Dynamic Economic Load Dispatch with Emission and Loss using GAMS

Dynamic Economic Load Dispatch with Emission and Loss using GAMS

Devendra Bisen

Department of Electrical Engineering Madhav Institute of Technology and Science Gwalior (M.P), India

Hari Mohan Dubey

Department of Electrical Engineering Madhav Institute of Technology and Science Gwalior (M.P), India

Abstract

Dynamic economic dispatch (DED) is a real time problem of electric power system. DED intends to schedule the online generators outputs with the predicted load demands over a certain period of time in order to operate an electric power system most economically within its security limits. This paper introduces a solution of the dynamic economic dispatch (DED) problem including the loss and emission is participated among all generating units over time interval for a system using General Algebraic Modeling System (GAMS). The objective of the collective problem can be expressed by taking the production cost including emission and losses into account with required constraints for 24 hour time interval of each generating unit. The general algebraic modeling system (GAMS) technique is guarantees the global optimality of the solution due to its look -further on capability. To validate practicability and robustness of the GAMS, it is tested on six generating unit system with different cases for determine minimum production cost of individual generating unit over a time period. In test case I only production cost without emission and loss, In test case II production cost with loss, In test case III production cost with including emission and without losses and In test case IV production cost including emission and losses for time interval of 24 hours.

Keyword Dynamic economic dispatch (DED), security limits, general algebraic modeling system (GAMS), production cost etc.

1. Introduction

   Economic dispatch problem is one of the most important proble ms in e lectric power system operation. Energy manage ment has to perform mo re complicated and timely system control function to operate a large power system re liably an effic iently.

   Electric utility system is interconnected to accomplish the benefits of min imu m production cost, ma ximu m reliability and superior operating conditions [1]. The economic scheduling is the on- line economic dispatch, in which it is required to distribute the load among the co mmitted generating units which are actually paralleled with the system, in such a way as to minimize the total cost of generation without violating constraints [2]. The Dyna mic Economic Dispatch (DED), which is an e xtension of the conventional economic dispatch problem, determines the optima l generation schedule of on -line generators, so as to meet the predicted load demand over a time horizon satisfying the constraint.

   The intention of economic load dispatch (ELD) is

   to work out the optima l a mount of the generated power for the fossil-based generating units in the system by minimizing the fuel cost. So operating at absolute minimu m cost may no longer be the only criterion for dispatching electric power due to increasing concern over the environmental considerations. In fact, the Clean Air Act Amend ments have been applied to reduce poisonous gases emissions from such power plants. Poisonous gases generate during power production in therma l stations by burn fossil fuels, due to poisonous gases (CO, CO2, SO2, NOX etc) effluent and these become a source of pollution for the environment. Lac k of planning while generating power puts the economica l aspect in difficulty. W ithin a plant, however, there is transmission loss ; also pollution e xists due to emission. The cost minimu m condition matchup to minimu m cost with considerable amount of loss and emission. Simila rly, the emission minimu m condition produces minimu m e mission with higher deviation fro m minimu m cost and loss. And also the loss minimu m condition produces minimu m loss with higher deviation from min imu m cost and emission. These three conditions cannot be imple mented simu ltaneously. Hence, the feasible optimu m corresponds to a small deviation in cost with an

   allo wable tolerance in loss and emission taking into account emission constraints and this type of economic load dispatch has been termed emission constrained economic d ispatch (ECED) wh ich comes under mult iobjective proble ms[3].

   The dynamic econo mic dispatch (DED), where optimization is done with respect to the dispatchable powers of the committed generation units for time particular g iven period and formulated as a minimizat ion problem of the total cost over the dispatch period under some constraints[4]. The development of DED is still going on; it has though

   reached a certain level of development in terms of academic thoughts. Various optimizat ion techniques

   simu lation studies are discussed in Section 4. Finally, Section 5 presents the conclusions.

2. Dispatch Problem Formulation

   The objective of solving the dynamic economic dispatch problem in electric power system is to determine the generation levels for all on-line units which minimize the total fuel cost and minimizing the losses and emission level of the system, while satisfying a set of constraints over a given dispatch period.

