Development of prototype software for steady state optimisation

Transcription

1 D3.2: Development of prototype software for steady state optimisation of the ETN WP3: Steady State Optimisation Proprietary Rights Statement This document contains information, which is proprietary to the "PEGASE" Consortium. Neither this document nor the information contained herein shall be used, duplicated or communicated by any means to any third party, in whole or in parts, except with prior written consent of the "PEGASE" Consortium. Grant Agreement Number: implemented as Large-scale Integrating Project Coordinator: GDF SUEZ Project Website:

2 Document Informationn Document Name: D3.2: Development of prototype software for steady state optimisation of the ETN ID: DEL_TE_WP3.3_D3.2_ _9 WP: 3 Task: 3 Revision: 9 Revision Date: 12/08/2011 Author: TE, RTE & ULg Diffusion list STB, SSB and Coordinator Approvals Author Task Leader WP Leader Name Company Date Ludovic Platbrood TE Stéphane Fliscounakis RTE Florin Capitanescu ULg Patrick Panciatici RTE Christian Merckx Miguel Ortega Vazquez TE UoM Ludovic Platbrood TE 12/08/2011 Christian Merckx TE 12/08/2011 Visa Documents history Revision Date Modification 0 01/06/2011 Initial version 1 30/06/2011 Comments and Chapter /07/2011 Comments and Chapter /07/2011 Modifications of Chapter 3 and Chapter 5 Author L.Platbrood F.Capitanescu, S.Fliscounakis,P.Panciatici F.Capitanescu, L.Platbrood, C.Merckx P.Panciatici, F.Capitanescu, Date: 12/08/2011 Page: 2 DEL_TE_WP3.3_D3.2_ _9

3 Document Informationn 4 to 7 14/07/2011 Chapter 6, Conclusions, Executive Summary 8 03/08/2011 Comments, Chapter 4 and Table of contents 9 12/08/2011 Comments L.Platbrood, C.Merckx M.Ortega-Vazquez, L.Platbrood, C.Merckx All F.Capitanescu, P.Panciatici, L.Platbrood, C.Merckx Date: 12/08/2011 Page: 3 DEL_TE_WP3.3_D3.2_ _9

4 Executive summary This report presents PEGASE project. the works performed in the context of task 3 of work-package 3 of the The report deals with the solution of Security-Constrained Optimal Power Flow (SCOPF) problems to provide appropriate anticipated operating points (or base cases) to Transmission System Operators (TSOs) of the European Transmission Network (ETN) the day-ahead of operation. The main topics investigated are, on the one hand, the ability to consider discrete variables, stemming from the model of the equipments or the decision variables involved in the process, in the SCOPF formulation, and, on the other hand, the ability to deal with the N-1 safety of such a large network as the ETN, especially given the high number of postulated contingencies. We first present the research performed on the handling of discrete variables in SCOPF problems. Three different kinds of problems where discrete variables intervene have been investigated: problems related to the model of physical equipments, problems involving the mitigation of corrective actions with respect to expensive preventive actions and a new approach to consider the increasing level of operational uncertainty due to the growing number and size of renewable energies. The use of MPEC approach for the first kind of problems applied to the ETN network shows that approximated solutions can be found. The two last kinds of problems are innovative approaches that have been illustrated through demonstrative tools on academic test cases. We then present prototype software embodying the result of the research on the handling of large number of contingency constraints. The prototype integrates two complementary approaches, namely iterative SCOPF with appropriate contingency filtering and network compression, to downsize the SCOPF problems to optimisation problems manageable by nowadays existing solvers. First results on a representative problem with respect to the size of most European national systems (2500 nodes and about 1300 contingencies) are presented. These very interesting results show that the algorithm converges to a solution which is very similar to the solution of the full problem, when it is possible to build and solve this latter. Although the computation time may still be prohibitive, the algorithm appears to be robust and demonstrates its ability to solve large scale problems formerly unattainable. Date: 12/08/2011 Page: 4 DEL_TE_WP3.3_D3.2_ _9

5 Table of content D3.2: Development of prototype software for steady state optimisation of the ETN... 1 WP3: Steady State Optimisation Introduction Deliverable organisation SCOPF problem formulation General SCOPF formulation Base case State immediately after a contingency Correctivee actions Modelling of corrective actions General formulation of the problem Practical SCOPF formulation Prototypes and demonstrator Modelling of discrete variables in SCOPF problems Huge number of contingencies Modelling of discrete variables in SCOPF problems Introduction: Mixed Integer Non Linear Programming ETN Related Problems Handling the integrality constraints modelling decision variables Problem formulation PST modelling Numerical simulations on the IEEE 30 bus system Handling the integrality constraints modelling phase shifting Problem formulation Two-step solution of the OPF problem Numerical simulations on the EHV PEGASE networks Integrality constraints to model connection/disconnection of compensation banks Problem definition Numerical simulations on the IEEE 30 bus system Integrality constraints to model start-up/shutdown of generators Problem definition More realistic scenario with smaller PSTs angle range on the network UCTE + TEIAS Combining IP and SQP methods on the network UCTE + TEIAS Date: 12/08/2011 Page: 5 DEL_TE_WP3.3_D3.2_ _9

