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Change of the cost rate values (feasible/infeasible) versus number of outer loop iterations for different initial points.
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... F-MSG algorithm has been already applied to non-convex economic dispatch problem [8][9]. Furthermore, power dispatch problem including limited energy supply thermal units [10][11], non-convex pumped-storage hydraulic unit scheduling problem [12][13] and short term hydrothermal coordination problem [14] were solved via F-MSG method. To our knowledge, the proposed algorithm has not been applied to the problem considered in this paper so far. ...
... Take ITER=0 Apply F-MSG to STHCP of each subinterval one by one without considering any hydraulic and limited energy fuel consumption constraints given by eq (8)(9)(10)(11)(12). At the solution point, calculate the followings and go to step 1. ...
In this paper, the modified subgradient algorithm based on feasible values (F-MSG) is applied to a short term hydrothermal coordination problem for a power system including limited energy supply thermal units. In the proposed method, all the constraints such as units’ power generation limits, transmission line capacities, bus voltage magnitude limits, hydraulic units’ minimum and maximum reservoir volume limits and hydraulic units’ starting and final reservoir water volumes are added into the optimization model. Actual transmission losses are inserted into the optimization model as equality constraints via load flow equations. It is also assumed that limited energy supply thermal units are fueled under take-or-pay agreement. The proposed method is tested on 16 bus test system which includes three normal thermal plants, two limited energy supply thermal plants and four serial-parallel hydraulically coupled hydro plants and better results are obtained in terms of optimal fuel cost values.
... F-MSG algorithm has already applied to non-convex economic dispatch problem [9][10]. Furthermore, power dispatch problem including limited energy supply thermal units [11][12] and non-convex pumped-storage hydraulic unit scheduling problem [13][14] were solved via F-MSG method. To our knowledge, the proposed algorithm has not been applied to the problem considered in this paper so far. ...
... In step 1, the hydraulic unit scheduling is done while optimal operation of thermal units are found in step 2 with already calculated hydraulic unit generation in step 1. It is obvious that the solutions calculated at the end of each iteration are the actual solutions to STHCP since the whole model described by equations (1)- (12) are considered in step 2. Therefore we carry out these iterations until there is no any further decrease on the optimal cost value. ...
... Nm represents there are Nm generating units in area m. In addition, the objective function is minimized and subjected to the following physical constraints [11,12]. 1) Real power balance constraints without considering transmission loss ...
The auxiliary problem principle is a powerful tool for solving multi-area economic dispatch problem. One of the main drawbacks of the auxiliary problem principle method is that the convergence performance depends on the selection of penalty parameter. In this paper, we propose a self-adaptive strategy to adjust penalty parameter based on the iterative information, the proposed approach is verified by two given test systems. The corresponding simulation results demonstrate that the proposed self-adaptive auxiliary problem principle iterative scheme is robust in terms of the selection of penalty parameter and has better convergence rate compared with the traditional auxiliary problem principle method.
... We applied the F-MSG method into non-convex security constrained dispatch problem in Ref. [9] . We also used the same method in solution of security constrained non-convex dispatch problem of an electric power area that includes limited energy supply thermal units in Ref. [10]. In Refs. ...
... In Refs. [9] and [10] , the outperformance of the F-MSG method against some evolutionary solution methods in terms of both solution time and total cost values is demonstrated on the example power systems that are frequently used in the literature. Application of the dispatch technique based on the F-MSG method and pseudo water price into pumped-storage hydraulic unit scheduling problem is given in Ref. [11] . ...
A security constrained environmental/economic power dispatch problem for a lossy electric power system area including a pumped-storage (p-s) hydraulic unit is formulated. The cost function is made up of weighted sum of total fuel cost and total emission cost of the thermal units in an operation cycle. An iterative solution method based on modified subgradient algorithm operating on feasible values (F-MSG) and pseudo water price for the p-s hydraulic unit is used to solve it. In the proposed solution method, the F-MSG algorithm is used to solve the dispatch problem of each subinterval, while the pseudo water price is employed to adjust the net amount of water spent by the p-s hydraulic unit during the considered operation period.
