Optimization of a plurality of multiple-fuel fired boilers using iterated linear programming
This application is a continuation of application Ser. No. 07/560,400, filed Jul. 31, 1990, now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention is directed to optimizing operation of a system having many interdependent variable operating conditions which may have non-linear relationships and, more particularly, to optimizing the operation of an energy producing system having a plurality of interrelated components. 
     2. Description of the Related Art 
     Optimization techniques are commonly used to maximize production or minimize cost of operating industrial processes, particularly with regard to the consumption of energy and the correlated production output. Several techniques have been applied to chemical engineering to establish operating conditions that yield a maximum return on investment while minimizing operating costs. These techniques include linear programming and evolutionary operation. Application of these techniques to optimizing cogeneration boiler systems for producing steam and electricity are disclosed in U.S. Pat. Nos. 4,604,714 and 4,805,114 to R. E. Putman et al. which are incorporated herein by reference. 
     Of the two techniques mentioned above, linear programming is fairly well known, while evolutionary operation is much less commonly used since it was originally developed as a manual system in the 1960&#39;s when powerful online computers were not available. Linear programming application software is available commercially, e.g., LP88 from Eastern Software Products, Inc. of Alexandria, Virginia. A particularly useful form of evolutionary operation was described in &#34;Process Improvement with SIMPLEX Self-Directing Evolutionary Operation&#34; by E. H. Carpenter and H. C. Sweeney, Chemical Engineering. Jul. 5, 1965, pp. 117-126. The evolutionary operation technique described in this article will be referred to herein as SSDEVOP. 
     The SSDEVOP technique modifies the setting of several key process variables to find a setting for an industrial process which results in maximum production or minimum costs, while taking into account predetermined constraints of the variables. The selected process variables are assigned different values in a set of experiments which in the original technique are conducted on the actual industrial process. The results are compared and a new base case is produced by averaging the setting of each process variable for all except the test which produced the worst case results, doubling this average and subtracting the worst case results. This base case is then used to produce a new set of tests by adding or subtracting small amounts to or from the setting of each variable. The results of the tests in a set of experiments is analyzed to find new best and worst cases and to produce a new base case and a new set of experiments from that base case. The process is repeated until there is little improvement in the best result. At this time, the amount of change from one test to another in a set of tests can be reduced for a new set of tests so that a more precise optimum setting for the process can be determined. The &#39;714 and &#39;114 patents teach application of SSDEVOP to the process of generating steam and electricity in a cogeneration system. At the end of the &#39;114 patent it is suggested that SSDEVOP be used to optimize load and fuel distribution in pairs of boilers which are part of a much larger system where the load and fuel distribution of the large system is determined by a linear programming matrix. However, there is no suggestion of how to optimize the performance of a process having many non-linear relationships between process variables. A linear programming matrix can handle many variables, but by definition does not handle non-linear relationships. The SSDEVOP technique handles non-linear relationships, but becomes unwieldy as the number of selected variables increases beyond four. 
     SUMMARY OF THE INVENTION 
     An object of the present invention is to provide a technique for optimizing processes having five or more non-linear relationships. 
     Another object of the present invention is to provide a technique for optimizing the operations of a plurality of multiple-fuel fired boilers which produce a total amount of steam. 
     Yet another object of the present invention is to determine appropriate coefficients for a model of a system including a combustion turbogenerator, a steam generator and a heat recovery steam generator. 
     The above objects are attained by providing a method for optimizing control of a process having interdependent operating conditions determined by a control unit, comprising the steps of: defining relationships between the operating conditions, all constraints on the process and a process variable to be optimized, in a linear programming matrix; assigning initial values to matrix elements in the liner programming matrix; executing a computer program to solve the linear programming matrix; modifying selected matrix elements representing a set of the operating conditions according to a test strategy and adjusting any unselected matrix elements that require change due to the modifying; executing the computer program to produce a solution of the linear programming matrix after completing the modifying; repeating the modifying and executing of the computer program on the modified linear programming matrix for each test defined by the test strategy until convergence of the solution of the linear programming matrix; and adjusting the control unit to establish the operating conditions indicated by the solution of the linear programming matrix resulting at the convergence 
     In one application of the present invention the test strategy used is simplex self-directing evolutionary operation. This method is particularly useful when the process to be optimized is the production of steam by a plurality of multiple-fuel fired boilers, such as a system having four boilers each capable of receiving three different types of fuel. 