6 4.7. Conclusion Information on standard MINLP solvers Extracts from the version 1.3 of BONMIN Users Manual (Pierre Bonami and Jon Lee) Extracts from the version of KNITRO Users Manual (Richard A. Waltz) Extracts from SNOPT : An SQP Algorithm for Large-Scale Constrained Optimization Software and Hardware Configuration Solvers Options Prototype for the treatment of potentially huge number of contingencies Functional requirements Assumptions regarding the use of the prototype Proposed approach Contingency filtering techniques Network Compression technique Proposed joint approach combining contingency filtering and network compression Peculiarities of the corrective approach Prototype description Main modules Global ISCOPF-NC scheme Solving SCOPF Security Assessment Contingency Filter Post-Contingency OPF Network Compression Update of the SCOPF with selected contingencies ISCOPF-NC Main Loop Factory tests Test case description Minimum active power deviation with preventive approach and Non Domination Minimum active power deviation with preventive approach and ranking based on Severity-Index Reactive power alignment with curative approach and Non Dominancy Reactive power alignment with curative approach and ranking based on Severity-Index Conclusions Perspectives on performances Identification of binding contingencies using Xpress-MP Date: 12/08/2011 Page: 6 DEL_TE_WP3.3_D3.2_ _9

7 6.1. Motivation Convex hull and active constraints Identification of binding constraints Mathematical formulation Security Criteria Graphical solution Solution as an optimisation problem Solution with Xpress-MP and contingencies identification Approaches to identify the binding contingencies Binding contingencies SCOPF solution (dc): binding contingencies identification from the set of binding constraintss Effect of the objective function Presolve analysis of the optimisation problem Effect of the objective function Primary positive and negative aspects of each approach Testing on the modified IEEE RTS system Analysis of the required computation capacities Notation Size of A, with a direct calculation Columns calculation Size of A for various systems Size of A for systems of different sizes with a proxy Files size Perspectives Conclusion Bibliography Date: 12/08/2011 Page: 7 DEL_TE_WP3.3_D3.2_ _9

8 1. Introductionn Optimal Power Flow (OPF) is a class of optimisation problems related to electric power systems. It includes equality constraints corresponding to the network equations and inequality constraints corresponding to operational limits (e.g., branch currents, bus voltage magnitudes) and physical limits of equipments (e.g., generator active/reactive powers, transformerr ratio/angle, etc.). The aim of an OPF problem is to provide an optimal operating point of the network according to some objective function while satisfying postulated constraints. Security Constrained OPF (SCOPF) is a generalization of the OPF problem where the optimal solution satisfies base case constraints and constraints relative to a given set of contingencies (generally, loss of one or many equipments). The latter constraints can be met either instantaneously (known as preventive approach) of after a restoration time was granted (known as corrective approach). In the practical context of operation planning and operation of power systems, decision making is carried out in an iterative fashion at different timeframes from day-ahead to minutes ahead. A classical SCOPF is generally used to this purpose. The objective of this SCOPF is to ensure system security at the lowest possible cost. However, in the more and more uncertain nowadays operation environment (e.g. due to intermittent renewable generation, load evolution, unforeseen topological changes in the internal/external system, etc.) a more recent strategy relies on the evaluation of possible future scenarios so as to identify the most difficult ones and to determine preventive measures that would be possible to maintain the security of the system under these scenarios. In this context a reasonable and commonly adopted strategy consists in identifying in advance potentially difficult scenarios, and in particular the worst-cases, and at the same time in postponing as long as possible the commitment of the most costly actions, to take advantage of the reduction of uncertainty over time and the possibility to assess again in due time whether or not these actions are indeed required. In day-ahead planning, the focus is on ascertaining whether in the worst cases (very extreme patterns of power injections and contingencies that could show up over the next day) the operator will still have sufficient controllability of the system to ensure the security of the system by combinations of preventive/corrective actions. The concept is a look-ahead preventive security assessment dealing with uncertainty, aimed at determining whether maintenance actions should be postponed or accelerated and assessing whether additional Mvar or MW reserves should be purchased for the next day. This problem is a feasibility problem rather than an optimization problem. In operation, progressively closer to real-time, the goal is to postpone until the last moment the implementation of expensive preventive actions. The problem becomes an optimization problem, with the objective to minimize cost of preventive actions by preparing for corrective actions. In emergency mode,, subsequently to a contingency, the job is to apply heroic actions to avoid cascades and subsequent blackouts. These corrective actions need to be prepared in advance, and possibly activated automatically, given the small amount of time available to apply them. To address these three types of problems coherently in the different timeframes, engineers establish at each decision stage a set of possible scenarios representing uncertainties around the best-guess scenario of future evolution, and they exploit these scenarios to first check that enough control resources will be available for the subsequent stages so as to cope even with the worst scenario during the future operation, and if this is not the case they decide on the actions that need to be taken at the current time step. In practice, the circumscription of uncertainties and worst cases is often decomposed based on the knowledge of structural weak points in the power system. Indeed, the domain experts know these weak points because they are related to local lack of generation, of transmission lines, of reactive support. This decomposition simplifies greatly the solution of the analysis problem. Indeed, for each weak point, the formulation of this problem may be reduced to a manageable optimization problem, first searching for the most extreme patterns of uncertainties, and then checking whether routine preventive/corrective actions can ensure the security of the system under these conditions. Date: 12/08/2011 Page: 8 DEL_TE_WP3.3_D3.2_ _9

