Project on economic load dispatch

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Project on economic load dispatch

  1. 1. Project on Economic Load dispatch<br />
  2. 2. Economic Load dispatch<br />What is economic dispatch?<br />The definition of economic dispatch is:<br />“The operation of generation facilities to produce energy at the lowest cost to reliably serve consumers, recognizing any operational limits of generation and transmission facilities”.<br /> Most electric power systems dispatch their own generating units and their own purchased power in a way that may be said to meet this definition.<br />The factors influencing power generation at minimum cost are :<br /><ul><li>operating efficiencies of generators
  3. 3. fuel cost and transmission losses</li></li></ul><li>Economic Load dispatch<br /> if the plant is located far from the load centre, transmission losses may be considerably higher and hence the plant may be overly uneconomical. Hence, the problem is to determine the generation of different plants whereby the total operating cost is minimum. The operating cost plays an important role in the economic scheduling .The input to the thermal plant is generally measured in Btu/h, and the output is measured in MW.<br />The input-output curve of a thermal unit known as heat-rate curve as shown in Fig. <br />
  4. 4. Heat-Rate Curve<br />(a)<br />fuel<br />input,<br />Btu/hr<br />Pi, MW<br />
  5. 5. Fuel-Cost Curve<br />(b)<br />The fuel cost is commonly express as a quadratic function<br />Ci=αi+ βiPi+ γiPi2 Rs/h <br />cost<br />Ci,<br />Rs/hr<br />Pi, MW<br />
  6. 6. incremental fuel-cost curve<br />The derivative is known as the incremental fuel cost.<br />λ<br />Rs/MWh<br />Pi, MW<br />
  7. 7. Economic Load dispatch<br />An economic dispatch schedule for assigning loads to each unit in a plant can be prepared by <br /><ul><li>assuming various values of total plant output
  8. 8. calculating the corresponding incremental fuel cost λ of the plant
  9. 9. substituting the value of λ for λi in the equation for the incremental fuel cost of each unit to calculate its output. </li></ul> For a plant with two units having no transmission losses operating under economic load distribution the λ of the plant equals λi of each unit, and so<br />λ= dC1/dP1 = 2γ1 P1+β1 ; λ= dC2/dP2 = 2γ2 P2+β2;and <br />
  10. 10. Economic Load dispatch<br />P1 + P2 = PD <br />or <br />where PD= total load demand<br />Solving for P1 and P2, we obtain<br />P1= (λ-β1)/2γ1 ; P2 = (λ-β2)/2γ2 <br />and for Pi<br />Pi= (λ-βi)/2γi (coordination equation) <br />Thus an analytical solution can be obtained for λ as<br /> λ = /<br />
  11. 11. Economic Load dispatch<br />For a system with k generating units <br />Ct= C1+ C2+ C3+…………………+ Ck= <br />Where Ct = total fuel cost for all generating units <br />The total MW power input to the network from all the units is the sum<br /> P1+P2+…………….+Pk= <br /> <br /> Where P1, P2,……Pk= the individual outputs of the units injected to the network. The total fuel cost C of the system is a function of all the power plant output <br />The economic dispatch problem including transmission losses is defined as<br />Min Ct=<br />Subject to PD+PL- = 0 PL is the total transmission system loss <br />
  12. 12. Economic Load dispatch<br />for including the effect of transmission losses is to express the total transmission loss as a quadratic function of the generator power outputs. The simplest function is<br />PL=<br /> The coefficients Bijare called loss coefficients or B-coefficients <br /> Making use of the Langrangian multiplier λ <br />L= Ct + λ (PD+PL- )<br />
  13. 13. Economic Load dispatch<br />For minimum cost we require the derivative of L with respect to each Pi to equal zero. <br />Since PD is fixed and the fuel cost of any one unit varies only of the power output of that unit is varied, equation yields<br />=<br />+ λ (PD+PL- )] = 0<br />
  14. 14. Economic Load dispatch<br />For each of the generating unit outputs P1, P2,……Pk. Because Ci depends on only Pi, the partial derivative of Ci can be replaced by the full derivative and above equation then gives<br /> <br />λ= {1/ }<br />for every value of i . This equation is often written in the form<br />λ=Li( )<br />where Li is called the penalty factor of plant i<br />
  15. 15. Economic Load dispatch with generator limit<br /><ul><li>The power output of any generator should not exceed its</li></ul> rating nor be below the value for stable boiler operation<br /><ul><li>Generators have a minimum and maximum real power output limits.