   It can be formu lated as follows:

   have been proposed by many researchers to deal with this multiobject ive programming proble m with varying degree of success. In the recent past,

   Minimize FTc

   Where

   T N

   i 1 i 1

   FTi

   (Pi )

   (1)

   stochastic search algorithms such as genetic

   algorith m (GA) [5], Part icle Swa rm Optimization

   FTi (Pi )

   Fi (Pi ) hi {Ei (Pi )}

   (2)

   (PSO) [6], evolutionary programming (EP) [7], simu lated annealing (SA) [8] and Differentia l Evolution (DE) [9] and artific ial immune system [10] methods are proven to be very effective in solving non-linear DED proble ms and provide a fast, reasonable nearly optima l solution.

   In this paper General Algebraic Modeling System (GAMS) approach has been proposed to solve the

   FTc = is the total operating cost over the whole dispatch period,

   FTi (Pi ) = is the Emission constrained fuel cost of

   ith unit at t imet

   i i

   F (P ) = is the total fuel cost of a ith generating unit

   E (P ) = is the total emission of a ith generating unit

   objective of the collective problem can be expressed i i

   by taking the total production cost, losses and total emission into account with required constraints for 24 hour time interval. General Algebra ic Modeling System (GAMS) is a high-level mode l development environment that supports the analysis and solution of linear, non linear and mixed integer optimization problems [11]. General Algebra ic Modeling System is especially useful for handling la rge dimension and comple x proble m easily and accurately.

   In this paper the effectiveness of the General

   Algebraic Modeling System (GAMS) is demonstrated using six generating unit system with diffe rent cases for determine minimu m production cost of individual generating unit over a time period In test case I only production cost without emission and loss, In test case II production cost with loss , In test case III production cost with including emission and without loss and In test case IV production cost

   hi = cost penalty factor

   i

   P = is a function of its real power output of ith at timet.

   T = is the number of hours in time horizon,

   N = is the number of d ispatch able units,

   The fuel cost Fi (Pi ) of generating unit i at any time ntervalt is norma lly e xpressed as a quadratic function.

   1. Objective Functions

      1. Fuel Cost Objecti ve

         The classical economic dispatch problem of finding the optima l comb ination of power generation, which minimizes the total fuel cost of a generating unit is usually described by a quadratic function of power output Pi while satisfying the total required demand can be mathe matica lly stated as follows:

         2

         including emission and loss for time interva l of 24 hours.

         Fi (Pi )

         Where

         ai Pi

         bi Pi

         c $/Hr (3)

         i

         The paper is organized as fo llows: Section 2 provides a brief description and mathe matica l formulat ion of DED proble ms. The concept of General A lgebraic Modeling System (GAMS) is discussed in Section 3. The performance of Genera l Algebraic Modeling System (GAMS) and the

         ai, bi and ci are the cost co-effic ient of unit i.

      2. Emission Objec ti ve

   The minimu m e mission dispatch optimizes the above classical economic dispatch including emission

   objective of a generating unit is usually described by a quadratic function of power output Pi as [12]:

   Each generating unit is constrained by its lower and upper limits of real power output to ensure stable

   Ei (Pi ) di Pi2

   ei Pi fi

   Kg/Hr (4)

   operation. The power generation of unit n should be between its minimu m and ma ximu m limits.

   Where

   di, ei and fi are the emission co-effic ient of unit i.

   Pi min Pi

   Where

   Pi max

   (11)

   2.1.2 Emission constrai ne d c ost e quation

   : Economic and emission dispatch problem is

   Pi min

   is the minimu m generation limit of unit i

   converted into single optimization proble m by introducing price penalty factor h [13]:

   The Emission constrained cost equation can now be formulated as:

   Pi max is the ma ximu m generation limit o f unit i

3. General Algebraic Modeling System (GAMS)

FTi

(P )

i

ai Pi2

bi Pi ci

hi (di Pi2

ei Pi f )

General Algebra ic Modeling System provides a high – level (algebra ic) language for the representation of

$/Hr (5)

Where

large and co mple x mode ls. It a llows for unamb iguous statements of algebraic re lations that define an

hi Fi max (Pi max )

Ei max (Pi max )

(6)

abstract system of variables and equations. It also

F (P

) a P2 b P

c Rs/Hr (7)

provides several mechanisms for data manage ment.

i max

i max

i i max

i i max i

The system performs appropriate data

E (P

) d P2 e P

f Kg/Hr (8)

transformations to create a specific instance of the

i max

i max

i i max

i i max i

The price penalty factor hi unifies the emission with the normal fue l costs and the total operating cost of the system (i.e., the cost of fuel + the imp lied cost of emission). Once the value of price penalty factor is determined, the problem reduces to a simp le economic dispatch problem. By proper scheduling of generating units, comparative reduction is achieved in both total fuel cost and emission.