9 2. Deliverable organisation The document is not dedicated to an in-depth description of each SCOPF solution algorithm but provides a global overview of the algorithm and the modelling works that have been carried out during the WP3.3 task. It also summarizes the main functionalities that can be expected by end-users. The required functionalities are justified by the specifications proposed by the works of WP3.1 and WP3.2 and presented in [22, 23] ]. The remaining of this report is organized as follows. Chapter 3 provides the description of the SCOPF problem. A first general description is given with additional explanations on the peculiarities of SCOPF problems. Then a more practical formulation is proposed. Chapter 4 presents the works realised on the modelling of SCOPF involving discrete variables. Such variables are used either to model decision variables or to model devices with discrete behaviour, such as transformer tap positions. Chapter 5 presents an algorithm to address SCOPF problems with a huge number of potential contingencies. The algorithm combines two methods in an iterative way in order to reduce the SCOPF problem size to comply with performances that existing solvers can nowadays hold, hardware memory limitations and running time constraints imposed by operational conditions. Chapter 6 gives an overview of investigations undertaken to develop an approach to reduce large SCOPF problems using DC approach and pre-solving capabilities of linear solvers. This approach was originally imagined as a possible extension or an alternative of the algorithm developed in chapter 5, but was finally abandoned due to poor expected results. Chapter 7 draws the final conclusions and gives perspectives on future works. 3. SCOPF problem formulation This chapter describes a very general formulation of the SCOPF problem, with all peculiarities of such problems to deal with preventive and corrective security. A more practical formulation is then proposed. This latter formulation will be used in the implementation of the prototype proposed in chapter General SCOPF formulation When optimizing the operation of power system, we consider different states of the system: The base case (or pre-contingency state), the state immediately after a contingency occurs and various states after contingency occur and after corrective actions (automatic or manual) take place Base case The base case state is considered as acceptable (safe) if the system can remain in this state indefinitely withoutt any problems for feeding consumption and allowing generation. It is generally defined using limits (L 0 ) of some quantities characterizing the state of the system (lines or transformers flows, voltage magnitude and phase angle...). This base case can be changed by controls (preventive) before any occurrence of contingency to ensure that the state of system is inside the domain defined by constraints (L 0 ), but also that the "post-contingency " and "postcontingency after corrective actions" states are acceptable State immediately after a contingency The situation after a contingency is determined by computing the effect of the loss of an equipment, taking into account automatic mechanisms that react in a very short time. Usually, in Date: 12/08/2011 Page: 9 DEL_TE_WP3.3_D3.2_ _9

10 the optimization problem, the fast controllers are not modelled but they are considered as ideal: for example, a generator erator is considered as a fix injector of active power with constant stator voltage and reactive e power limits without any detailed models of AVR or primary power / frequency governor. Given the time required for the implementation of other corrective actions (changes all the best and 10 seconds time constant of the order of a minute), it is necessary that the system state after a contingency is acceptable during this certain duration (about 1 min.). This state is defined by meeting the new constraints L 2 Figure 1: Security thresholds Corrective actions Each corrective action (u c ) required a certain time to be implemented, so the state before the implementation of this action must remain acceptable, i.e. respect the limits L 1,c If the threshold L 1,,c is exceeded, the corrective actions must bring the system below this threshold. These actions are automatic or manual, their implementation times vary. Typically, AGC takes from 10 to 15 minutes; start-up of a gas turbine takes 20 min. Note that the same is true for manual switching actions (transfer of charge, bus splitting/merging ) The three security thresholds L 0, L 1, L 2 are illustrated in Figure Modelling of corrective actions Both manual and automatic corrective actions are usually triggered when threshold L 1 is exceeded. Modelling in an optimization problem of this mechanism is quite complex and requires the use of discrete variables to model this conditional action. An approximation that greatly simplifies the problem is usually used to associate the corrective action to a contingency. The main problem is to define a priori this association because in reality, it is usually noncompliance of a constraint that triggers corrective action. The second problem is that this action is not associated with a specific constraint but it can be implemented to meet all constraints. There e is a specific issue about transformers with on load tap changers of transformers adjusters. A standard approximation consists in adding the variable ratio" in the problem with these min- max bounds. This implies that the set point voltage would then be set to the value of the voltage obtained at the optimum which is not the case in reality. Strictly speaking, we should add the constraint on voltage magnitude of the controlled side of the OLTC (V= V c ) in the problem where V c is given and fixed but it may create difficult problems of feasibility if the ratio hits its limits. If OLTC are used (in this quite optimistic way) to check the state immediately after a Date: 12/08/2011 Page: 10 DEL_TE_WP3.3_D3.2_ _9