     In another application of the present invention the process to be optimized is the production of steam and electricity by a system having a gas turbogenerator, a steam generator and a heat recovery steam generator. In this application the linear programming matrix includes energy balance equations for the system. The selected matrix elements are modified by calculating first values representing a first steady state model based on the initial solution of the linear programming matrix and second values representing a second steady state model based upon slight modifications to the initial solution The first values are assigned to the selected matrix elements representing the first steady state model and local linear models determined as a function of the assignment made by the first steady state model are assigned to the selected matrix elements which represent effects of change from conditions in the first steady state model. The modification and execution steps are repeated until an accurate model is being used for the system to produce steam and electricity with optimal settings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a tabulation of variables using simplex self-directing evolutionary operation; 
     FIG. 2 is a block diagram of a steam producing system having a plurality of multiple-fuel fired boilers; 
     FIG. 3 is a flowchart of a method according to the present invention; 
     FIG. 4 is a graph of heat produced versus load for different fuels in a multiple-fuel fired boiler; 
     FIGS. 5 and 7 are flowcharts of test strategies used in the first and second embodiments, respectively; and 
     FIG. 6 is a block diagram of a steam and electricity cogeneration plant to which the present invention can be applied. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The simplex self-directing evolutionary operation technique is summarized in the &#39;714 patent as a method based on a matrix of several process variables disturbed by a predetermined increment successively in accordance with a series of tests revealing an improved situation, after which, based on the best results attained, a new set of values for the process variables is in turn disturbed, and this is repeated until the results converge on an optimum. While originally the SSDEVOP technique was applied to the process itself, the &#39;714 patent discloses that the SSDEVOP technique can be applied to a set of equations which simulate a process. 
     With reference to an illustration of the SSDEVOP process in FIG. 1, initially some assumptions are made regarding the operation of the process. These assumptions are used to produce an initial base case for SSDEVOP. The assumptions may be based upon current settings if the process is already in operation, or that all components are operating at maximum efficiency (this is typically impossible or improbable, but at least provides a starting point for a system under design), or simply a best guess of the values. The base case is then used to produce n+1 test cases where n is the number of process variables in the optimizing set which are permitted to change. The test cases preferably use a pattern of changes like those illustrated in the middle block of FIG. 1. 
     The values of the process variables which are changed are used as a set of experiments or simulation runs to produce responses RESP 1  -RESP 4 . The mean of each variable in the three better cases is doubled and the value of the corresponding variable in the worst case is subtracted to produce a new base case. When the response of the best case from one iteration to the next does not change significantly, the response of the best case in the last two iterations can be used as the settings for the process or as a base case for a new sequence of tests using smaller changes, i.e., smaller values of Δ1-Δ3. 
     By carefully selecting the variables modified by the SSDEVOP technique, i.e., the boiler load on each of two dual-fuel fired boilers and one of the two fuels supplied to each of the boilers, it is possible to apply SSDEVOP to a system having two dual-fuel fired boilers, as disclosed in the &#39;114 patent. However, if the number of boilers or the number of fuels supplied to each boiler is increased, it is no longer possible to use the SSDEVOP technique for four variables illustrated in FIG. 1. As the number of variables increase, the number of tests that must be run increases and the selection of a proper SSDEVOP test strategy, i.e., the pattern of changes from one test to the next, can become very difficult. 
     An example of a system having a plurality of multiple-fuel fired boilers which would be difficult to optimize using SSDEVOP is illustrated in FIG. 2. Four boilers 12a-12d are capable of outputting steam via steam lines 14a-14d, respectively, to a common header 16 which supplies the steam to a steam turbogenerator or other processes using steam in a cogeneration system. Control of the overall operation of the system, or at least the boilers 12a-12d, is provided by a distributed control system 18 which includes a computer system 20. Data is provided to the distributed control system 18 via signal lines 22, 24a-24d from a pressure transducer 26 mounted on the common header 16 and pressure transducers 28a-28d corresponding to the steam lines 14a-14d. 
     Each of the boilers in the embodiment illustrated in FIG. 2 can receive three fuels, e.g., coke oven gas, blast furnace gas and oil. As a result, there are twelve interdependent variables, i.e., fuel/boiler combinations, in the system illustrated in FIG. 2. Due to the similarities of the components controlling fuel supplied to the boilers, the following discussion will describe control of a boiler without use of the letter identifying which boiler is being described. Each fuel is supplied to the boiler 12 under the control of control loops 30, 32, 34. Communication between the boiler 12 and the control loops 30, 32, 34 as well as delivery of fuel is provided by bi-directional links 36, 38, 40, respectively. Each fuel control loop includes a manual/automatic transfer control element 42, 44 or 46 which permits manual override of automatic operation provided by the distributed control system 18 in a conventional manner. Communication between the control loops 30, 32, 34 and the distributed control system 18 is provided by communication lines 46, 48, 50, respectively. 