  16. 16. The problem is to find the real power generation for each plant such that cost are minimized, subject to:
  17. 17. Meeting load demand - equality constraints
  18. 18. Constrained by the generator limits - inequality constraints</li></li></ul><li>Economic Load dispatch with generator limit<br />The Kuhn-Tucker conditions<br />λ , Pi(min)<Pi<Pi(max)<br /> <= λ, Pi=Pi(max)<br /> >= λ, Pi=Pi(min)<br />
  19. 19. Unit commitment<br />System Constraints<br /><ul><li>The total outputof all the generating units must be equal tothe forecast value of the system demandat each time-point.
  20. 20. The total spinning reservefrom all the generating units must be greater than or equal to the spinning-reserve requirement of the system. This can be either a fixed requirement in MW or a specified percentage of the largest output of any generating unit.
  21. 21. Minimum up time : Once the unit is running, it should not be turned off immediately.
  22. 22. Minimum down time: Once the unit is de-committed (off), there is a minimum time before it can be recommitted.
  23. 23. The output powerof the generating units must be greater or equalto the minimum powerof the generating units.
  24. 24. The output powerof the generating units must be smaller or equalto the maximum powerof the generating units.</li></li></ul><li>Priority list method<br /><ul><li>Simplest of all methods, but at the same time approximate also. Units are ranked according to their full load production cost rate and committed accordingly. Has to follow a shut down algorithm for the satisfaction of the minimum uptime and down time constraints and spinning reserve constraints.</li></li></ul><li>limits of using priority list method<br /><ul><li>No load cost is zero.
  25. 25. Unit input-output characteristics is linear between zero output and full load.
  26. 26. Start up cost are fixed amount.
  27. 27. There are no other constraints. </li></ul> <br />
  28. 28.  Economic Dispatch – Summary<br /><ul><li>Economic dispatch determines the best way to minimize the current generator operating costs.
  29. 29. The lambda-iteration method is a good approach for solving the economic dispatch problem.
  30. 30. generator limits are easily handled
  31. 31. penalty factors are used to consider the impact of losses
  32. 32. Economic dispatch is not concerned with determining which units to turn on/off (this is the unit commitment problem).
  33. 33. Economic dispatch ignores the transmission system limitations.</li></li></ul><li>program to solve economic dispatch problem <br />Problem 4.3.<br />Bi0 and B00 are neglected. Assume three units are on-line and have the following characteristics:<br /> <br />Unit 1: <br />H1 = 312.5 + 8.25P1 + 0.005P2 MBtu/h<br />50 ≤ P1 ≤ 250 MW<br />Fuel Cost = 1.05 Rs/MBtu<br /> <br />Unit 2: <br />H2 = 112.5 + 8.25P2 + 0.005P2 MBtu/h<br />5≤ P2 ≤150 MW<br />Fuel Cost = 1.217 Rs/MBtu<br /> <br />Unit 3: <br />H3 = 50 + 8.25P3 + 0.005P2 MBtu/h<br />5≤ P2 ≤150 MW<br />Fuel Cost = 1.1831Rs/MBtu<br />
