the static problem with the classical specifications in certain context and with the specification of the uncertainties and their modeling. The notion of quality of a scheduling must be moreover defined at this step. This step has been partially done for the addressed problem in the previous section. In fact the performances to be reached have not been yet defined. - step 1: taking into account the perturbations, a set of solutions is calculated by a static algorithm: i.e. all the necessary data are available by the algorithm to compute the solution(s).

- step 2: during the execution, a single solution is calculated by a dynamic algorithm: i.e. the algorithm works with progressive available data. The found solution will be based or not on the set of solutions of step 1. Actually there are three approaches to deal with uncertainties. They are based on the previous process and can be defined as follows: (1) The process includes only the steps 0 and 1. Thus the approach of resolution will be a proactive one: i.e. the knowledge about uncertainty is used by the static algorithm to build a robust solution. (2) The process includes only the steps 0 and 2. Thus the approach of resolution is a reactive one: i.e. the solution is dynamically computed using the progressive knowledge of the data during the production. (3) The process includes the three steps. Thus the approach of resolution is proactive/reactive: i.e. using the progressive knowledge of the data and eventually following a perturbation, either the dynamic algorithm chooses the best solution among those proposed by the static algorithm, or it repairs the current solution, or it calculates a new solution when the data of the initial problem are too much perturbed. The chosen approach of resolution will be the result of a well thought out choice which is strongly linked with the modeling in the step 0.

∗ where σN is the optimal value of the objective function for the instance N . - the relative robustness: ∗ σ(Q(N )) − σN λ(Q, σ, P ) = max ∗ N ∈P σN

The authors of (Billaut et al., 2005) give the other usual robustness criterion used in the literature. On the other hand, the author of (Rossi, 2003) gives a mathematical definition of the robustness as follows: “Being given an objective function to minimize σ, a robustness criterion λ, and a performance level to guarantee L, the solution Q is said to be robust on the risk to be covered P if and only if λ(Q, σ, P ) ≤ L.” This definition allows to preserve several robust solutions which can prove to be useful if a reactive approach follows the proactive approach. Moreover the definition in (Rossi, 2003) puts forward the fact that: 1) the robust character of a solution Q completely depends on the chosen robustness criterion lambda. 2) the solution Q will be robust only for the risk which was selected to be covered P . 3) the solution will guarantee a performance at least so high than the performance level L. Using this definition, Rossi proposes 3 problems of robustness. They can be defined as follows: (1) “knowing Q and P what is the value of L such that λ(Q, σ, P ) ≤ L?” (2) “knowing Q and L what is the risk which is covered P such that λ(Q, σ, P ) ≤ L?” (3) “knowing P and L what is the solution Q such that λ(Q, σ, P ) ≤ L?”

3.1 About robustness in scheduling When the chosen approach is proactive, the way to deal with uncertainties is to bring robustness: i.e. the performance of the computed solution must be not very sensitive to the perturbations. So the robustness could be simply defined as guarantee of performance for a risk to be covered. The necessity to define the performance and the risk which must be covered clearly appears here. To value the robustness of a solution, two definitions can be used: (Kouvelis and Yu, 1997) and (Rossi, 2003). The definition in (Kouvelis and Yu, 1997) is a minmax approach. In this approach the solution Q is robust only if it minimizes the robustness criterion λ on the set P and λ can be: - the absolute robustness: λ(Q, σ, P ) = maxσ(Q(N )) N ∈P

- the robust deviation: ∗ λ(Q, σ, P ) = max(σ(Q(N )) − σN ) N ∈P

3.2 About flexibility in scheduling When the chosen approach is proactive/reactive, the dynamic algorithm can have to repair easily a solution. To build solutions which are easy repairable, one regularly used tool is flexibility. Flexibility can be seen like the ability of a solution to undergo modifications involving an acceptable loss of performances. The ability is valued by the cost due to the modifications. This cost can be financial but not necessarily. Flexibility must be given during the proactive step. The authors of (Billaut et al., 2005) talk about flexibility on the times, on the sequence, on the resources and on the procedures. 4. DEALING WITH UNCERTAINTIES FOR THE CONFIGURATION OF A POWER DISTRIBUTION NETWORK As seen in the previous section, researches were already carried out on the dealing with uncer-

tainties in the field of scheduling. How drawn profit of these reflections and of the principles already stated in scheduling to apply them to the addressed problem?