     The distributed control system 18 determines which boilers are operating and which fuels in what amounts are being supplied to the operating boilers to produce a requested steam load in the common header 16. This control is accomplished with the aid of additional control devices (not shown) associated with the boilers, e.g., master control loops. In addition to receiving data from the transducers and control loops, the distributed control system 18 receives instructions and displays status of the system from and to a man-machine interface 52 via a bi-directional communication link 54. Typically, the man-machine interface 52 is provided by a keyboard and video terminal. 
     According to the present invention, the computer system 20 in the distributed control system 18 determines an optimal solution by applying the method illustrated in FIG. 3. Prior to controlling the system illustrated in FIG. 2, it is necessary to define linear programming matrix equations for the system. The step of defining 60 is performed manually and while unique for each boiler, should not be difficult to accomplish by anyone of ordinary skill in the art of designing or operating steam generator systems of the type illustrated in FIG. 2. The equations preferably include the cost of using each of the fuels or some other process variable to be optimized. In addition, each boiler should have an equation which is a sum of each fuel multiplied by a corresponding heating value coefficient as equal to an efficiency coefficient times the steam produced by that boiler, e.g. equation (1), 
     
         a.sub.bfg *F.sub.bfg +a.sub.cog *F.sub.cog +a.sub.oil *F.sub.oil -EFF.sub.*M =0                                            (1) 
    
     where (i) is a fuel/boiler combination, a.sub.(i) is the heating value coefficient, F.sub.(i) is the fuel flow rate, EFF is the efficiency coefficient and H is enthalpy acquired by the steam. The heating value coefficient a.sub.(i) may be calculated using equation (2), 
     
         a.sub.(i) =CV.sub.(i) *ETA.sub.(i)                         (2) 
    
     where CV is the fuel heating value per fuel unit, e.g., Kcal/NM3, and ETA is the combustion efficiency. The efficiency coefficient EFF may be calculated using equation (3), 
     
         EFF=-1*ETA.sub.ref /(ETA.sub.load *EFF.sub.foul)           (3) 
    
     where ETA ref  is the combustion efficiency of a reference fuel, e.g., oil, ETA load  is the combustion efficiency of the boiler for the load assigned to the boiler and EFF foul  is a constant with a value between 0 and -1 throughout the load range of the boiler and represents efficiency degradation due to fouling. Other equations can be added to the linear programming matrix representing constraints on the process, such as the maximum load, maximum total amount of coke oven gas and blast furnace gas available, etc. 
     Initially, values are assigned 62 to the matrix elements assuming, e.g., maximum efficiency in each of the boilers. More accurate heating value and efficiency coefficients can be determined by storing a table indicating the heat produced versus load for each boiler. A graphical representation of the data which would be stored in such a table is illustrated in FIG. 4. Such data is typically available from the manufacturer of a boiler based upon design parameters and can be verified by experimentation on a boiler. 
     Once values have been assigned to the matrix elements, the matrix can be solved 64 by executing a computer program. Programs for solving such matrices are available commercially, including LP88 from Eastern Software Products, Inc. of Alexandria, Va. An example of data input to and output from such a program to find a solution for a matrix representing four boilers, each supplied with three fuels provided in the tables entitled Boiler Optimization Matrix, Starting solution, Augmented Matrix and Optimal Solution. On the in first run of the program, it would be impossible to determine whether an optimum solution has been found, i.e., whether the solution has converged 66. Therefore, the matrix elements are modified 68 according to a test strategy. 
     In the first embodiment of the present invention, the test strategy uses the SSDEVOP technique illustrated by the flowchart in FIG. 5 which corresponds to the tabulation in FIG. 1. In the case of the system illustrated in FIG. 2, the loads on three of the boilers in the solution from the linear programming matrix may be used to determine 70 the base case variables V 10 , V 20  and V 30 . If one of the boilers is not in use during the first solution of the matrix, the load on that boiler should not be used in the base case. If all boilers are in use, any boiler can be left out of the base case; its load will be equal to the difference between the total requested load and the sum of the other three boilers&#39; loads. The perturbation amounts, i.e., Δ 1 , Δ 2  and Δ 3 , may be 1%-5% of the corresponding load or may all be the same value, e.g., 1% of the largest load. The test cases #1-#4 are then generated 72 as indicated in FIG. 1 and these loads are used to determine 74 the heating value coefficients and efficiency coefficients for each fuel/boiler combination. 