  34. 34. program to solve economic dispatch problem <br />a. No Losses Used in Scheduling <br />i. Calculate the optimum dispatch and total cost neglecting losses for PD = 190 MW. <br />ii. Using dispatch and the loss formula, calculate the system losses. This can also be solved using the following Matlab script file and function file which make use of the fsolveMatlab function to solve the system of equations:<br />b. Losses Included in Scheduling<br /> i. Find the optimum dispatch for a total generation of PD = 190 MW using the coordination equations and the loss formula.<br /> ii. Calculate the cost rate. <br />iii. Calculate the total losses using the loss formula. <br />iv. Calculate the resulting load supplied. This can also be solved using the following Matlab script file and function file which make use of the fsolveMatlab function to solve the system of equations, the program is also capable of computing the incremental losses and the penalty factors which are calculated in each iteration within the function file:<br /> <br />
  35. 35. program to solve economic dispatch problem <br />% PART A OF THE PROBLEM<br />% set global variables<br />global P1 P2 P3 Lambda<br />global P1min P2min P3min P1max P2max P3max<br />% define Pi_min and Pi_max<br />P1min = 50;<br />P2min = 5;<br />P3min = 15;<br />P1max = 250;<br />P2max = 150;<br />P3max = 100;<br />% set initial guess of powers and system lambda<br />P10 = 142;<br />P20 = 15;<br />P30 = 32;<br />Lambda0 = 10;<br />% solve equations in function ednoloss_eqs_p43 using fsolve<br />z1 = fsolve(@ednoloss_eqs_p43,[P10 P20 P30 Lambda0],optimset(’MaxFunEvals’,10ˆ2,…<br />’MaxIter’,10ˆ2));%,’Display’,’iter’));<br />clc;<br />
  36. 36. program to solve economic dispatch problem <br />% compute cost of scheduling each unit<br />F1 = 308.125 + 8.6625*P1 + 0.0052500*P1ˆ2;<br />F2 = 136.9125 + 10.04025*P2 + 0.0060850*P2ˆ2;<br />F3 = 59.155 + 9.760575*P3 + 0.0059155*P3ˆ2;<br />FT = F1+F2+F3;<br />% output<br />disp(’Problem 4.3.’)<br />disp(’------------’)<br />disp(’Part a - i’)<br />disp(’P1 P2 P3 Lambda’)<br />disp(num2str(z1))<br />disp([’Cost of Dispatching Unit 1: ’ num2str(F1)])<br />disp([’Cost of Dispatching Unit 2: ’ num2str(F2)])<br />disp([’Cost of Dispatching Unit 3: ’ num2str(F3)])<br />disp([’Total Cost of ED: ’ num2str(F1+F2+F3)])<br />
  37. 37. program to solve economic dispatch problem <br />% PART ii OF THE PROBLEM<br />P = [P1; P2; P3];<br />B = [ 1.36255*10ˆ(-4) 1.75300*10ˆ(-5) 1.83940*10ˆ(-4);<br />1.75400*10ˆ(-5) 1.54480*10ˆ(-4) 2.82765*10ˆ(-5);<br />1.83940*10ˆ(-4) 2.82765*10ˆ(-4) 1.61470*10ˆ(-3)];<br />PLOSS = P’*B*P<br /> <br />Function File:<br /> <br />function F = ednoloss_eqs_p43(z)<br />global P1 P2 P3 Lambda<br />global P1min P2min P3min P1max P2max P3max<br />% Unknown variables<br />P1 = z(1);<br />P2 = z(2);<br />P3 = z(3);<br />Lambda = z(4);<br />if P3 < P3min<br />P3 = P3min;<br />else<br />if P3 > P3max<br />P3 = P3max;<br />end<br />end<br />
  38. 38. program to solve economic dispatch problem <br />if P2 < P2min<br />P2 = P2min;<br />else<br />if P2 > P2max<br />P2 = P2max;<br />end<br />end<br />if P1 < P1min<br />P1 = P1min;<br />else<br />if P1 > P1max<br />P1 = P1max;<br />end<br />end<br />% dL/dPi and dL/dLambda<br />F(1,1) = 8.6625 + 0.0105*P1 - Lambda;<br />F(2,1) = 10.04025 + 0.01217*P2 - Lambda;<br />F(3,1) = 9.760575 + 0.011831*P3 - Lambda;<br />F(4,1) = P1 + P2 + P3 - 190;<br />