4.1 About the uncertainties on the demand: towards a proactive approach The customers demand varies all the time: to open the refrigerator, to start the heating, the electric furnace, the washing machine have a lot of incidences on the load curve. So how taking into account this uncertainty on the customers demand to configure the power distribution network? What robustness criterion to use to measure the guarantee of performance offered by the configuration? What are the performances to guarantee by the configuration despite the uncertainty on the demand? Answering these questions is the goal of this section. By reconsidering the definition of robustness given to the previous section and in the context of the addressed problem, P will be the set of customers demands Wc to take into account. The set P can be determined using the average load curves of each customer. But what could be the objective function σ and robustness criterion λ? • Definition of the objective function σ As seen in section 2.3, several objective functions are regularly used in the literature. This objective function has to measure the quality of a configuration for a known static customers demand Wc . In the following of this paper, the objective function σ which will be used, is the power losses because this objective function is the most used in the literature and is really sensitive to the variation of the demand. Let Rij value the resistance of the line (Vi , Vj ) for the solution Q and for the demand Wc . The current Iij on the line (Vi , Vj ) can be deduced from the spanning forest defined by the solution Q (Iij is a flow which depends on Q and Wc ). Iij is nil when the switch represented by the edge (Vi , Vj ) is opened or when this switch does not exist (Aij = 0). Thus: X X σ(Q(Wc )) = Rij × |Iij (Q, Wc )|2 i/Vi ∈V j/Vj ∈V

• Definition of the robustness criterion λ If an expert is able to give a maximum power losses level (which will represent L), an interesting robustness criterion to use could be the absolute robustness: λ(Q, σ, P ) = max σ(Q(Wc )). Wc ∈P

Thus the robustness criterion λ will measure the maximal power losses on all the demand Wc of P for the configuration Q.

λ and σ are now defined, so how to answer to the three problems defined in section 3.1 ? • About the resolution of problem (1) Solving problem (1) means to answer to the question “having a fixed configuration Q and knowing the set P of possible customers demands Wc , what is the value of the maximum power losses level L such that λ(Q, σ, P ) ≤ L?”. If P is a set of explicit scenarios for Wc , there exists a tool in electrotechnical engineering (called Load Flow ) which computes the power losses for a fixed customers demand (Elgerd, 1982). To apply this tool to all the demands of P permits to find the maximum power losses of the configuration Q on P and so to determine the maximum power losses level L. However considering P as a set of explicit scenarios for Wc is not really realistic because of the number of possibilities of scenarios for a real network. What could be considered is that P will be a neighborhood around a predictive average demand Wcref . Taking into account these considerations, this problem remains an opened one. • About the resolution of problem (2) Solving problem (2) means to answer to the question “having a configuration Q and knowing the maximum power losses level L, what is the set P of demands Wc such that λ(Q, σ, P ) ≤ L?”. The answer to this question could result in the determination of a stability radius within the meaning of (Sotskov et al., 1998). This stability radius would value the maximal amplitude of a perturbation on an average reference demand Wcref for which the maximum power losses would not be exceeded. The robustness of this configuration will thus be determined by the predictive demand Wcref and a neighborhood around this demand. • About the resolution of problem (3) Solving problem (3) means to answer to the question “the set of possible customers demands P and the maximum power losses level L what is the configuration Q of the power distribution network such that λ(Q, σ, P ) ≤ L?”. This problem is certainly the hardest one. As seen before, the problem of computing the configuration of a power distribution network which minimizes power losses for a fixed customers demand is a hard one. In chapter 15 of (Billaut et al., 2005), the author has already determined the complexity of some finely perturbed problems when the initial problem was NP-hard. He proved that these problems remain NP-hard. Naturally this conclusion cannot be generalized to all the NP-Hard problems, but it could be a challenging question for the addressed problem in this paper. A first answer for this problem could be: on the

basis of an existing configuration, to develop a method which is able to modify the solution so that a robust configuration is obtained. This problem seems simpler because the configuration exnihilo must start from nothing.

