Comparison of Conventional and Meta-Heuristic Methods for Security-Constrained OPF Analysis

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1 Comparison of Conventional and Meta-Heuristic Methods for Security-Constrained OPF Analysis Jagadeesh Gunda, Sasa Djokic School of Engineering The University of Edinburgh Edinburgh, Scotland, UK Abstract Development and implementation of accurate, robust and computationally efficient analytical and modelling tolls is very important for the anticipated transformation of existing networks into the future smart grids. These tools for network analysis are used at both planning and operating stages, in order to ensure optimal design and configuration of power supply systems, in terms of the requirements for higher flexibility, increased security and improved overall techno-economic performance of modelled networks. In this context, particularly important are smart grid applications requiring (close to) realtime controls of large and interconnected power supply systems under serious contingency scenarios and other highly stressed network operating conditions. This paper provides a detailed discussion and analysis of both conventional and meta-heuristic methods for security-constrained optimal power flow (SCOPF) studies. The comparison of performance of two conventional SCOPF methods and three meta-heuristic SCOPF algorithms is illustrated on IEEE 14-bus and IEEE 30-bus test networks. The analysis and optimization of objective functions in considered SCOPF methods include minimization of constraint violations in post-contingency states, as well as minimization of fuel costs, active power losses, and CO2 emissions. Keywords conventional and meta-heuristic methods; optimal power flow analysis; power supply networks; security; NOMENCLATURE Variables:, State and control variables, Real and reactive power output of generating unit, Real and reactive demand at bus, Voltage magnitude and phase angle at bus Contingency index, zero for base case Set of credible contingencies Functions: Objective function, Equality and inequality constraint functions, Total fuel cost and total emission Total active power loss Penalized objective function ɸ,ɸ Penalty functions for equality/inequality constraints Constants:,, Fuel cost coefficients of generating unit,, Emission coefficients of generating unit,, Number of buses, generators and branches Conductance of a line connecting buses and,, Roberto Langella, Alfredo Testa Dept. of Industrial and Information Eng. The Second University of Naples Aversa, Italy roberto.langella@unina2.it Penalty for violating bus voltage constraints Penalty for violating active power generation limit Penalty for violating reactive power generation limit Penalty for violating branch MVA constraints Number of particles or populations Maximum number of iterations or generations Acceleration coefficients for PSO Initial and final inertia weight for PSO Crossover probability for GA Initial temperature for SA I. INTRODUCTION Planning and operation of modern and future power supply systems (so-called smart grids ) are becoming increasingly complex tasks, as network designers have to analyze large interconnected networks for a number of relevant technical and non-technical operating conditions during the design stage, while network operators should control and operate their networks closer to their operational security limits and for higher loading conditions than before, in order to meet the requirements of the deregulated markets. Therefore, it is very important to develop and implement computationally efficient, accurate and robust analytical and modelling tolls, in order to ensure optimal design and operation of power supply systems in terms of the requirements for higher flexibility, increased security and improved overall techno-economic performance of analyzed networks. In this context, particularly important are smart grid applications requiring (close to) real-time controls of large and heavily interconnected power supply systems under serious contingency scenarios and other highly stressed network operating conditions. This paper considers optimal power flow (OPF) and security-constrained OPF (SCOPF) studies, which are amongst the very basic tools for the assessment of network performance during both planning and operational stages. A detailed analysis and discussion of two conventional and three meta-heuristic SCOPF methods is presented, with the comparison of their performance illustrated on IEEE 14-bus and IEEE 30-bus test networks. Generally building on the previous work and results in [1]-[3], the analysis and optimization of objective functions in this paper consider SCOPF methods that include minimization of constraint violations in post-contingency states, as well as minimization of fuel costs, active power losses and CO2 emissions.