11 contingency, one must pay attention to the value of the minimum threshold L 2 in order to reflect the time of action of these tap changers General formulation of the problem The idea is to prioritize corrective actions with respect to preventive actions which are much more expensive. Indeed, even if corrective actions are expensive, they are conditioned by the violation of a constraint (or the occurrence of a fault) that leads to lower average costs. So preventive actions are implemented before any contingency to ensure that the system state is inside the domain defined by the constraints (L 0 ) but also that the "post-contingency and "post contingency and corrective actions" states are acceptable and first that the global problem is feasible. min,, 0,,, 0,,,,,,,, 0,,, Where: C : set of postulated contingencies 0 : subscript meaning that the concerned variables belong to the pre-contingency state k : subscript meaning that the concerned variables belong to the k th post-contingency state D : set of corrective actions c : subscript meaning that the concerned variables belong to the c th post-corrective actions state k, c : subscriptss meaning that the concerned variables belong to the k th post-contingency and c th post-corrective actions state x : set of state variables u : set of control variables f ( x, u) : objective function g ( x, u) : set of equality constraints h ( x, u) : set of inequality constraints An important issue is the optimization of the balance between preventive and corrective actions. The objective is to minimize the costly preventive actions by taking into account all the possible corrective actions. A new original formulation of the problem of minimization of preventive actions taking into account corrective actions has been proposed. Date: 12/08/2011 Page: 11 DEL_TE_WP3.3_D3.2_ _9

12 Boolean decision variables trigger corrective actions only if at least one violation of the thresholds L1,c occurs while the objective function tends to minimise the use of preventive actions to respect the thresholds L0 and L2. Eventually, preventive actions are used to ensure that corrective actions can respect L1,c limits. According to this general definition of the previous section, the relationship between the controls in the pre- and post- contingency conditions defines two kinds of security strategies: Preventive security: no post-contingency rescheduling of control variables is allowed, except those due to the automatic response of the system. Hence, post-contingency operating constraints with limits L2 must be satisfied without any further action. Corrective security: in contrast to preventive security, some control variables are allowed to be rescheduledd in a given range to remove any violated operating constraint in post- actions can be contingency states with limits L1. From the here above definition, the concept of preventive and corrective summarized in the following way: Preventive actions are all modifications of control variables in the pre-contingency state required to ensure the respect of both pre-contingency and post-contingency constraints, either after the automatic response of the system for the preventive security or after the rescheduling of control variables for the corrective security. It should be mentioned that all modifications of control variables destined to the optimisation of the pre-contingency state also belong to preventive actions. Corrective actions are rescheduling of control variables in the post-contingency states to remove any violated constraints. Chapter 4 will provide an illustrative example with this approach on the IEEE 30 bus system. Date: 12/08/2011 Page: 12 DEL_TE_WP3.3_D3.2_ _9

13 Practical SCOPF formulation As explained in the previous paragraph, the general preventive/corrective SCOPF is very complex and some approximations must be done to allow a practical implementation. We propose here to associate a set of corrective action to any contingency in order to avoid any conditional action which leads to Boolean decision variables. The proposed practical formulation of SCOPF is Where: uk min f ( x g( x0, u0 ) = 0 h ( x0, u0 ) 0 s. t g k ( xk, uk ) = 0, k C h k ( xk, uk ) 0, k C duk τ rest uk u dτ max : the change in control variables due to the automatic control of the system following th the k contingency (e.g. primary frequency regulation). τ rest du k dτ : allowed restoration time (in particular, max : maximum allowable rate of change of control variables * * * * The optimal solution ( x, u, x k, u ) of a SCOPF can be interpreted as the best pre- * * * * contingency operating point ( x, ) able to reach the steady-state operating point ( x, ) within the maximal time, u ) contingency is obtained by setting k 0 u0 rest 0 u τ rest k τ rest duk dτ max = 0 in the preventive approach). τ when contingency k occurs. The sta τ rest equals to zero.,k C k u k ate immediately after Observe that, for the sake of simplicity with respect to the SCOPF formulation of the previous sub-section, we do not model together the states immediately after contingency and after contingency and corrective actions. This practical approach will be used in the concepts developed in Chapter 5. Date: 12/08/2011 Page: 13 DEL_TE_WP3.3_D3.2_ _9

14 3.2. Prototypes and demonstrator The purposes of the prototypes developed in WP3.3 task are two-fold Modelling of discrete variables in SCOPF problems The first task of WP3.3 is to solve realistic SCOPF problems that involve both discrete and continuous variables and belong to the class of optimisation problems known as Mixed Integer Linear Programmingg (MILP) or Mixed Integer Non Linear Programmingg (MINLP). Three types of problems have been addressed Devices with discrete behaviours The first prototype will focus on formulating the problems with discrete variables for the modelling of devices with discrete behaviour (PST, capacitor/reactor banks, start up/ stop generators,) for a very large system.. Optimisation of preventive actions taking into account corrective actions Stemming from the general formulation of the SCOPF, the problem is to determine optimal preventive actions to cover postulated contingencies (at least one) while taking into account corrective actions. Boolean decision variables are used to ensure that corrective actions appear only when at least one violation of the thresholds L1 occur. Discrete variables involved in the determination of a worst-case situation with respect of a contingency The uncertainty on the anticipated state of the system is so high that assessing the security of the most probable state is no longer sufficient. A new approach has been proposed to deal with this difficulty. Further details on this approach can be found in [25] and [27]. The treatment of discrete variables leads to a very large Mixed Integer Non Linear Programming (MINLP) problem. The state of the art shows that existing MINLP solvers couldn t find a solution in most cases in reasonable computation time (say less than 1 min.). Therefore, some simplifications are required to deal with this problem. Tests on the most well-known MINLP solvers will be presented to confirm this point. The alternative is to use an approximate solution based on MPEC constraints. In this case, binary constraints b {0, 1} are substituted by MPEC constraints min (b, 1-b) =0. For this project, it has been decided to model the optimization problems with the AMPL [11] language. The use of this modeling language is justified by two important advantages in the perspective of the various tests we were planning to do. Firstly, with AMPL, the modeling is open and can be easily modified. As the modeling of the discrete variables was one of the goals of our work, this was a very interesting feature. AMPL includes an automatic differentiation functionality which avoids the laborious coding of the derivatives for continuous variables. Secondly, AMPL is supported by most of the professional solvers. It thus also avoids the implementation of an interface for each tested solver. AMPL performances are good on small and middle size networks, but on very large networks as the ETN network, the required memory and the computation time becomee very large. This is the price to pay to have a very flexible tool and all approaches based on automatic differentiation have the same drawback. Our tests have demonstrated that there is currently no MINLP solver which allows an operational implementation of a full SCOPF (multi contingencies) with discrete Date: 12/08/2011 Page: 14 DEL_TE_WP3.3_D3.2_ _9