4.2 About the risk to lose a line or the necessity to offer flexibility The uncertainties on the customers demand are not the only perturbations. To lose an electrical junction can also occur as shown in the section 2.4. But what can be the consequences of such a loss? And especially what is the relevance to keep on considering the power losses as objective function? Considering the configuration of figure 3, if the line represented by the arc (T 5, C11) is broken then the customers represented by (C7, C8, C9, C10 and C11) are no more served. It is naturally intolerable from the point of view of the customer. The first emergency is not to rebuild a configuration which would minimize the power losses but to serve away the customers as fast as possible. Now from the point of view of the distributor network provider, this reconfiguration must be the least expensive as possible. The cost of the reconfiguration could be valued by the number of switch changeovers. Thus, to deal with the risk to lose a line means to have a proactive configuration which is able to easily commutate to another configuration. So the configuration of the step 1 has to be as flexible as possible in term of switch changeovers. The flexibility presented here preserves same philosophy as the one presented in section 3.2 even if the both do not relate to the same data. Indeed, the stake will be here to build a solution able to undergo modifications (switch changeovers) by accepting a loss of the performance (the increase of the power losses) when a line is broken.

5. CONCLUSION The goal of this paper was to define the problem of the configuration under uncertainty of a power distribution network using the terminology of scheduling under uncertainty. The paper tried to answer to the question: “How transpose the concepts of flexibility and robustness in scheduling to the field of power distribution network?” Thus some guidelines have been carried out to deal with the uncertainties by offering robustness and flexibility to the configuration of the power distribution network. A very close attention was paid to use the terminologies suggested by (Billaut et al., 2005).

Various challenging problems concerning robustness and flexibility have been highlighted in this paper. Now methods have to be developed to answer these problems. However, whereas the study of these problems is yet only with its stammering, the reflection carried out around this subject should make it possible to obtain encouraging results soon.

REFERENCES Ahuja, R., T. Magnanti and J. Orlin (1993). Network flows: Theory, Algorithms, and Applications. Prentice-Hall. Upper Saddle River, New Jersey. Billaut, J-C., A. Moukrim and E. Sanlaville (2005). Flexibilit´e et robustesse en ordonnancement. Herm`es. Paris. Duan, G. and Y. Yu (2003). Power distribution system optimiszation by an algorithm for capacitated steiner tree problems with complex flows and arbitrary cost functions. Electrical Power and Energy Systems 25, 515–523. Elgerd, O. (1982). Electric Energy Systems Theory: An Introduction. Mc Graw Hill. New York. Gotzig, B., N. Hadjsaid, R. Jeannot and R. Feuillet (1997). Optimization of large scale distribution systems in normal and emergency state for real time application. In: IFAC/CIGRE Symposium on Control of Power Systems and Power Plants. Beijing. Kouvelis, P. and G. Yu (1997). Robust Discrete Optimization and its applications. Kluwer Academic Publisher. Dordrecht. Quinquempoix, O., S. Fliscounakis and E. Bourgade (2004). Pr´evision des pertes ´electriques sur le r´eseau THT et HT fran¸cais. Available on www. rte − f rance. com/htm/f r/vie /vie previ perte. jsp. Ramos, E., A. Exp´osito, J. Santos and F. Iborra (2005). Path-based distribution network modeling: Application to reconfiguration for loss reduction. IEEE Transactions on Power Systems 20, 556–564. Rossi, A. (2003). Ordonnancement en milieu incertain, mise en œuvre d’une d´emarche robuste. PhD thesis. Institut National Polytechnique de Grenoble. Sotskov, Y.N., A.P.M. Wagelmans and F. Werner (1998). On the calculation of the stability radius of an optimal or an approximate schedule. Annals of Operational Research 83, 213– 252. T¨ urkay, B. and T. Arta¸c (2005). Optimal distribution network design using genetic algorithms. Electric Power Components and Systems 33, 513–524.

ABOUT CONFIGURATION UNDER UNCERTAINTY OF ...

composed of several power sources, electrical lines ... an electrical line with one switch. ... T3. T2. S2. S1. T5. T1. T4. T6. Fig. 2. Network modeling of Figure 1.

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