     The program is then executed to solve 64 the linear programming matrix for each of the test cases. The best of the test cases can then be compared with the previous best solution to determine whether the SSDEVOP process has converged 66. Assuming non-convergence, the SSDEVOP technique requires that the mean of the three better cases for each of the loads be found and a new base case calculated 70 by doubling the mean of the better tests and subtracting the load of the worst test. The test caseloads are then generated 72 as discussed above for the initial case and the heating value and efficiency coefficients are determined 74. 
     When the best case in the new set of tests does not significantly improve on the best case of the previous set of tests, convergence will be found 66 and the controls can be adjusted 76 using the settings in the best solution. As known in the art and described in the &#39;714 and &#39;114 patents, an additional loop can be added to the method illustrated in FIG. 3 where instead of immediately adjusting controls after convergence is found, the value of the perturbations, i.e., Δ 1 , Δ 2  and Δ 3  is reduced, e.g., from 5% to 1% of the corresponding load, and the matrix elements are again modified 68. 
     In the second embodiment of the present invention, the test strategy uses results from the solution of the linear programming matrix to correct equations in the matrix. This test strategy can be used where the linear programming matrix contains equations that are linear approximations of highly nonlinear relationships. An example of an electricity/steam cogeneration system is illustrated in FIG. 6. A combustion turbogenerator 80 outputs exhaust gas 82 to a heat recovery steam generator 84. Low pressure steam 86 from the heat recovery steam generator 84 is supplied to heaters 87 and if there is a surplus, supplied along with high pressure steam 88 to a steam turbogenerator 90. Electrical supply lines 92, 94 from the combustion 80 and steam 90 turbogenerators supply 115 kV power 96 to, e.g., an electrical utility and r also may be reduced to, e.g., 34.5 kV power 98 by a transformer 100 to be supplied to an industrial user. 
     The high pressure steam 88 from the heat recovery steam generator 84 can also be combined with the high pressure steam 102 from the steam turbogenerator 90 to be supplied to the combustion turbogenerator 80 to reduce nitrous oxide (NOx) emissions, after passing through a desuperheater 104. The resulting NOx steam 106 increases the power generated by the combustion turbogenerator 80. To prevent damage to the combustion turbogenerator 80, the ratio of NOx steam flow to fuel flow must not exceed a maximum value. Steam 108 from the steam turbogenerator 90 may also be supplied as process steam 110 to an industrial user after passing through a desuperheater 112. Some of the high pressure steam 88 from the heat recovery steam generator 84 may also be supplied as process steam 110 after passing through a desuperheater 114. The remaining steam 116 from the steam turbogenerator 90 and any remaining high pressure steam 88 from the heat recovery steam generator 84 is supplied to a condenser 118. The system illustrated in FIG. 6 will be termed a cogeneration system herein, since the steam is used for more than just producing electricity, even if only to reduce NOx emissions. 
     The type of system illustrated in FIG. 6 is very difficult to model due to the large number of nonlinear and interacting (closely coupled) relationships. One of ordinary skill in the art of designing such systems can provide models for the operation of individual components, based either on design parameters or experimentation on an operating unit. As illustrated in FIG. 7 and described in the section entitled On-line Optimization Matrix and the following sections, these component models can be used to ca a coefficients of equations in the linear programming matrix which represent a first steady state model of the cogeneration system by using a solution of the linear programming model. As discussed above, the initial values of the matrix elements can be based upon assumptions of how the system is operating and the model of the system represented by the linear programming matrix will become more accurate by applying the present invention as discussed below. 
     As illustrated in FIG. 7, coefficients for a second steady state model are calculated 122 by slightly modifying the solution of the linear programming matrix used to calculate 120 the first steady state model coefficients. The rate of change between the first and second steady state models can be easily calculated from the first and second steady state model solution values. A linear approximation of the nonlinear relationships in the cogeneration system can thus be represented 124 in the linear programming matrix by assigning 84 local linearized model coefficient values to matrix elements representing effects of changes from conditions in steady state operation of the cogeneration system. 
     The linear programming matrix which results 68 (FIG. 3) from the method illustrated in FIG. 7 and described above, can be solved 64 to provide a new set of data to calculate 120, 122, 124 a more accurate linear approximation of the cogeneration system for a requested steam load. When there is no significant change in the solution of the linear programming matrix, typically within three iterations, the settings of the components represented by the solution to the linear programming matrix can be used to adjust 76 the controls of the cogeneration system. 
     The many features and advantages of the present invention are apparent from the detailed specification, and thus it is intended by the appended claims to cover all such features and advantages of the procedure which fall within the true spirit and scope of the invention. Further, since numerous modifications and changes will readily occur to those skilled in the art from the disclosure of this invention, it is not desired to limit the invention to the exact construction and operation illustrated and described, accordingly, suitable modifications and equivalents may be resorted to, all falling within the scope and spirit of the invention. 