2.2 Transmission Loss

The transmission losses Pt can be found using Bmn coeffic ients

model. The Genera l Algebraic Modeling System (GAMS) is a high-leve l model specially designed for modeling linear, nonlinear and mixed integer optimization proble ms [11]. GAMS can easily handle large and comple x proble ms. It is especially useful for handling large comple x proble ms, which may require much revision to establish an accurate model. Conversion of linear to nonlinear optimization is also very simple. Models can be developed, solved and documented simu ltaneously, maintain ing the same GAM S model file. The basic structure of a mathe matica l mode l coded in GAM S has the

components: sets, data, variable, equation, model and

n n

PL i 1j 1 Pi Bij Pi

n

B P B

i 1 oj i oo

(9)

output [14] and the solution procedure are shown below in Figure 1.

A GAMS model is a collection of statements in

Where

B

,

ij

and

B

oj

Boo are the transmission line

the GAMS language. This statement define the variables of the model, specify the symbolic

coeffic ients

2.3. Constraints

2.3.1 Power Bal ance Constraints

The real power balance between generation and the load must be ma intained at all t imes, while assuming the load at any time to be constant. The total supply must be equal to power de mand.

relationships between them in the form of equations specify data structures and assign values to them and instructs the computer to generate and solve the model. Other GAMS statements are used to handle output [14].

Pi PD PL

Where

(10)

PD is the load demand

1. Ge nerator limit Constr aints

   Figure 1. GAM S modeling and solution procedure STEPS FOR PROBLEM Formu lation WITH GAM S

   GAM S formu lation follows the basic format as given below:

   1. SETS

      Decla ration

      Assignment of me mbers

   2. Data (PARAM ETERS, TABLES, SCA LA RS)

      Decla ration

      Assignment of values

   3. VA RIABLES Decla ration

      Assignment of type

      Assignment of bounds and/or initia l va lues (optional)

   4. EQUATIONS Decla ration

      Definition

   5. MODEL and SOLVE statements

   6. DISPLA Y statements (optional)

1. Result and Discussion

   The GAMS approach has been tested on four diffe rent test cases of a single six unit system of the dynamic economic dispatch problem. In test case I only production cost without taking loss and emission, In test case II production cost with loss and without e mission, In test case III production cost with including e mission and without loss and In test case IV total production cost including e mission and loss for hourly time interval of 24 hours . The imple mentation model on GAMS with system configuration Core 2 Duo processor and 3GB RAM.

   Table I. Cost and emission co-efficient of six unit system

   Linear Coeffic ient of B losses are:

   B0j = [0.00003, 0.00009, 0.00012, 0.00007, 0.000085,

   0.00011]

   1. Test case I

      The generator cost coefficients and generation limits of six units system are taken fro m Table I. For this test case emission coeffic ients and losses coefficients are not considered. In this test case production cost of six-unit system is calculated at different power demand over time period of 24 hours using GAMS. Table II shows the optimal solution of power output and production cost of each generator at different power demand over time period of 24 hours obtained using GAMS.

      Table II. The optimal solution of total generation cost of each unit

   2. Test case II

      The generator cost coefficients, loss coefficients and generation limits of six units system are taken fro m Table I of six-unit system. For this test case emission coeffic ients are not considered. In this test case production cost with losses of six-unit system is calculated at different power de mand over time period of 24 hours using GAMS. Tab le III shows the optima l solution of power output and production cost of each generator at different power demand over time period of 24 hours obtained using GAMS.

      Table III. The optima l solution of total generation cost with power loss of each unit

   3. Test case III

      The generator cost coefficients, emission coefficients and generation limits of six un its system are taken fro m Table I of six-unit system. For this test case loss coefficients are not considered. In this test case production cost including emission of six-unit system is calculated at different power demand over time

      period of 24 hours using GAMS. Table VI shows the optima lsolution of power output and production cost of each generator at different power demand over time period of 24 hours obtained using GAMS.