2 II. THEORETICAL BACKGROUND The secure operation of large interconnected power supply systems is not possible without implementing required controls. Economic dispatch (ED) of generating power plants, which is also considered as a tertiary frequency control, is commonly used as a higher level decision-making tool in dispatching centers for supply-demand balancing and for minimizing the operating cost of generation. In this case, the analysis of the power supply network is effectively reduced to a single equality constraint (e.g. [4]), as ED provides only an optimal scheduling of active power outputs of generators, neglecting adjustment of their voltage set points and other available network controls. To provide a more detailed analytical tool, so-called traditional OPF formulation is introduced as an extension to ED, allowing to explicitly model all network components and all available network controls. The main objective of OPF is again to determine optimal dispatching of generating plants to meet demand plus losses, but with considering flows of complex powers in the network and determining optimal settings of controllable network components (e.g. shunt and series capacitors, FACTS devices, tap changing transformers, phase shifters, etc.), so that specified constraints are satisfied and certain performance or objective function is optimized. Since its introduction, OPF studies have evolved to include diverse objective functions, various controls and different constraints, mainly depending on target applications, or stipulated operational/security requirements [5]. While traditional OPF studies warrant satisfying operating constraints only under normal operating conditions, SCOPF studies consider both normal operating conditions and credible contingencies [6]. In analytical terms, incorporating diverse objective functions and different constraints in SCOPF studies is formulated as a large-scale, non-linear, non-convex, multimodal and highly constrained optimization problem. Network designers and operators typically rely on use of conventional methods, including: Newton-Raphson method, Quadratic programming (QP), interior point algorithm (IPA), etc., to solve OPF/SCOPF problem. These conventional algorithms typically use either line search, or trust region methods for iterative exploration of the search space for optimum solutions. As both line search and trust region methods use gradient-based information to make decisions on the search direction and length, conventional algorithms are generally sensitive to the selection of initial values, with possible convergence problems. As mentioned, OPF problem is in general case a non-convex and nonlinear optimization problem, but many of the conventional algorithms assume that the problem is convex, typically resulting in a local optimum convergence, rather than in finding a global optimum solutions. In addition, due to the introduction of valve-point effects, associated with conventional generating units, and prohibited operating zones, the search space of the problem may become piecewise continuous, which then again limits the applicability of the conventional methods. Nevertheless, conventional OPF algorithms are widely used due to computational efficiency and strong theoretical background. Recently, many meta-heuristic algorithms are proposed to solve various power system optimization problems, as they are generally insensitive to the selection of initial values, have a good performance in finding (near) optimal solutions and can be applied to a wide range of practical problems. Despite their advantages, however, none of the meta-heuristic algorithms has seen a practical implementation in power industry, even for off-line analysis of electrical networks. The two most likely reasons for this are: a) from algorithm implementation perspective, meta-heuristic algorithms are computationally more intensive and generally lack a clear theoretical/analytical understanding and ensuring of convergence criteria, and b) from practical implementation perspective, there is still no clear set of instructions that could inform network planners and operators for exactly what system studies and under what system operating conditions meta-heuristic algorithms could provide realistic benefits over conventional algorithms. In order to fill that gap, this paper investigates general applicability of three well-known meta-heuristic methods (genetic algorithm, GA, particle swarm optimization, PSO, and simulated annealing, SA) for OPF and SCOPF analysis. Their performance is compared against two conventional methods and