Optimal Operation of Thermal Electric Power Production System without Transmission Losses: An Alternative Solution using Artificial eural etworks based on External Penalty Functions G.J. TSEKOURAS, F.D. KAELLOS, C.D.TSIREKIS 3,.E. MASTORAKIS Production Engineering & Management Department, Technical University of Crete Department of Electrical & Computer Science, Hellenic aval Academy Terma Hatzikyriakou, 8539, Piraeus 3 Independent Power Transmission Operator Technical University Campus, Chania, 7300 GREECE Email: tsekouras_george_@yahoo.gr, kanellos@mail.ntua.gr, ktsirekis@desmie.gr, mastor@wseas.org 89 Dyrrachiou, Athens, 0443 Abstract: - The optimal economic operation of a power system is a crucial problem usually solved with Lagrange method. Alternatively, artificial neural networks (A can be used for continuous, nonlinear, constrained optimization problems like the one examined in this paper. In this paper, a special A is developed for the determination of the optimal operation of an electric power production system without transmission losses. A specific case study is studied via the proposed method and the advantages and disadvantages of the method are discussed. Key-Words: - A, economic dispatch, thermal power system, system marginal cost Introduction owadays, world increasingly concerns about climate changes, air quality and greenhouse gas emissions. This has led to stricter emissions regulations for electric power systems mainly concerning the thermal electric power generating units (coal, oil and gas. The use of the thermal units is inevitable because of their controllability and availability in contrast to alternative energy sources (i.e. wind-turbines, photovoltaic panels, hydroturbines, etc characterized by intermittent power production. However, for environment protection purposes financial penalties are imposed for Green House Gas (GHG emissions offering a substantial economic advantage to clean alternative sources. Penalties vary for each thermal unit according to the GHG emissions and it is added to the operation economic cost of the unit. Taking into consideration this additional cost the optimal operation of the thermal units, known as economic dispatch, can be solved using the technique of Lagrange multipliers [-]. The obective of this paper is to present an alternative solution based on artificial neural networks specialized in the solution of continuous, nonlinear constrained optimization problems, based on external penalty functions [3]. In section the mathematical base of the optimal operation of a thermal electric power system is formed. In section 3 the mathematical base of the developed A is presented and in section 4 the indicative case studies are presented. Finally, in section 5 the advantages and the disadvantages of the classic Lagrange technique and the presented method are commented. Mathematical Bases for Optimal Operation of All-Thermal Power System without Losses The optimum operation of the thermal and of the hydro-thermal continental power systems (with or without transmission losses has been already thoroughly analyzed in the literature [-]. In this section a short review of the optimal operation of the thermal units without considering transmission system losses is presented, as in Fig.. G ~3 G ~3 bus G ~3 load Fig.. Power system with thermal units in the same bus (without transmission loss. It is assumed that the thermal system consists of thermal generating units connected to the same bus. Let the th unit produces output active power P TH, bounded by the technically minimum active ISB: 978--6804-6-3 37
power, P minth, and the technically maximum active power P maxth of the unit. The respective constraints are formulated as: Pmin TH PTH Pmax TH ( Where =,,,. The respective fuel cost of each unit is given by the function F TH (P TH, which is usually a polynomial of second or third order of P TH. This means that: 3 FTH ( PTH = a b PTH c PTH d PTH ( Where a, b, c, d are the proper economic coefficients and they are considered known next. For each time interval DT with constant load demand P D the total fuel cost F tot of the power system is the sum of the individual fuel costs of the units, which is calculated as: tot TH ( TH F = F P (3 = Active power balance without considering transmission losses can be written for time interval DT, as: PTH = PD (4 = The target is to determine the generating levels of the units such that the total fuel cost F tot, is minimized and active power balance constraint is fulfilled. This means that the total fuel cost F tot should be minimized taking into consideration the respective constraint of eq. (4. This can be solved by including the eq. (4 in eq. (3 by using an unknown Lagrange multiplier, i.e. λ. Then eq. (3 is modified as: Ltot = FTH ( PTH λ PTH PD (5 = = The optimality conditions are obtained by setting the partial derivatives of L tot with respect to P TH equal to 0: L dfth ( PTH tot = 0 λ = 0 TH ( PTH TH TH dfth λ = for i=,,..., ( ( dfth PTH dfth PTH λ = =... = (6 TH TH This means that during time interval DT thermal units should share the load in a way that their respective incremental costs are equal to Lagrange coefficient, λ. Lagrange coefficient λ is known as system marginal cost (SMC and differs between time intervals with different loads. After the determination of λ, the generating levels of the units P TH can be calculated. If the function cost is a polynomial of second order (d =0 for all i, eq. (6 is simplified as: dfth ( PTH λ = = b c PTH for =,,..., (7 TH Resolving eq. (7 the generating level of the th unit can be calculated as a function of the respective system marginal cost λ: λ b PTH = for =,,..., (8 c System marginal cost λ is obtained if the active power balance (eq. (4 constraint is used as following: λ b (4 & (8 = PD = c b λ PD c c = = = b λ = PD = c (9 = c If the function cost is a polynomial of third order, eq. (6 is simplified as: λ = b c P 3d P for =,,..., (0 TH From eq. (0 the generating level of the th unit can be calculated as a function of the respective system marginal cost λ: c c 3d( λ b PTH = for =,,..., ( 3d Using Gauss-Seidel classic technique the determination of λ is achieved as following: Initialization of λ (0 and λ (. Determination of the generating levels of the units P THi based on eq. (. 3 Calculation of the active power imbalance ε (k : ( k ( k ε = PTHi P D ( i= 4 Step to next iteration: k = k (3 5 Check of the first iteration: if k=, step is executed else continue with step 6. 6 Check for convergence: if ε (k convergence limit then step 8 is executed else we continue with step 7. 7 Determination of λ (k : Based on ewton- Raphson method the λ (k is determined as: ( k ( k ( k ( k λ λ ( k λ = λ ε (4 ( k ( k ε ε Afterwards, step is executed. 8 After the determination of λ and the generating TH ISB: 978--6804-6-3 38
levels of the units, P THi, the inequalities from the technical constraints should be checked. If a violation of an inequality exists, i.e. P TH < P minth, the output active power is set to the respective value of the violated constraint, i.e. P TH = P minth. Afterwards the rest load demand P D is calculated as P / D = P D - P TH and the optimization process is repeated from step without considering the respective unit, i.e. the unit. The Gauss-Seidel method can be also applied in case of second order polynomial function cost and in the third iteration the active power imbalance error becomes zero. 3 Mathematical Base for Continuous, onlinear, Constrained Optimization Problems with Exterior Penalty Functions solved with As 3. General A constrained minimization problem can be stated T n as the following one: Find x = ( x, x,..., x n which minimizes the continuous, non-linear scalar function f = f ( x, x,..., xn subect to the inequality constraints gi 0 for i =,,..., m. It is noted that inequality constraint w 0 can be replaced by the constraint inequality w 0, while each equality constraint can be represented equivalently by two inequality constraints, i.e. h( x = 0 can be replaced by the two inequalities h 0 and h 0. The aforementioned problem can be transformed to an unconstrained problem having the following energy pseudo-cost function: m E x = f x k p g x (5 ( ( i ig i( i= Where k i are the multiplication penalty factors (k i >0 and pig ( gi ( x are the quadric penalty function terms defined by: ( ( ( min { 0, p ( } ig gi x = gi x (6 0, if gi 0 pig ( gi = > 0, if gi < 0 If the first partial derivatives of the constraints g x n are continuous in then this will also i ( happen for the respective penalty functions of eq. (6, as: pig ( gi ( x i = ( min{ 0, gi } (7 The minimization of the energy function is ( succeeded by using a gradient technique [3]. The respective system of differential equations =,,.., n is: dx m f i = μ kisigi dt x i= x (8 Where, - t is the auxiliary parameter of time, - μ is the A learning rate lower limited by a constant positive number μ 0 defined by the following equation: t μ ( t = max μinit, exp, μ0, (9 T μ 0, μ > 0, μ > 0, > 0 with init, 0, T μ 0, - k i is the multiplication penalty factor upper limited by a constant positive number k 0 defined by the following equation: t ki( t = min kinit, i exp, k0, i (0 T k0, i k > 0, k > 0, T > 0 with init, i 0, i k0, i - S i is a proper step function, which implements the minimization problem of (6: 0, if gi 0 Si( gi = (, if gi < 0 The block diagram of a constrained nonlinear optimization A based on the penalty method is shown in Fig. and is proposed by Chua et al [6]. 3. Optimal operation of all-thermal power system without losses The target is to minimize the total fuel cost F tot of the power system given by eq. (3. In this minimization problem the unknown variable vector x consists of the generating levels of the units P TH while the function f ( x represents total fuel cost function, F tot : x = P, P,..., P T ( ( TH TH TH, 3 ( = ( a b PTH c PTH d PTH f x = (3 The technical constraints of the thermal units of eq. ( are transformed to the following inequalities: gi = Pmax THi PTH i 0 (4 gi = PTHi Pmin THi 0 (5 Where i=,,,. The equality of the active power balance without considering the losses of the transmission system given by eq. (4 is transformed to the following to inequalities: ISB: 978--6804-6-3 39
g ( x - S Control network g ( x x gm ( - - S m S k Computing network for x m k k m k m (- x x... x... x m x (0 Fig.. Block diagram of a constrained nonlinear optimization A based on the penalty method. g x = P P 0 ( TH = g x = P P 0 D ( D TH = (6 (7 For generating units inequality constraints are obtained. The first partial derivatives of the minimization function and of the constraints are the following: f = b c PTH 3dPTH (8 TH i TH, = i = 0, i i=,,..., (9 i TH, = 0, = i i TH TH i=,..., (30 = = (3 (3 Where =,,,. The A technique used in this study is an iterative process, where eq. (8 can easily be transformed into the following discrete-time iterative equation: ISB: 978--6804-6-3 40
( ( f x ( PTH μ ( μ ( ( P TH PTH = (33 ( m i ( ( x ki( Si( gi( x i= TH Where, l is the discrete variable for time or alternatively the training epochs of A technique. The proposed A training process will be terminated if one of the following two stopping criterions is reached: the generating levels of the units P TH are stabilized (the variation between two epochs should be smaller than limit_convergence the maximum number of epochs is exceeded (larger than max_ epochs. 4 Application of the Proposed Method 4. Case study A simple thermal electric power system without transmission losses is assumed. It consists of three thermal units with the following fuel cost functions and technical constraints: 700 00PTH, m.u. FTH, ( PTH, = 4 3 0,03 PTH, 0 P TH, h 8760 75PTH, m.u. FTH, ( PTH, = 5 3 0, P h TH, 5 0 P TH, 4800 90 PTH,3 m.u. FTH,3 ( PTH,3 = 5 3 0,03 PTH,3 8 0 P TH,3 h 50 PTH, 350 MW 50 PTH, 350 MW 50 PTH,3 450 MW Where m.u. is the monetary unit. 4. Application of the A method Two load levels are examined: 460 MW and 60 MW. The last one activates the technical constraints of the thermal units. In all cases of A training the following parameters values have been used after a few tests so that the algorithm remains stable: 3 μinit, = 0 μ 0, = 0.06 T μ 0, = 00 for =,, 3 k init, i = 0 k 0, i = 00 T k0, i = 5 for i=,, 8 limit_convergence = 0 4 max_ epochs = 00000 The initialization of the generating levels of the units is done as following: Pmin TH Pmax TH PTH = for =,,3 (34 In case of 460 MW demand load the generating levels of the units are obtained after 650 epochs, approximately. The analytical results of the generating levels, the power balance error (difference between power produced by the units and load demand, the epochs and the total cost of the system are registered in Table. Unfortunately, the active power balance is approximately satisfied, as the respective error is almost 0.8 MW. This happens as the limit of convergence depends on the variation of the generating levels and not on balance of equality, as happening in the classical Lagrange method of section. In Figures 3 and 4 the generating levels of the thermal units and the total fuel cost with respect to the training epochs are presented. It is proven by these Figs that the method is stable for the chosen parameters values. Fig. 3. Case of demand load 460 MW: the generating levels of the thermal units with respect to the number of epochs. Fig. 4. Case of demand load 460 MW: the total fuel cost with respect to the number of epochs. In case of 60 MW load demand the generating levels of the units are obtained after 80 epochs, ISB: 978--6804-6-3 4