     
         __________________________________________________________________________
BOILER OPTIMIZATION MATRIX                                                
STARTING MATRIX                                                           
__________________________________________________________________________
EREG1      OBJECTIVE: MIN                                                 
                       VARIABLES: 20                                      
                                 DATE 06-22-1989                          
BASIS: NONE                                                               
           CONSTRAINTS: 27                                                
                       SLACKS: 16                                         
                                 TIME 10:16:18                            
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    SW1  SW2  SW3  SW4  BFG1 COG1 OIL1                                    
COST                    8.0500                                            
                             41.650                                       
                                  153.40   COST                           
SW.1                                                                      
    1.0000                                 SW.1                           
SW.2     1.0000                            SW.2                           
SW.3          1.0000                       SW.3                           
SW.4               1.0000                  SW.4                           
Y.1                     1.0090                                            
                             6.0810                                       
                                  13.830   Y.1                            
Y.2                                        Y.2                            
Y.3                                        Y.3                            
Y.4                                        Y.4                            
BFG                     1.0000             BFG                            
COG                          1.000         COG                            
STM                                        STM                            
Y.8 -40.000             1.0000             Y.8                            
Y.9 -6.0000                  1.0000        Y.9                            
Y.10                                                                      
    -90.000                                Y.10                           
Y.11                                                                      
    -15.000                                Y.11                           
Y.12     -40.000                           Y.12                           
Y.13     -6.0000                           Y.13                           
Y.14     -90.000                           Y.14                           
Y.15     -15.000                           Y.15                           
Y.16          -70.000                      Y.16                           
Y.17          -11.000                      Y.17                           
Y.18          -160.00                      Y.18                           
Y.19          -20.000                      Y.19                           
Y.20               -70.000                 Y.20                           
Y.21               -11.000                 Y.21                           
Y.22               -160.00                 Y.22                           
Y.23               -20.000                 Y.23                           
    SW1  SW2  SW3  SW4  BFG1 COG1 OIL1                                    
__________________________________________________________________________
    STM1 BFG2 COG2 OIL2 STM2 BFG3 COG3                                    
COST     8.0500                                                           
              41.650                                                      
                   153.40    8.0500                                       
                                  41.650   COST                           
SW.1                                       SW.1                           
SW.2                                       SW.2                           
SW.3                                       SW.3                           
SW.4                                       SW.4                           
Y.1 -1.0000                                Y.1                            
Y.2      .99000                                                           
              5.9600                                                      
                   13.550                                                 
                        -1.0000            Y.2                            
Y.3                          .96000                                       
                                  5.7800   Y.3                            
Y.4                                        Y.4                            
BFG      1.0000              1.0000        BFG                            
COG           1.0000              1.0000   COG                            
STM 1.0000              1.0000             STM                            
Y.8                                        Y.8                            
Y.9                                        Y.9                            
Y.10                                                                      
    1.0000                                 Y.10                           
Y.11                                                                      
    1.0000                                 Y.11                           
Y.12     1.0000                            Y.12                           
Y.13          1.0000                       Y.13                           
Y.14                    1.0000             Y.14                           
Y.15                    1.0000             Y.15                           
Y.16                         1.0000        Y.16                           
Y.17                              1.0000   Y.17                           
Y.18                                       Y.18                           
Y.19                                       Y.19                           
Y.20                                       Y.20                           
Y.21                                       Y.21                           
Y.22                                       Y.22                           
Y.23                                       Y.23                           
    STM1 BFG2 COG2 OIL2 STM2 BFG3 COG3                                    
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    OIL3 STM3 BFG4 COG4 OIL4 STM4                                         
COST                                                                      
    153.40    8.0500                                                      
                   41.650                                                 
                        153.40             COST                           
SW.1                                       SW.1                           
SW.2                                       SW.2                           
SW.3                                       SW.3                           
SW.4                                       SW.4                           
Y.1                                        Y.1                            
Y.2                                        Y.2                            
Y.3 13.140                                                                
         -1.0000                           Y.3                            
Y.4           .94000                                                      
                   5.6600                                                 
                        12.860                                            
                             -1.0000       Y.4                            
BFG           1.0000                       BFG                            
COG                1.0000                  COG                            
STM      1.0000              1.0000        STM                            
Y.8                                        Y.8                            
Y.9                                        Y.9                            
Y.10                                       Y.10                           
Y.11                                       Y.11                           
Y.12                                       Y.12                           
Y.13                                       Y.13                           
Y.14                                       Y.14                           
Y.15                                       Y.15                           
Y.16                                       Y.16                           
Y.17                                       Y.17                           
Y.18     1.0000                            Y.18                           
Y.19     1.0000                            Y.19                           
Y.20          1.0000                       Y.20                           
Y.21               1.0000                  Y.21                           
Y.22                         1.0000        Y.22                           