illustrated using IEEE 14-bus and IEEE 30-bus test network. The presented analysis and optimization of objective functions in all considered SCOPF methods include minimization of constraint violations in post-contingency states, as well as minimization of fuel costs, active power losses, and CO2 emissions. III. TEST NETWORKS USED FOR THE ANALYSIS Two test networks are used for the analysis and comparison of various OPF/SCOPF approaches presented in this paper. The first one is IEEE 14-bus network, depicted in Fig.1. It has five generators supplying total demand of 259 MW and 81.3MVAr, [7]-[8]. The second is IEEE 30-bus network, illustrated in Fig.2. It has six generators supplying total demand of MW and MVAr, [6]-[9]. During the analysis, all generators are considered as conventional synchronous machines. Generator power limits, fuel cost and emission coefficients are given in Table I. The limits for allowed variations of bus voltages are 0.95 pu and 1.1 pu for voltage controlled buses, and 0.95 pu and 1.05 pu for load buses. Fig.1 IEEE 14-bus test network

3 Fig. 2 IEEE 30-bus test network TABLE I GENERATION LIMITS, FUEL COST AND EMISSION COEFFICIENTS IEEE-14 Bus network Bus Pg Pg Qg Qg No max min max min a b c d*10-6 e*10-6 f IEEE-30 Bus network Bus Pg Pg Qg Qg a b c d*10-6 e*10-6 f No max min max min IV. OPF PROBLEM FORMULATION Solving an OPF/SCOPF problem is aimed at finding optimal settings of electrical control variables (generator outputs, bus voltages, tap settings, etc.), in order to minimize separately or simultaneously one or more objective functions, while satisfying related equality and inequality constraints. The SCOPF problem can be formulated as: Minimize:, (1) Subject to:, 0 (2), 0, 0,1,2, (3) Different OPF problem formulations can include various objective functions, (1), in order to meet various technoeconomic and environmental requirements. In this paper, three most frequently used objective functions are optimized separately: minimization of fuel cost, (4), minimization of emissions, (5), and minimization of active power losses, (6). $/ (4) / (5), 2 (6) Equality constraints, (2), are represented by the power flow balance, (7)-(8). Inequality constraints, (3), represent equipment operating limits: generator real and reactive power limits, (9)-(10), transformer tap setting limits, (11), and branch thermal rating limits, (12), as well as bus voltage limits, (13). 0 (7) 0 (8),1,2, (9),1,2, (10),1,2, (11),1,2, (12),1,2, (13) In meta-heuristic approaches, constrained optimization problem, (1)-(3), should be converted to an unconstrained optimization problem by using one of two penalty function (14), [10]: exterior penalty functions, which penalize infeasible solutions (used in this paper), and interior penalty functions, which penalize feasible solutions. (14),,,, V. CONVENTIONAL AND META-HEURISTIC METHODS Meta-heuristic methods are general-purpose search algorithms which can be applied to a wide variety of optimization problems, including OPF/SCOPF studies. While conventional algorithms follow deterministic rules to explore search space, meta-heuristic algorithms use guided stochastic search and can be considered as black-box or problemindependent and plug-and-play type algorithms, as they rarely rely on knowledge of the nature of considered problem. In general, meta-heuristic algorithms can be classified into three groups: evolutionary computation, swarm intelligence and physics-inspired algorithms. The analysis in this paper considers one meta-heuristic algorithm from each group (GA, PSO and SA), as well as two conventional algorithms: interior point algorithm (IPA) from [11] and OPF solver from [12]. Meta-heuristic algorithms initially randomly generate a feasible single decision vector, or set of vectors, which are called population or swarm. Afterwards, they apply various stochastic manipulations or operations (depending on formulated algorithm), in order to find next feasible point(s) in the search space. This process is repeated until the predefined thresholds or tolerance criteria are met, or until the maximum number of iterations is completed. The basic flow charts for the implementation of GA and PSO algorithms are shown in Fig. 3, while a more detailed discussion is given in [13]-[14]. In Fig. 3, fitness evaluation block involves implementation of a full custom-built Newton-Raphson power flow (NRPF). The algorithm for SA can be described as follows, [15]: 1) Input initial temperature (T 0 ) and (random) initial feasible solution ( ) 2) Calculate fitness at, i.e. Loop-1 begin, Loop-2 Begin 3) Generate another random solution ( ) and calculate fitness at, i.e. 4) If : (minimization of objective) 5) If, generate a random number, 0,1 and calculate acceptance probability ( ) 6) If : Loop-2 End 7) Reduce the temperature: Loop-2 End