Since all equality and inequality constraints in our nonlinear optimization model are functions of bus voltage magnitudes and phase angles, the off-nominal tap settings of tap changing transformers and susceptances values of svar systems, they are taken as independent variables. Load flow equations are added into the model as equality constraints. Therefore, the actual transmission loss is used in solution of the considered dispatch problem. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints and svar system susceptance value constraints are added into the optimization problem as inequality constraints. Since the F-MSG algorithm requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by a method, which does not add any extra independent variable into the model, before application of the F-MSG algorithm to the optimization problem. Since the method does not add any extra independent variable into the model, the solution time of the optimization problem is reduced further.
The proposed dispatch technique is demonstrated on an example power system. Pareto optimal solutions for the power system without any p-s unit are calculated first. Later on, the same Pareto optimal solutions for the power system with the p-s unit are recalculated, and the obtained savings in the sum of optimal total fuel cost and total emission cost, due to the employment of the p-s unit, are presented. We also applied the F-MSG method to the dispatch problem with p-s unit directly. We demonstrated that the proposed solution method, where the F-MSG method is employed to solve an interval’s dispatch problem, gives less solution time than the one obtained from the direct application of the F-MSG method to the dispatch problem with the p-s unit although both methods give very close sum of total fuel and emission cost values.
Keywords: Environmental/economic pumped-storage hydraulic unit scheduling problem; Security constraints; The modified subgradient algorithm based on feasible values; Pseudo water price.
... We applied the F-MSG method into non-convex security constrained dispatch problem in Ref. [9] . We also used the same method in solution of security constrained non-convex dispatch problem of an electric power area that includes limited energy supply thermal units in Ref. [10]. In Refs. ...
... In Refs. [9] and [10] , the outperformance of the F-MSG method against some evolutionary solution methods in terms of both solution time and total cost values is demonstrated on the example power systems that are frequently used in the literature. Application of the dispatch technique based on the F-MSG method and pseudo water price into pumped-storage hydraulic unit scheduling problem is given in Ref. [11] . ...
A security constrained environmental/economical power dispatch problem for a lossy electric power system area including a pumped-storage (p-s) hydraulic unit is formulated. The cost function is made up of the weighted sum of the total fuel cost and the total emission cost of the thermal units in an operation cycle. An iterative solution method based on modified subgradient algorithm operating on feasible values (F-MSG) and pseudo water price for the p-s hydraulic unit is used to solve it. In the iterative proposed solution method, the F-MSG algorithm is used to solve the dispatch problem of each subinterval, while the pseudo water price for the p-s hydraulic unit is employed to adjust the net amount of water spent by the p-s hydraulic unit during the considered operation period.
Since all equality and inequality constraints in our nonlinear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles, the off-nominal tap settings of tap changing transformers and susceptances values of svar systems, they are taken as independent variables. Load flow equations are added into the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints and svar system susceptance value constraints are added into the optimization problem as inequality constraints. Since the F-MSG algorithm requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method, which does not add any extra independent variable into the model, before application of the F-MSG algorithm to the optimization problem. Since the method does not add any extra independent variable into the model, the solution time of the optimization problem is reduced further.
The proposed dispatch technique is demonstrated on an example power system. Pareto optimal solutions for the power system without any p-s unit are calculated first. Later on, the same Pareto optimal solutions for the power system with the p-s unit are recalculated, and the obtained saving in the sum of optimal total fuel cost and total emission cost, due to the employment of the p-s unit, is presented. We also applied the F-MSG method to the dispatch problem with p-s unit directly. We demonstrated that the proposed solution method, where the F-MSG method is employed to solve an interval’s dispatch problem, gives less solution time than the one obtained from the direct application of the F-MSG method to the dispatch problem with the p-s unit although both methods give very close sum of total fuel and emission cost values.
Keywords: Environmental/economical pumped-storage hydraulic unit scheduling problem; Security constraints; The modified subgradient algorithm based on feasible values; Pseudo water price
... Application of the modified subgradient algorithm based on feasible values into non-convex security constrained dispatch problem is given in Ref. [14]. The F-MSG method is used in solution of security constrained non-convex economic dispatch problem of an electric power area that includes limited energy supply thermal units in Ref. [15]. In Refs. ...