Y.23                         1.0000        Y.23                           
    OIL3 STM3 BFG4 COG4 OIL4 STM4                                         
__________________________________________________________________________
                                  RHS                                     
COST                              .0000000 COST                           
SW.1                              = 1.000000                              
                                           SW.1                           
SW.2                              = 1.000000                              
                                           SW.2                           
SW.3                              = 1.000000                              
                                           SW.3                           
SW.4                              = 1.000000                              
                                           SW.4                           
Y.1                               = .0000000                              
                                           Y.1                            
Y.2                               = .0000000                              
                                           Y.2                            
Y.3                               = .0000000                              
                                           Y.3                            
Y.4                               = .0000000                              
                                           Y.4                            
BFG                               = 150.0000                              
                                           BFG                            
COG                               = 15.00000                              
                                           COG                            
STM                               = 435.0000                              
                                           STM                            
Y.8                               &lt;= .0000000                             
                                           Y.8                            
Y.9                               &lt;= .0000000                             
                                           Y.9                            
Y.10                              &lt;= .0000000                             
                                           Y.10                           
Y.11                              &gt;= .0000000                             
                                           Y.11                           
Y.12                              &lt;= .0000000                             
                                           Y.12                           
Y.13                              &lt;= .0000000                             
                                           Y.13                           
Y.14                              &lt;= .0000000                             
                                           Y.14                           
Y.15                              &gt;= .0000000                             
                                           Y.15                           
Y.16                              &lt;= .0000000                             
                                           Y.16                           
Y.17                              &lt;= .0000000                             
                                           Y.17                           
Y.18                              &lt;= .0000000                             
                                           Y.18                           
Y.19                              &gt;= .0000000                             
                                           Y.19                           
Y.20                              &lt;= .0000000                             
                                           Y.20                           
Y.21                              &lt;= .0000000                             
                                           Y.21                           
Y.22                              &lt;= .0000000                             
                                           Y.22                           
Y.23                              &gt;= .0000000                             
                                           Y.23                           
                                  RHS                                     
__________________________________________________________________________
 
    
     
         __________________________________________________________________________
BOILER OPTIMIZATION MATRIX                                                
STARTING MATRIX for SSDEVOP                                               
__________________________________________________________________________
EREG11     OBJECTIVE: MIN                                                 
                       VARIABLES: 20                                      
                                 DATE 06-22-1989                          
BASIS: NONE                                                               
           CONSTRAINTS: 30                                                
                       SLACKS: 16                                         
                                 TIME 10:23:16                            
__________________________________________________________________________
    SW1  SW2  SW3  SW4  BFG1 COG1 OIL1                                    
COST                    8.0500                                            
                             41.650                                       
                                  153.40   COST                           
SW.1                                                                      
    1.0000                                 SW.1                           
SW.2     1.0000                            SW.2                           
SW.3          1.0000                       SW.3                           
SW.4               1.0000                  SW.4                           
Y.1                     1.0090                                            
                             6.0810                                       
                                  13.830   Y.1                            
Y.2                                        Y.2                            
Y.3                                        Y.3                            
Y.4                                        Y.4                            
BFG                     1.0000             BFG                            
COG                          1.0000        COG                            
STM                                        STM                            
Y.8 -40.000             1.0000             Y.8                            
Y.9 -6.0000                  1.0000        Y.9                            
Y.10                                                                      
    -90.000                                Y.10                           
Y.11                                                                      
    -15.000                                Y.11                           
Y.12     -40.000                           Y.12                           
Y.13     -6.0000                           Y.13                           
Y.14     -90.000                           Y.14                           
Y.15     -15.000                           Y.15                           
Y.16          -70.000                      Y.16                           
Y.17          -11.000                      Y.17                           
Y.18          -160.00                      Y.18                           
Y.19          -20.000                      Y.19                           
Y.20               -70.000                 Y.20                           
Y.21               -11.000                 Y.21                           
Y.22               -160.00                 Y.22                           
Y.23               -20.000                 Y.23                           
Y.24                                                                      
    AUGMENTED MATRIX                     Y.24                             
Y.25                                     Y.25                             
Y.26                                     Y.26                             
    SW1  SW2  SW3  SW4  BFG1 COG1 OILI                                    
__________________________________________________________________________
    STM1 BFG2 COG2 OIL2 STM2 BFG3 COG3                                    
COST     8.0500                                                           
              41.650                                                      
                   153.40    8.0500                                       
                                  41.650   COST                           
SW.1                                       SW.1                           
SW.2                                       SW.2                           
SW.3                                       SW.3                           
SW.4                                       SW.4                           