      Table IV. The optimal solution of total generation cost including e mission of each unit

   4. Test case IV

      The generator cost coefficients, loss coeffic ients, emission coefficients and generation limits of six units system are taken fro m Table I of six-unit system. In this test case production cost including emission with power losses of six-unit system is calculated at different power de mand over time period of 24 hours using GAMS. Table V shows the optima l solution of power output and production cost of each generator at different power demand over time period of 24 hours obtained using GAMS.

2. Conclusion

In this paper, General Algebraic Modeling System (GAMS) for optimization have been used for solving

Table V. The optima l solution of total generating cost including emission and power loss of each unit

dynamic power d ispatch problems. Three different test cases of a single six unit system of the dynamic economic d ispatch problem are taken. In test case I only production cost without emission and loss, In test case II production cost with loss, In test case III production cost with including emission and without loss and In test case IV production cost including emission and loss for time interval o f 24 hours.

An effic ient economic dispatch algorithm for dealing with nonlinear functions such as the thermal cost, transmission loss and emission constraint is developed. The quality of the solutions generated by the General Algebra ic Modeling System (GAMS) offers e xcellent approach to solve the dynamic therma l power dispatch problem. The solution is analytic in nature with high accuracy and it is used for any online application. The result shows that GAM S performs better so far for the above

mentioned test cases. The GAMS algorith m has superior features, including quality of solution and good computational effic iency. Therefore, this results shows that GAMS is a promising technique for solving complicated proble ms in power system.

Acknowledgme nt

The authors are thankful to Director, Madhav Institute of Technology & Science, Gwa lior (M .P) India for prov iding support and facilities to carry out this research work

References

1. J. Wood and B. F. Wollenberg, Powe r Generation, Operat ion and Control, Wiley., Ne w Yo rk 2nd ed, 1996.

2. H. H. Happ, Optima l power d ispatches a comprehensive survey, IEEE Trans. Power Apparatus Syst., 1971, PAS-96, pp. 841-854.

3. Palanicha my and N. S. Babu, Analytica l solution for combined economic and emissions dispatch, Electric Power Systems Research, 2008, vol. 78, pp. 1129 1137.

4. X. S. Han, H. B. Gooi, and S. Kirschen Danie l, Dyna mic economic dispatch: feasible and optima l solutions, IEEE Trans Power Syst., 2001, vol. 16(1), pp. 228.

5. D. C. Wa lters and G. B. Sheble, Genetic algorith m solution of economic dispatch with valve point loadings , IEEE Trans Power Syst, 1993, vol. 8(3), pp. 132531.

6. Gaing Zwe-Lee, Part icle swarm optimization to solving the economic dispatch considering the generator constraints, IEEE Trans Power Syst, 2003, vol. 18(3), pp. 118795.

7. T. Jayabharathi, K. Jayaprakash, N. Jeyakuma r and T. Raghunathan, Evolutionary programming techniques for different kinds of economic dispatch problems, Elect Power Syst , 2005, Res vol. 73(2), pp. 16976.

8. C. K. Panigrahi, P. K. Chattopadhyay, R. N. Chakrabarti, and M. Basu, Simulated annealing technique for dynamic economic d ispatch, Electric Power Compon Syst., 2006, vol. 34(5), pp. 57786.

9. Z. F. Hao, G. H. Guo, and H. Huang, A particle swa rm optimizat ion algorithm with diffe rential evolution, IEEE Int Conf Syst Man Cybern, 2007, vol. 2, pp. 10311035.

10. M. Basu, Artificia l immune system for dynamic economic d ispatch, Int J Electr Power Energy Syst, 2011, vol. 33(1), pp. 131136.

11. Debabrata Chattopadhyay, Application of General a lgebraic modeling system to power system optimization, IEEE Trans. on Power Systems, February 1999, vol. 14, no. 1.

12. R. Yo koyama , S. H. Bae, T. Morita and H. Sasaki, Mult iobjective Optima l Generation Dispatch based on Probability Security Criter ia, IEEE Transactions on Power Systems, 1988, vol. 3, No. 1, pp. 317-324.

13. P. Venkatesh, R. Gnanadass and Narayana Prasad. Padhy, Co mparison and application of evolutionary Programming techniques to combined economic Emission dispatch with line flo w constraints, IEEE Trans. Power Syst. 2003, 18(2), pp. 688-697.

14. Sichard E. Rosenthal, GAMS, A Users Gu ide, Tutorial GAMS Development Corporation , Washington, 2011.