... In Refs. [14,15], the outperformance of the F-MSG method against some evolutionary solution methods in Nomenclature w weighting factor n j emission price penalty factor for subinterval j (R/ kg) R a fictitious monetary unit R set of real numbers N T set that contains all limited energy supply thermal units N S set that contains all normal thermal units N Bi set that contains all buses directly connected to bus i N tap set that contains all tap changing transformers in the network N svar set that contains all svar systems in the network L set that contains all lines in the network t j length of time interval j (h) U i,j voltage magnitude of bus i in the jth subinterval (pu or kV) d i,j phase angle of bus i in the jth subinterval (rad) r i,j + jx i,j series impedance of the line between buses i and j (pu or X) g ij + jb ij series admittance of the line between buses i and j (pu or S) g shi + jb shi = g shi + j(b capi + b svari ) sum of the half line charging admittance and external shunt susceptances (svar system) if any at bus i (pu or S) b svari,j susceptance value of the svar system being connected to bus i in the jth subinterval ai,j off-nominal tap setting value of tap setting facility at bus i in the jth subinterval p ik,j ,q ik,j active and reactive power flows from bus i to bus k at bus i border in the jth subinterval, respectively (pu or MW, MVar) Àp ki,j , Àq ki,j active and reactive power flows from bus i to bus k at bus k border in the jth subinterval, respectively (pu or MW, MVar) p l,j active power flow on line l in the jth subinterval (pu or MW) P Gi,j , Q Gi,j active and reactive power generations of the ith unit in the jth subinterval, respectively (pu or MW, MVar) P Load i,j , Q Load i,j active and reactive loads of the ith bus in the jth subinterval, respectively (pu or MW, MVar) P LOSS,j total active power loss in the network in the jth subinterval (pu or MW) E i (P Gi,j ) emission rate of the ith generation unit in the jth vector obtained at the mth iteration of the inner loop of the nth outer loop iteration u n m ; c n m dual variables calculated at the mth iteration of the inner loop of the nth iteration of the outer loop s m positive step size parameter calculated at the mth iteration e F n T;j modified weighted total cost rate value of the jth subinterval which will be checked in the nth outer loop (R) D n decrement or increment on e F n T;j value, at the end of nth outer loop, according to whether e F n T;j is feasible or not (R) e 1 , e 2 tolerance values for kh(x)k and D n , respectively terms of both solution time and total cost values is demonstrated on the example power systems that are frequently used in the literature . Security constrained non-convex pumped-storage hydraulic unit scheduling problem is solved via the F-MSG method and pseudo water price in Ref. [16]. ...
... , 6. The load schedule of the example power system is given in Ref. [15]. The simulation program was coded in MATLAB 6.1 (The Math- Works, Natick, Massachusetts, USA). ...
... Application of the modified subgradient algorithm based on feasible values into non-convex security constrained dispatch problem is given in Ref. [14]. The F-MSG method is used in solution of security constrained non-convex economic dispatch problem of an electric power area that includes limited energy supply thermal units in Ref. [15]. In Refs. ...
... In Refs. [14,15], the outperformance of the F-MSG method against some evolutionary solution methods in Nomenclature w weighting factor n j emission price penalty factor for subinterval j (R/ kg) R a fictitious monetary unit R set of real numbers N T set that contains all limited energy supply thermal units N S set that contains all normal thermal units N Bi set that contains all buses directly connected to bus i N tap set that contains all tap changing transformers in the network N svar set that contains all svar systems in the network L set that contains all lines in the network t j length of time interval j (h) U i,j voltage magnitude of bus i in the jth subinterval (pu or kV) d i,j phase angle of bus i in the jth subinterval (rad) r i,j + jx i,j series impedance of the line between buses i and j (pu or X) g ij + jb ij series admittance of the line between buses i and j (pu or S) g shi + jb shi = g shi + j(b capi + b svari ) sum of the half line charging admittance and external shunt susceptances (svar system) if any at bus i (pu or S) b svari,j susceptance value of the svar system being connected to bus i in the jth subinterval ai,j off-nominal tap setting value of tap setting facility at bus i in the jth subinterval p ik,j ,q ik,j active and reactive power flows from bus i to bus k at bus i border in the jth subinterval, respectively (pu or MW, MVar) Àp ki,j , Àq ki,j active and reactive power flows from bus i to bus k at bus k border in the jth subinterval, respectively (pu or MW, MVar) p l,j active power flow on line l in the jth subinterval (pu or MW) P Gi,j , Q Gi,j active and reactive power generations of the ith unit in the jth subinterval, respectively (pu or MW, MVar) P Load i,j , Q Load i,j active and reactive loads of the ith bus in the jth subinterval, respectively (pu or MW, MVar) P LOSS,j total active power loss in the network in the jth subinterval (pu or MW) E i (P Gi,j ) emission rate of the ith generation unit in the jth vector obtained at the mth iteration of the inner loop of the nth outer loop iteration u n m ; c n m dual variables calculated at the mth iteration of the inner loop of the nth iteration of the outer loop s m positive step size parameter calculated at the mth iteration e F n T;j modified weighted total cost rate value of the jth subinterval which will be checked in the nth outer loop (R) D n decrement or increment on e F n T;j value, at the end of nth outer loop, according to whether e F n T;j is feasible or not (R) e 1 , e 2 tolerance values for kh(x)k and D n , respectively terms of both solution time and total cost values is demonstrated on the example power systems that are frequently used in the literature . Security constrained non-convex pumped-storage hydraulic unit scheduling problem is solved via the F-MSG method and pseudo water price in Ref. [16]. ...