4 A. Base case OPF Analysis This section presents results of OPF analysis for a precontingency network (i.e. base case ) using conventional and meta-heuristic methods. The optimum generation dispatch for three considered objective functions is shown in Tables III and IV for the two test networks. Bus voltages and branch power flows for IEEE 14-bus network are shown in Figs. 4 and 5, while bus voltages for IEEE-30-bus network are illustrated in Fig. 6, all for fuel minimization objective function. It can be seen that for the base case meta-heuristic methods and conventional OPF solvers are producing almost the same objective function values, with similar generation schedule. Furthermore, the differences in voltage profiles and generation schedules indicate that the search process for SA algorithm is different from PSO and GA methods. a) PSO b) GA Fig. 3 PSO and GA flowchart VI. RESULTS AND DISCUSSION The presented results are divided into two main sections. First, OPF problem is solved using conventional and metaheuristic methods for the pre-contingency network (no outage of any network component). Afterwards, SCOPF problem is solved using conventional and meta-heuristic methods for a number of single contingencies. In both cases, the OPF analysis is carried out with three objective functions denoted as (F) (minimization of fuel cost), (E) (minimization of CO2 emission) and (L) (minimization of network losses). Component modelling: System loads are represented by constant power load type, while transformers are modelled with fixed tap ratios, in order to directly compare results calculated with different OPF solvers. Program settings: All programs are executed on a 64-bit Intel Core i7-3770, 3.4 GHz CPU desktop PC. Conventional OPF/SCOPF algorithms are implemented using [8] (marked in figures as IPA ) and [9] (marked in figures as PSSE ), while three meta-heuristic approaches are coded in [16]. The power mismatch tolerances used for each solution were 0.01MW and 0.01MVAr for the base case and 0.5MW and 0.5MVAr for the contingency cases [6]. Parameter settings: These are listed in Table II for GA, PSO and SA methods, together with applied constraint violation penalties. The maximum population size and maximum number of iterations are both selected to achieve 100% success rate and strictly enforce reactive power and slack bus active power limits. TABLE II PARAMETER SETTINGS AND PENALTIES FOR GA, PSO AND SA GA settings PSO settings SA settings Penalty settings =40 : 20 =300 : 1000 =400 =400 =10000 : 100 Selection: Tournament =2, =2 N/A : 500 =0.8 wi=0.6 N/A 100 Mutation: Gaussian wf=0.1 N/A N/A TABLE III ACTIVE (PG) AND REACTIVE (QG) GENERATION FOR IEEE 14 BUS Fuel Cost Minimization (F) Bus IPA-AC(F) PSSE(F) GA(F) PSO(F) SA(F) Total cost Emission minimization (E) Bus IPA-AC(E) PSSE(E) GA(E) PSO(E) SA(E) Total Emiss Active power loss minimization (L) Bus IPA-AC(L) PSSE(L) GA(L) PSO(L) SA(L) Total loss Fig. 4. Bus voltages for IEEE 14-bus system

5 MVA flow (% of rated MVA) Bus voltage (pu) IPA(F) PSSE(F) GA(F) PSO(F) SA(F) Branch No Fig. 5. Branch flows for IEEE 14-bus system IPA(F) PSSE(F) GA(F) PSO(F) SA(F) Bus No Fig. 6. Bus voltages for IEEE 30 bus system TABLE IV ACTIVE (P G ) AND REACTIVE (Q G ) GENERATION FOR IEEE 30 BUS Fuel Cost Minimization (F) Bus IPA-AC(F) PSSE(F) GA(F) PSO(F) SA(F) Emission minimization (E) Bus IPA-AC(E) PSSE(E) GA(E) PSO(E) SA(E) Active power loss minimization (L) Bus IPA-AC(L) PSSE(L) GA(L) PSO(L) SA(L) B. SCOPF Analysis of N-1 Contingencies SCOPF analysis should provide solutions for generation dispatch and control settings which ensure stable postcontingency operation, prevent constraint violations and still allow to minimize considered objective function. For the two sets of N-1 contingencies selected for two IEEE networks, the optimum objective function values found by conventional and meta-heuristic methods are shown in Tables V and VI. Branch power flows for IEEE 30-bus network for loss minimization with an outage of transformer (T27-28) are shown in Fig. 7. TABLE V COMPARISON OF OBJECTIVE FUNCTION VALUES FOR IEEE 14-BUS Fuel Cost ( /hr) T L L Emission (ton/hr) T L L Loss (MW) T L L TABLE VI COMPARISON OF OBJECTIVE FUNCTION VALUES FOR IEEE 30-BUS Fuel Cost ( /hr) L L L L T27-28 X 1 X Emission (ton/hr) L L L L T27-28 X X Loss (MW) L L L L T27-28 X X Unable to converge MVA flow (% of rated MVA) GA(L) PSO(L) SA(L) Branch No Fig. 7. Branch MVA flows for IEEE 30-bus with transformer (T27-28) outage