Y.1 -1.0300                                Y.1                            
Y.2      .99000                                                           
              5.9600                                                      
                   13.550                                                 
                        -1.0300            Y.2                            
Y.3                          .96000                                       
                                  5.7800   Y.3                            
Y.4                                        Y.4                            
BFG      1.0000              1.0000        BFG                            
COG           1.0000              1.0000   COG                            
STM 1.0000              1.0000             STM                            
Y.8                                        Y.8                            
Y.9                                        Y.9                            
Y.10                                                                      
    1.0000                                 Y.10                           
Y.11                                                                      
    1.0000                                 Y.11                           
Y.12     1.0000                            Y.12                           
Y.13          1.0000                       Y.13                           
Y.14                    1.0000             Y.14                           
Y.15                    1.0000             Y.15                           
Y.16                         1.0000        Y.16                           
Y.17                              1.0000   Y.17                           
Y.18                                       Y.18                           
Y.19                                       Y.19                           
Y.20                                       Y.20                           
Y.21                                       Y.21                           
Y.22                                       Y.22                           
Y.23                                       Y.23                           
Y.24                                                                      
    1.0000                                 Y.24                           
Y.25                    1.0000             Y.25                           
Y.26                                       Y.26                           
    STM1 BFG2 COG2 OIL2 STM2 BFG3 COG3                                    
__________________________________________________________________________
    OIL3 STM3 BFG4 COG4 OIL4 STM4                                         
COST                                                                      
    153.40    8.0500                                                      
                   41.650                                                 
                        153.40             COST                           
SW.1                                       SW.1                           
SW.2                                       SW.2                           
SW.3                                       SW.3                           
SW.4                                       SW.4                           
Y.1                                        Y.1                            
Y.2                                        Y.2                            
Y.3 13.140                                                                
         -1.0800                           Y.3                            
Y.4           .94000                                                      
                   5.6600                                                 
                        12.860                                            
                             -1.0100       Y.4                            
BFG           1.0000                       BFG                            
COG                1.0000                  COG                            
STM      1.0000              1.0000        STM                            
Y.8                                        Y.8                            
Y.9                                        Y.9                            
Y.10                                       Y.10                           
Y.11                                       Y.11                           
Y.12                                       Y.12                           
Y.13                                       Y.13                           
Y.14                                       Y.14                           
Y.15                                       Y.15                           
Y.16                                       Y.16                           
Y.17                                       Y.17                           
Y.18     1.0000                            Y.18                           
Y.19     1.0000                            Y.19                           
Y.20          1.0000                       Y.20                           
Y.21               1.0000                  Y.21                           
Y.22                         1.0000        Y.22                           
Y.23                         1.0000        Y.23                           
Y.24                                       Y.24                           
Y.25                                       Y.25                           
Y.26     1.0000                            Y.26                           
    OIL3 STM3 BFG4 COG4 OIL4 STM4                                         
__________________________________________________________________________
                                  RHS                                     
COST                               .0000000                               
                                           COST                           
SW.1                              = 1.000000                              
                                           SW.1                           
SW.2                              = 1.000000                              
                                           SW.2                           
SW.3                              = 1.000000                              
                                           SW.3                           
SW.4                              = 1.000000                              
                                           SW.4                           
Y.1                               = .0000000                              
                                           Y.1                            
Y.2                               = .0000000                              
                                           Y.2                            
Y.3                               = .0000000                              
                                           Y.3                            
Y.4                               = .0000000                              
                                           Y.4                            
BFG                               = 150.0000                              
                                           BFG                            
COG                               = 15.00000                              
                                           COG                            
STM                               = 435.0000                              
                                           STM                            
Y.8                               &lt;= .0000000                             
                                           Y.8                            
Y.9                               &lt;= .0000000                             
                                           Y.9                            
Y.10                              &lt;= .0000000                             
                                           Y.10                           
Y.11                              &gt;= .0000000                             
                                           Y.11                           
Y.12                              &lt;= .0000000                             
                                           Y.12                           
Y.13                              &lt;= .0000000                             
                                           Y.13                           
Y.14                              &lt;= .0000000                             
                                           Y.14                           
Y.15                              &gt;= .0000000                             
                                           Y.15                           
Y.16                              &lt;= .0000000                             
                                           Y.16                           
Y.17                              &lt;= .0000000                             
                                           Y.17                           
Y.18                              &lt;= .0000000                             
                                           Y.18                           
Y.19                              &gt;= .0000000                             