  39. 39. program to solve economic dispatch problem <br />Execution of the Matlab Script file yields the solution for the problem as described below:<br /> <br />Problem 4.3.<br />------------<br />Part a - i<br />P1 P2 P3 Lambda<br />143.9936 11.02567 34.98076 10.17443<br />Cost of Dispatching Unit 1: 1664.3235<br />Cost of Dispatching Unit 2: 248.3527<br />Cost of Dispatching Unit 3: 407.8259<br />Total Cost of ED: 2320.5021<br />PLOSS =<br />6.8484<br />
  40. 40. program to solve economic dispatch problem <br />% PART B OF THE PROBLEM<br />% set global variables<br />global P1 P2 P3 Lambda IL1 IL2 IL3 PF1 PF2 PF3 PLOSS B PLOAD<br />global P1min P2min P3min P1max P2max P3max<br />% define Pi_min and Pi_max<br />P1min = 50;<br />P2min = 5;<br />P3min = 15;<br />P1max = 250;<br />P2max = 150;<br />P3max = 100;<br />% define system load<br />PLOAD = 190;<br />% define b-matrix<br />B = [ 1.36255*10ˆ(-4) 1.75300*10ˆ(-5) 1.83940*10ˆ(-4);<br />1.75400*10ˆ(-5) 1.54480*10ˆ(-4) 2.82765*10ˆ(-5);<br />1.83940*10ˆ(-4) 2.82765*10ˆ(-4) 1.61470*10ˆ(-3)];<br />
  41. 41. program to solve economic dispatch problem <br />% set initial guess of powers and system lambda<br />% this values are the results from part one of the problem<br />P10 = 143.9936;<br />P20 = 11.02567;<br />P30 = 34.98076;<br />Lambda0 = 10.17443;<br />% solve equations in function ednoloss_eqs_p43 using fsolve<br />z1 = fsolve(@dispatchloss_eqs_p43,...<br /> [P10 P20 P30 Lambda0 1 1 1 1 1 1 1 1 1 1 1 1],optimset(’MaxFunEvals’,10ˆ2,...<br />’MaxIter’,10ˆ2));%,’Display’,’iter’));<br />clc;<br />% compute cost of scheduling each unit<br />F1 = 308.125 + 8.6625*P1 + 0.0052500*P1ˆ2;<br />F2 = 136.9125 + 10.04025*P2 + 0.0060850*P2ˆ2;<br />F3 = 59.155 + 9.760575*P3 + 0.0059155*P3ˆ2;<br />FT = F1+F2+F3;<br />% output<br />
  42. 42. program to solve economic dispatch problem <br />disp(’Problem 4.3.’)<br />disp(’...................................................................’)<br />disp(’Part b - i’)<br />disp(’----------’)<br />disp(’P1 P2 P3 Lambda’)<br />disp(num2str([z1(1,1) z1(1,2) z1(1,3) z1(1,4)]))<br />disp([’Cost of Dispatching Unit 1: ’ num2str(F1)])<br />disp([’Cost of Dispatching Unit 2: ’ num2str(F2)])<br />disp([’Cost of Dispatching Unit 3: ’ num2str(F3)])<br />disp([’Total Cost of ED: ’ num2str(F1+F2+F3)])<br />disp(’...................................................................’)<br /> <br />% PART ii of Part B of the Problem<br />CR1 = Lambda*(1/PF1);<br />CR2 = Lambda*(1/PF2);<br />CR3 = Lambda*(1/PF3);<br />disp(’Part b - ii’)<br />
  43. 43. program to solve economic dispatch problem <br />disp(’-----------’)<br />disp([’Penalty Factor of unit 1:’ num2str(PF1)])<br />disp([’Penalty Factor of unit 2:’ num2str(PF2)])<br />disp([’Penalty Factor of unit 3:’ num2str(PF3)])<br />disp([’Cost Rate of unit 1 :’ num2str(CR1)])<br />disp([’Cost Rate of unit 2 :’ num2str(CR2)])<br />disp([’Cost Rate of unit 3 :’ num2str(CR3)])<br />disp(’...................................................................’)<br />disp(’Part b - iii’)<br />disp(’------------’)<br />P = [z1(1,1); z1(1,2); z1(1,3)];<br />PLOSS1 = P’*B*P<br />disp(’...................................................................’)<br />disp(’Part b - iv’)<br />disp(’------------’)<br />disp(’Resulting load supplied: Psupplied = Ploss_calculated + Pload’)<br />Psupplied = PLOSS1 + PLOAD<br /> <br />