6 C. Discussion of Results The following conclusions can be drawn from the results presented in Tables V-VI: - When a conventional algorithm converges for a selected contingency, both conventional and meta-heuristic algorithms are able to provide solutions that prevent constraint violations and results in almost the same objective function values. - Neglecting the small differences in objective function values, it is very difficult to advocate benefits of using meta-heuristic methods over the conventional ones, as both provide similar solutions after the convergence. - From computational requirements perspective, metaheuristic methods are inferior to conventional algorithms, as they require hundreds or even thousands of function evaluations before they can find a near optimal value. - The benefits of meta-heuristic algorithms can be clearly identified when conventional algorithms fails to converge for severe contingencies. This is related to the outage of transformer T27-28, Table VI, for which conventional algorithms do not converge, but meta-heuristic methods provide solution. In these cases, meta-heuristic methods are also able to minimize three objective functions with only a few relatively low constraint violations. The information obtained from meta-heuristic methods in cases when conventional methods cannot converge may be used by network planners and operators, either to aid the convergence process of conventional algorithms, or to help with some other applications (e.g. management of system outages/faults, optimal load shedding, etc.). This will be discussed in the future work (e.g. [17]). VII. CONCLUSIONS Future smart grids will see profound changes in levels and nature of system-load interactions, shifting actual system operating and loading conditions well outside the traditionally assumed ranges, limits and physical boundaries. The main drivers for these changes are introduction of new technologies and formulation of the more stringent techno-economic requirements for higher flexibility, increased security and improved network performance. As a consequence, existing analytical and modelling tolls will become inadequate, where of particular concern are new smart grid applications requiring (close to) real-time control of large interconnected networks under serious contingency scenarios and other highly stressed system operating conditions. To answer these challenges, accurate, robust and computationally efficient methods should be developed and implemented for analysis of existing networks and future smart grids. This paper provides a detailed discussion and analysis of conventional and meta-heuristic methods for OPF and SCOPF studies. These studies are frequently used by system planners and operators for optimal design and secure network operation, as well as for assessment of network performance in various target applications. Despite their advantages, however, none of meta-heuristic algorithms has so far seen implementation in practice, even for off-line network analysis. In order to investigate benefits of meta-heuristic methods, three popular meta-heuristic algorithms (GA, PSO and SA), and two conventional methods are used for OPF/SCOPF study of two IEEE test networks and for various objective functions. The presented results for normal operating conditions and less severe outages show that meta-heuristic methods provide same solutions as conventional methods, but require longer times to find optimal solutions. However, when conventional methods fail to converge, meta-heuristic approaches could provide solution and further useful information to network operators and planners for handling severe network contingencies. This will be investigated in more detail in the future work. VIII. ACKNOWLEDGEMENTS The authors gratefully acknowledge support from the University of Edinburgh PCD scholarship and POLIGRID project, funded by Italian Campania Region. REFERENCES [1] E. Chiodo, F. Gagliardi, D. Menniti, A. Testa, A Pattern Recognition Approach for Steady-state Security Evaluation of an Electrical Power System, European Trans. on El. Power Eng., Vol. 1, No. 5, [2] E. Chiodo, D. Menniti, C. Picardi, A. Testa, Steady State Security Evaluation of Electrical Power Systems by Means of an Artificial Neural Network, European Trans. on El. Power Eng., Vol. 5, No. 2, [3] D. Menniti, C. Picardi, N. Sorrentino, A. Testa, Steady-state Security in Presence of Load Uncertainty, European Trans. on El. 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