... , 6. The load schedule of the example power system is given in Ref. [15]. The simulation program was coded in MATLAB 6.1 (The Math- Works, Natick, Massachusetts, USA). ...
... Application of the modified subgradient algorithm based on feasible values (F-MSG) into the non-convex security constrained dispatch problem was given in [7]. The F-MSG method was used in the solution of a security-constrained non-convex dispatch problem of an electric power area that includes limited energy supply thermal units in [8]. In those works, the outperformance of the F-MSG method against some evolutionary solution methods given in recent literature, in terms of both solution time and total cost values, were demonstrated on a selected sample power system. ...
... Research is currently underway on the application of the F-MSG method to some other non-convex security-constrained economic power dispatch problems, such as the short-term hydrothermal scheduling problem and the economic/environmental dispatch problem of a power area with fuel-constrained thermal units. Downloaded by [ Please see Nomenclature the meaning of the symbols used in the following equations [8]: ...
A security-constrained power dispatch problem with a non-convex cost function for a lossy electric power system area including a pumped-storage hydraulic unit is formulated. Then, an iterative solution method based on a modified subgradient algorithm operating on feasible values and pseudo water price for the pumped-storage hydraulic unit is used to solve it. In the iterative proposed solution method, the modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the pseudo water price for the pumped-storage hydraulic unit is employed to adjust the net amount of water used by the pumped-storage hydraulic unit during the considered operation period. Since all equality and inequality constraints in the proposed non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles, the off-nominal tap settings, and susceptances values of SVAR systems, they are taken as independent variables. Load flow equations are added into the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints, and SVAR system susceptance value constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method, which does not add any extra independent variable into the model, before its application to the optimization model. Since the method does not add any extra independent variable into the model, the solution time is reduced further. The proposed dispatch technique is tested on a sample power system that has 12 buses with 5 thermal units and a pumped-storage hydraulic unit. Optimal total cost value for the power system without any pumped-storage unit is calculated first. Later, the same optimal total cost value for the power system with the pumped-storage unit is recalculated, and the obtained saving in the optimal total cost value, due to the employment of the pumped-storage unit, is presented. The solution times for the dispatch problems with and without the pumped-storage unit is also presented. At the end, the modified subgradient algorithm based on feasible values is applied directly to the dispatch problem with a pumped-storage unit. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem with the pumped-storage unit.
A security constrained economic dispatch problem with prohibited operation zones for a lossy electric power system is formulated. An iterative solution method that is based on modified subgradient algorithm operating on feasible values is employed to solve it. Bus voltage magnitudes and phase angles, off-nominal tap settings and susceptance values of svar systems are taken as independent (decision) variables in the solution algorithm. Since load flow equations are added into the model as equality constraints, actual power system loss is used in solution of the optimization model. The proposed technique is tested on IEEE 30-bus test systems. The minimum total cost rates and the solution times obtained from F-MSG algorithm and from the other techniques are compared, and the outperformance of the F-MSG algorithm with respect to the other methods in each test system is demonstrated.