approximately. The respective results are registered in Table, while in Figs. 5 and 6 the graphics of the thermal units generating levels and the total fuel cost with respect to the epochs are shown, respectively. The active power balance error is approximately 0. MW which is smaller than in the previous case. Fig. 5. Case of demand load 60 MW: the generating levels of the thermal units with respect to the number of epochs. 4.3 Comparison with classical technique In Table the results of the generating levels of the thermal units, the power balance error (difference between produced power by the units and the load demand, the epochs and the total cost of the system are registered for both of previously examined cases (460 MW and 60 MW of load demand and both minimization techniques (A and the classical Lagrange method of section. It is concluded that: Both methods give practically the same results. The A based method needs significant more epochs for convergence than the classical technique. The A based method does not satisfy exactly the active power balance constraint. This happens as the limit of convergence depends on the variation of the generating levels and not on balance of equality, as happening in the classical Lagrange method The A based method seems to be slightly superior to the classical technique in terms of fuel cost minimization. This happens because in both cases the total production of the generating of the units is slightly smaller than the total load demand. Fig. 6. Case of demand load 60 MW: the total fuel cost with respect to the number of epochs. TABLE COMPARISO OF A BASED METHOD & CLASSICAL LAGRAGE OPTIMIZATIO METHOD Method A Classic A Classic Target Load 460 MW 60 MW P TH, [MW] 0,97 0,58 49,99 50,00 P TH, [MW] 57,50 57,83 09,7 08,78 P TH,3 [MW] 90,80 9,58 00,5 0, Error [MW] -0,73 0,0-0, 0,00 Epochs[-] 647 8 84 3 Cost[m.u./h] 67 65 403 4054 5 Conclusions This paper presents an A based optimization method applied to a thermal electric power system without considering transmission losses. The proposed method is specialized in continuous, nonlinear, constrained optimization problems with external penalty functions. The respective results are satisfactory if the proper parameters of learning rate and penalty multiplicative factors are properly selected. However, the proposed method is characterized by the three following drawbacks: ( the number of iterations (epochs needed is significant larger than those required in classical Lagrange technique, ( active power balance constraint is not satisfied completely, (3 the system marginal cost (SMC, which is a crucial parameter for the operation of the power system is not directly obtained. It should be calculated afterwards in contrast to the classical Lagrange optimization technique where it is determined within the optimization process (see eq. (4. In future work, the proposed method can be improved by using augmented Lagrange multiplier A method, which can easily remove the last two disadvantages. ISB: 978--6804-6-3 4
References: [] J. A. Momoh. Electric Power System Applications of Optimization, Marcen Dekker Inc., ew York, USA, 00. [] F. Saccomanno. Electric Power Systems, Analysis and Control. Wiley Interscience, IEEE press, Piscataway, USA, 003. [3] A. Cichocki, R. Unbehauen. eural etworks for Optimization and Signal Processing. John Wiley & Sons, Stuttgart, 993. [4] S. Osowski. eural networks for nonlinear programming with linear constraints. Int. J. Circuit Theory and Application, vol. 0, 99, pp. 93-98. [5] A. Cichocki, R. Unbehauen. Switchedcapacitor neural networks for differential optimization. Int. J. Circuit Theory and Application, vol. 9, 99, pp. 6-87. [6] L.O. Chua, G.. Lin. onlinear programming without computation. IEEE Trans. Circuits and Systems, vol. CAS-3, 984, pp. 8-88. ISB: 978--6804-6-3 43