15 variables even on middle size networks, and AMPL is not the main limiting factor as it will be explained later. Indeed, the approach presented in section 4.2 requires the duplication of post-contingency constraints in order to insure the priority of the corrective actions on the preventive ones. When considering discrete variables, this leads to a problem of a complexity which is not addressable by actual MINLP solvers. Ideally, we should have dealt with all these problems together for the pan-european grid with a large number of contingencies but for the here above exposed reasons, this approach is intractable. In orderr to reduce the problem complexity, we propose an approach based on optimization for a single contingency with corrective actions associated to constraints violations. An external loop could be designed to solve multi-contingency problems. This external loop is required only if preventive actions are needed. The implementation of this approach could be very efficient due the possible very high level of parallel computations (by contingency). Nevertheless, this problem is still very complex in presence of other discrete variables. Consequently, in order to be able to address the ETN network in our tests, we choose in the prototype to focus on modelling devices implying discrete variables. Concerning the optimization of preventive actions taking into account corrective actions and the computation of the worst case situation related to a single contingency and its associated corrective actions, no acceptable industrial solution have been found for large systems. Hence, these functionalities will not be implemented in a prototype. Nonetheless, in order to show the potential of the worst case method, a demonstrator will be made available Huge number of contingencies The second task of WP3.3 is to develop a SCOPF solver able to optimize a realistic model of the European Transmission Network (ETN), considering only continuous variables 1. The size of the ETN and the resulting huge number of contingencies require the solution of a very large optimisation problem that prevents the use of classical SCOPF or generic solvers. A second proposed prototype will combine an iterative approach that reduces the size of the SCOPF by limiting the number of contingencies included, and a method that reduces the size of the post-contingency state itself by focusing on the most affected areas of the network. The model of the ETN proposed by WP6 contains about nodes and branches. Hence, if we consider that each contingency represents the loss of a single piece of equipment, about contingencies must be considered. Building a SCOPF problem with several thousands of contingencies is nowadays impossible due, at least, to memory limitations and unacceptable computation time without any guarantee that the algorithm could be able to manage such a number of variables and constraints. For current SCOPFs, a maximumm of ten contingencies could be considered and the time needed should be even larger than the 15 minutes requested. 1 Discrete variables are treated as continuous, approach known as continuous relaxation Date: 12/08/2011 Page: 15 DEL_TE_WP3.3_D3.2_ _9

16 4 Modelling of discrete variables in SCOPF problems 4.1 Introduction : Mixed Integer Non Linear Programming ETN Related Problems The mixed integer nonlinear programming (MINLP) problems can be seen as belonging to the more general class of optimization problems called mathematical programming with equilibrium (or complementarity) constraints (MPEC) [1]. However, while for convex MINLP problems of reasonable size mature solvers exist, the handling of complementarity constraints 1 (MPEC) as NLPs has been commonly regarded as numerically unsafe, due to the failure of the Mangasarian- Fromovitz constraint qualication (MFCQ) at any feasible point. In the context of power systems, the use of MPEC formulation in OPF has not been deemed for long time as satisfactorily reliable. Up to now very few problems in the area of electric power systems have been formulated as MPEC [2,3,4,5,6,7,8,9]. Most of these problems relate to gaming within electricity markets and use a linear model [2,3,4,5]. The application of MPEC in the context of nonlinear OPF has received much less attention so far [6,7,8,9]. Most of these approaches solve the MPEC problem using commercial solvers e.g. LOQO [10]. However, the MPEC eld has ourished in the last years and more and more MPEC solvers emerged [15]. The aim of this report is to compare various MINLP and MPEC commercial solvers accessible through the modeling language AMPL [11]. To this end we use four MINLP optimal power ow (OPF) test problems in which discrete/integer variables can be also modeled by complementarity constraints. We rst present in section 4.2 simulations on the IEEE 30 bus network focusing on decision variables 2 as binary variables. We then present in section 4.3 simulations on the very large EHV PEGASE network focusing on phase shifting transformers (PST) taps as discrete variables. Next we illustrate in section 4.4 the handling of integrality constraints stemming from the connection/disconnection of compensation banks using the IEEE30 bus system. Finally we illustrate in section 4.5 the handling of integrality constraints stemming from the the start-up/shut down of generators using the very large EHV PEGASE network. To make easier and understandable the comparison of MPEC and MINLP approaches, each binary variable x in a MINLP solver is simply replaced by the constraint min(x, 1 x) = 0 in a MPEC NLP solver. Note that since the OPF problems considered in this report include both active power controls (e.g. generators active power, PST) and reactive power controls (e.g. on-load tap changer (OLTC) transformers, PST [12]) together with their associated constraints (e.g. limits on branch currents, voltage magnitudes, and controls) and that discrete variables (e.g. PST taps) have a signicant impact on both active and reactive power ows, a problem decomposition into active power and reactive power subproblems is invalid. Hence, the use of the popular DC approximation and MILP solvers is prohibited. Furthermore the OPF problems are non-convex even if integrality constraints are dropped. The options of various MINLP and MPEC solvers investigated in this report are provided in section Handling the integrality constraints modeling decision variables Problem formulation A drawback of the conventional security constrained OPF (SCOPF) is that it does not model the system operator operating rules which associate a pre-dened set of corrective actions with a given post-contingency constraint violation, the corrective actions being activated only if the 1 i.e. pairs of inequalities, where in each pair at least one inequality must hold with equality. 2 A decision variable is a binary variable used to activate a corrective action if and only if a monitored constraint (e.g. a branch current limit) is violated. Date: 12/08/2011 Page: 16 DEL_TE_WP33_D3.2_ _9