  44. 44. program to solve economic dispatch problem <br />Function File:<br /> <br />function F = dispatchloss_eqs_p43(z)<br />global P1 P2 P3 Lambda IL1 IL2 IL3 PF1 PF2 PF3 PLOSS B PLOAD<br />global P1min P2min P3min P1max P2max P3max<br />% Unknown variables<br />P1 = z(1);<br />P2 = z(2);<br />P3 = z(3);<br />Lambda = z(4);<br />IL1 = z(5);<br />IL2 = z(6);<br />IL3 = z(7);<br />PF1 = z(8);<br />PF2 = z(9);<br />PF3 = z(10);<br />PLOSS = z(11);<br />% bound powers to their limits<br />if P3 < P3min<br />P3 = P3min;<br />else<br />
  45. 45. program to solve economic dispatch problem <br />if P3 > P3max<br />P3 = P3max;<br />end<br />end<br />if P2 < P2min<br />P2 = P2min;<br />else<br />if P2 > P2max<br />P2 = P2max;<br />end<br />end<br />if P1 < P1min<br />P1 = P1min;<br />else<br />if P1 > P1max<br />P1 = P1max;<br />end<br />end<br />
  46. 46. program to solve economic dispatch problem <br />IL1 = 2*(B(1,1)*P1 + B(1,2)*P2 + B(1,3)*P3);<br />IL2 = 2*(B(2,1)*P1 + B(2,2)*P2 + B(2,3)*P3);<br />IL3 = 2*(B(3,1)*P1 + B(3,2)*P2 + B(3,3)*P3);<br />PF1 = 1/(1 - IL1);<br />PF2 = 1/(1 - IL2);<br />PF3 = 1/(1 - IL3);<br />PLOSS = (P1ˆ2)*B(1,1) + P1*B(1,2)*P2 + P1*B(1,3)*P3+ ...<br />P2*B(2,1)*P1 + (P2ˆ2)*B(2,2) + P2*B(2,3)*P3+ ...<br />P3*B(3,1)*P1 + P3*B(3,2)*P2 + (P3ˆ2)*B(3,3);<br />F(1,1) = PF1*(8.6625 + 0.0105*P1) - Lambda;<br />F(2,1) = PF2*(10.04025 + 0.004*P2) - Lambda;<br />F(3,1) = PF3*(9.760575 + 0.011831*P3) - Lambda;<br />F(4,1) = PLOAD + PLOSS - (P1+P2+P3) ;<br /> <br />
  47. 47. program to solve economic dispatch problem <br />The results of the execution of the script file, shown below, yield the solution to the problem:<br /> <br />Problem 4.3.<br />...................................................................<br />Part b - i<br />----------<br />P1 P2 P3 Lambda<br />143.8125 35.70524 14.94276 10.778<br />Cost of Dispatching Unit 1: 1662.4798<br />Cost of Dispatching Unit 2: 503.1612<br />Cost of Dispatching Unit 3: 206.8946<br />Total Cost of ED: 2372.5356<br />...................................................................<br />Part b - ii<br />-----------<br />Penalty Factor of unit 1:1.0482<br />Penalty Factor of unit 2:1.0172<br />Penalty Factor of unit 3:1.1384<br />Cost Rate of unit 1 :10.2826<br />Cost Rate of unit 2 :10.5956<br />Cost Rate of unit 3 :9.468<br />
  48. 48. program to solve economic dispatch problem <br />...................................................................<br /> <br /> <br />Part b - iii<br />------------<br />PLOSS1 =<br />4.5121<br />...................................................................<br />Part b - iv<br />------------<br />Resulting load supplied: Psupplied = Ploss_calculated + Pload<br />Psupplied =<br />194.51<br />
  49. 49. Economic Load dispatch<br />Does including loss formula matter? <br /> <br /> The inclusion of losses does matter; it can be seen from the results above that the consideration of losses in the scheduling changes the optimum economic dispatch, increases the total generation cost (since the total demand will now include the power dissipated in the resistive elements) and also the system lambda; the inclusion of losses it’s also a more realistic representation of what is happening in the real power system, which has losses.<br />
  50. 50. Thank you<br />

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