17 constraints are not satised by preventive actions. The activation of these corrective actions can be modeled using binary decision variables. We focus on the solution of the SCOPF problem with decision variables which can be formulated as follows: F (V, u o ) = 0 min u o ū o 2 (1) where s.t. C(V, u o ) L o F post_c (V post_c, e post_c, u o ) = 0 C post_c (V post_c, e post_c, u o ) L 2 F post_c+ca (V post_c+ca, e post_c+ca, u o, u e ) = 0 C post_c+ca (V post_c+ca, e post_c+ca, u o, u e ) L 1 Controled T hresholds (L 1 ) l b l (C post_c (V post_c, e post_c, u o )) l (L 1 ) l + λ l b l l CT u e ū e j u max e u min e j b l l CT (u e ) j = x k,j d k,j x k,j = 1 k k Integrality of variables b and x u o and u e and denote respectively the vectors of preventive and corrective control decisions ; the constant vectors L 0, L 1, and L 2 denote the security thresholds respectively associated to the "base" case, the "post-contingency before corrective actions" case (subscript post_c ) and the "post-contingency after corrective actions" state (subscript post_c+ca ); V, V post_c, V post_c+ca stand for the complex voltages vectors associated to the three states; F, F post_c, F post_c+ca are nodal power balances associated to the three states; C, C post_c, C post_c+ca are security constraints (positive functions) associated to the three states; e post_c, e post_c+ca are bounded scalar variables necessary to satisfy active power balances; the constant vector λ is xed such that L 2 L 1 + λ; CT is the set of Controled Thresholds. Thanks to the decision variables b, corrective actions are activated only when a violation of the thresholds L 1 occur for at least one index l in CT. Note that thresholds L 2 are satised by preventive actions only. For this study, we incorporate all the current limits L 1 in the set CT. Preventive actions are modeled as modications of generators real power and corrective actions are dened as adjustments of phase shifter tap position. Note that solving the SCOPF problem (1) for very large systems such as the PEGASE network models of next sections is intractable on current computers due to the duplication of postcontingency constraints. Date: 12/08/2011 Page: 17 DEL_TE_WP33_D3.2_ _9

18 4.2.2 PST modeling Figs. 1 and 2 describe the technology and model of PSTs. The phase shift regulation may be on the shunt or the series transformer. Figure 1: In-phase regulating auto-transformer When adjusting transformer tap settings, the current ratio of the in-phase transformer r and the current phase shift α become discrete variables, X(r, α), the sum of the in-phase transformer and phase shifter reactances, and ρ vary as a consequence. Figure 2: One phase diagram V 2 + jx(r, α)i = rv 1 (1 + e jθ tan α sin θ tan α cos θ ) (2) If φ pst max, Xmax pst and X pst 0 design respectively the maximal phase shift, the equivalent PST reactances at maximal phase shift and at zero phase shift, the hypothetical parameter values adopted for this test are : φ pst max = 10 X pst max = 0.15 p.u. X pst 0 = 0.1 p.u. (3) Assuming the reactance of the regulating winding varies as the square of the turns, the current equivalent PST reactance is given by [12]: X l (φ pst l ) = X pst 0 + (Xmax pst X pst 0 ) tan2 φ pst l tan 2 φ pst max (4) where φ pst l is the current phase shift of the PST l. The discrete control device is illustrated by enforcing integer values in the range [ φ pst max, +φ pst max] for the quantity φ l expressed in degrees. Date: 12/08/2011 Page: 18 DEL_TE_WP33_D3.2_ _9

19 The module ρ l (φ pst l ) of the complex turn ratio is represented as function of φ pst l and is given by : ρ l (φ pst l ) cos φ pst l = 1.0 (5) We believe that for this very complex real-life model of PST the popular decoupling of the SCOPF problem in active and reactive power sub-problems is not valid Numerical simulations on the IEEE 30 bus system Fig. 3 shows the one-line diagram of the modied IEEE 30 bus system. Note that three identical quadrature booster PSTs have been installed in series with the initial lines on the branches 15-18, and Figure 3: Modied IEEE 30-bus The three levels of branch current limits L 0, L 1, L 2 are set to ratemv A/V init, 2.0, and 3.0 p.u, respectively. The lower and upper voltage magnitude limits at all buses are xed to 0.9 and 1.1 p.u. Table 1 provides the main characteristics of the IEEE 30 bus system. Table 1: IEEE 30 bus system voltage level (kv) number of nodes number of lines Table 2 provides the solvers' performances for the solution of SCOPF problem (1) considering constraints only for a single contingency (the outage of line 14-15). Date: 12/08/2011 Page: 19 DEL_TE_WP33_D3.2_ _9

20 Table 2: Solvers' performances for the solution of SCOPF problem (1) (contingency: outage of line 14-15) solver diagnostic cpu time objective value KNITRO MPEC Locally optimal solution found KNITRO BOOLEAN Locally optimal solution found (383 nodes) BONMIN Optimal (9557 iterations and 18 nodes) COUENNE After 7700 nodes, 3622 on tree, no solution e +50 SNOPT+NRS functions Optimal solution found In this table NRS acronym denotes the use of the smooth natural residual function. The latter can be dened as: φ NRS (a, b) = 1 2 (a + b (a b) 2 + ab ) (6) σ NR for a xed parameter σ NR > 1 2. Since the property a, b 0 and φ NRS (a, b) 0 min(a, b) = 0 holds, the separate complementarity constraints min(x 1i, x 2i ) = 0 can be replaced by the lumped version x 1, x 2 0 and e T Φ(x 1, x 2 ) <= 0 where Φ(x 1, x 2 ) is a vector composed of functions φ NRS (x 1i, x 2i ). For this test σ NR is xed to 1. Table 3: Base-case reactive power generations bus knitro mpec knitro boolean bonmin snopt + NRS functions qmax qmin Table 4: Post-contingency reactive power generations bus knitro mpec knitro boolean bonmin snopt + NRS functions qmax qmin We notice that KNITRO MPEC and BOOLEAN, BONMIN and SNOPT NRS nd exactly the same solution in terms of discrete variables and generators' active power, i.e p.u. at bus 1 and p.u. at bus 2. Furthermore they also identify the same binding reactive power limits after the contingency. The slightly negative nal integrality gap of KNITRO BOOLEAN (abs/rel= -5.96e-05/-5.96e- 05) can be interpreted as an indication of either a non convex problem or an insucient accuracy of continuous NLP subproblems (see section 4.8.2). The column SNOPT NRS of the Tables 2, 3, 4 show the excellent results of the sequential quadratic programming method implemented in SNOPT. This are attributable to the theoretical reasons explained in [13]. Date: 12/08/2011 Page: 20 DEL_TE_WP33_D3.2_ _9

21 4.3 Handling the integrality constraints modeling phase shifting transformers Problem formulation In real-life networks nding the tap positions of PSTs is a dicult daily task for system operators. In order to investigate the ability of various solvers to handle the integrality constraints stemming form the real-life model of PSTs [12] we consider the following MINLP OPF problem [8]: where s.t. Q min i Pi min min i G p 2 i (7) i G Pi init + p i = φ i (θ, V, ρ, α, Y, a) ψ i (θ, V, ρ, α, Y, a) Q max i Pi init + p i Pi max i N G Pi init = φ i (θ, V, ρ, α, Y, a) Q init i = ψ i (θ, V, ρ, α, Y, a) V oltage Limits : i N Vi min V i Vi max On Load T ap Changer Limits : o OLT C ρ min o ρ o ρ max o Current Limits : l L I l (θ, V, ρ, α, Y, a) Il max P hase Shift T ransformer Limits : d P ST x t,d = 1 t A(d) α d = ρ d = Y d = a d = t A(d) t A(d) t A(d) t A(d) x t,d α t,d x t,d ρ t,d x t,d Y t,d x t,d a t,d Integrality of variable x : x t,d {0, 1} N, G, L, OLTC, PST denote the sets of buses, generators, branches, OLTCs, and PSTs, respectively; θ and V are the vectors of voltage angles and magnitudes; φ and ψ are the vectors of the computed active and reactive injections; I is the vector of the currents through the lines and transformers; Date: 12/08/2011 Page: 21 DEL_TE_WP33_D3.2_ _9

22 ρ is the vector of OLTCs ratios. Note that we have not shown explicitely that these variables are discrete because, as will be explained in subsection 4.3.2, we treat these variables by continuous relaxation; V min i and V max i are the voltage magnitude limits; P init, Q init are the initial active and reactive power injections; P min, P max, Q min, Q max denote the active and reactive power limits of generators; for the PST d, α d is the phase shift, ρ d the turn ratio, Y d is the admittance, a d is the loss angle, A(d) is the set of tap positions, and t is the tap position. Note that in the variables α d, ρ d, Y d, and a d depend on the tap position t of the variable x t,d. The control variables of this OPF problems are: generators' active power, generators' terminal voltage, ratio of OLTCs and the internal variables of PSTs. Note that when this MINLP OPF problem is solved by a MPEC solver the intergrality constraints on variable x (x t,d {0, 1}) are replaced by: min x t,d (1 x t,d ) = 0, d P ST, t A(d). (8) A feature of OPF problem (7) is that, even if the discrete variables x were xed, it would remain non-convex. This requires the choice of the branch and bound method and a specic interpretation of stopping criterions for the MINLP solver as described in sections and Two-step solution of the OPF problem We solve this OPF problem is two steps distinguishing between the treatment of discrete variables stemming from OLTCs and PSTs. This is due to unlike the PSTs, the OLTCs ratios concern a large number of entities and require only a discrete value per transformer. This procedure is as follows. In the rst step the OPF problem is solved considering the ratios of OLTCs as continuous. At the solution of this OPF the ratio of OLTCs are rounded-o. In the second step, the optimization assumes xed values for the OLTCs ratios. In this step the key point is the respect of all the integrality constraints in such a way that the process can be stopped. Figure 4 summarizes the two-step solution of OPF problem (7). First step Coordination of generators + OLTC + PST Continuous OLTC Pinit provided by initial data Second step Coordination of generators + PST Fixed OLTC and Pinit provided by the rst step Figure 4: Two-step solution of the OPF problem (7) Numerical simulations on the EHV PEGASE networks Assumptions and description of the test system At the date of this report, there are still some unsolved features concerning the data of PEGASE networks such as: absence of generators' active power limits; absence of location and size of shunt compensation banks; absence of realistic branch thermal limits; Date: 12/08/2011 Page: 22 DEL_TE_WP33_D3.2_ _9

23 very large phase shifter angle ranges; presence of negative resistances; These features signify that nal versions of PEGASE networks are not today available. At the time of performing the numerical simulations of this report we have made some assumptions concerning these aspects that are detailed hereafter. Tables 5 and 6 provide the main characteristics of the EHV PEGASE and UCTE + TEIAS networks. Table 5: UCTE network (8387 buses) voltage level (kv) number of buses number of lines Table 6: UCTE + TEIAS network (9241 buses) voltage level (kv) number of buses number of lines Tables 7 and 8 provide the transformer characteristics of the EHV PEGASE and UCTE + TEIAS networks. Table 7: UCTE transformers number type xed 2030 OLTC PST 1 Table 8: UCTE + TEIAS transformers (+ step-up transformers) number type xed 2237 (6705) OLTC 1940 (5906) 80 PST 1 Table 9 provides the number of discrete variables associated to each PEGASE network. The rst number is related to the full PSTs range and the second number corresponds to a tighter PST range, limited to the interval [ 5, 5 ]. We have observed that the 79 PSTs with variable Date: 12/08/2011 Page: 23 DEL_TE_WP33_D3.2_ _9

24 angle have very dierent characteristics (e.g. angle ranges, number of taps, angle shift, etc.). We furthermore consider that the PST impedance varies with the tap position as explained in [12]. Table 9: Number of binary variables and complementary constraints associated to Phase Shifter Transformers test case full PSTs range tighter PSTs range UCTE UCTE + TEIAS UCTE + TEIAS + step-up transformers Since branch thermal limits were not available when performing he simulations, we adopted two scenarios on the EHV part of the PEGASE network: a uniform scenario where we assume a common current limit xed to 5 ka; a more realistic scenario where the current limits are xed to 4.6 ka on the 400 and 380 kv levels, to 1.6 ka on the 225 kv level, and to 5 ka elsewhere. All voltages are enforced in the range [0.9,1.1] p.u. The very large size of this OPF problem for the PEGASE data prohibits the use of the NEOS server to examine the performances of various solvers. In order to complete this comparison we have installed the solvers KNITRO and BONMIN on the computer conguration detailed in the section "Uniform" Scenario Tables 10 and 11 report the performances of various solvers for the two-step solution of OPF problem. Table 10: Solvers' performances for the solution of problem (7) before discretization of OLTCs solver diagnostic cpu time objective value KNITRO MPEC Locally optimal solution found KNITRO BOOLEAN Iteration limit reached BONMIN Infeasible problem Table 11: Solvers' performances for the solution of problem (7) after discretization of OLTCs solver diagnostic cpu time objective value KNITRO MPEC Locally optimal solution found Note that the KNITRO MPEC is the single solver that converges to a solution. The computational time obtained with this solver is acceptable given the very large size and complexity of the problem as well as that it has not been tunned for this particular power systems problem. By looking closely at these two Tables one can observe also that the computational time of this solver decreases with the problem complexity. For comparison purposes, note that KNITRO BOOLEAN and BONMIN solvers spent much more time without providing a feasible solution. This behaviour is due to the branh-and bound method used by these solvers involve a large number of large size NLP relaxation subproblems, which is very time-consuming. These MINLP solvers may in particular run in trouble on non convex problems because the solution of the NLP relaxation subproblem may not provide a lower bound on the solution of the original problem. Date: 12/08/2011 Page: 24 DEL_TE_WP33_D3.2_ _9