                                           Y.19                           
Y.20                              &lt;= .0000000                             
                                           Y.20                           
Y.21                              &lt;= .0000000                             
                                           Y.21                           
Y.22                              &lt;= .0000000                             
                                           Y.22                           
Y.23                              &gt;= .0000000                             
                                           Y.23                           
Y.24                              = 85.00000                              
                                           Y.24                           
Y.25                              = 85.00000                              
                                           Y.25                           
Y.26                              = 155.0000                              
                                           Y.26                           
                                  RHS                                     
__________________________________________________________________________
 
    
     
         __________________________________________________________________________
STARTING SOLUTION                                                         
MINIMUM              ENTERS: COG3  BASIS X: 16 VARIABLES: 20              
PIVOTS: 28           LEAVES: COG2  BASIS S: 11 SLACKS: 16                 
LAST INV: 0            DELTA -.0039632          COST 4156.618             
CONSTRAINTS: 27                                                           
__________________________________________________________________________
BASIS STM1  STM2   STM3  STM4  S.25   BFG2  BFG3  S.12   BFG4             
COG4  S.27  S.15   S.13  OIL1  SW1    S.19  COG3  OIL2   SW2              
S.23  S.21  DIL3   SW3   S.24  S.17   S.26  SW4                           
PRIMAL                                                                    
      90.000                                                              
            90.000 160.00                                                 
                         95.000                                           
                               .85866 40.000                              
                                            70.000                        
                                                  40.000 40.000           
10.141                                                                    
      75.000                                                              
            75.000 6.0000                                                 
                         6.5076                                           
                               1.0000 75.000                              
                                            4.8587                        
                                                  3.7196 1.0000           
140.00                                                                    
      6.1413                                                              
            4.9252 1.0000                                                 
                         30.000                                           
                               6.0000 65.000                              
                                            1.0000                        
DUAL  -74.696                                                             
            -54.122                                                       
                   -39.660                                                
                         .00000                                           
                               11.092 11.321                              
                                            11.674                        
                                                  11.922 -3.1565          
.827  11.922                                                              
            .00000 .00000                                                 
                         -.82996                                          
                               .00000 -.00134                             
                                            .00000                        
                                                  -.60075                 
                                                         .00000           
.00083                                                                    
      .00000                                                              
            -.24751                                                       
                   .00000                                                 
                         .00000                                           
                               .00000 .00000                              
                                            .00000                        
__________________________________________________________________________
AUGMENTED MATRIX INITIAL SOLUTION                                         
MINIMUM              ENTERS: COG3  BASIS X: 16 VARIABLES: 20              
PIVOTS: 25           LEAVES: COG2  BASIS S: 14 SLACKS: 16                 
LAST INV: 0            DELTA -.0039632                                    
                                    COST 4380.047                         
                                                CONSTRAINTS:              
__________________________________________________________________________
                                                30                        
BASIS S.14  S.18   S.22  STM4  OIL1   OIL2  OIL3  S.20   BFG4             
COG4  S.27  S.12   S.13  STM1  S.15   BFG2  COG3  STM2   S.17             
BFG3  S.21  STM3   S.23  S.24  S.17   S.26  SW4   SW1    SW2              
SW3                                                                       
PRIMAL                                                                    
      5.0000                                                              
            5.0000 5.0000                                                 
                         110.00                                           
                               6.3304 3.5387                              
                                            6.7397                        
                                                  11.957 51.95            
11.000                                                                    
      90.000                                                              
            40.000 6.0000                                                 
                         85.000                                           
                               70.000 40.000                              
                                            4.0000                        
                                                  85.000 70.00            
58.043                                                                    
      7.0000                                                              
            155.00 135.00                                                 
                         18.043                                           
                               6.0000 50.000                              
                                            1.0000                        
                                                  1.0000 1.0000           
DUAL  .0000 -.02068                                                       
                   .00000                                                 
                         -.05465                                          
                               11.092 11.321                              
                                            11.674                        
                                                  11.923 -3.157           
1.0000                                                                    
-25.827                                                                   
      12.042                                                              
            .00000 .00000                                                 
                         .00000                                           
                               .00000 -.00052                             
                                            .00000                        
                                                  .00000 .0000            
.00000                                                                    
      .00000                                                              
            .00000 .00000                                                 
                         .00000                                           
                               -.00497                                    
                                      .00000                              
                                            .00000                        
                                                  -.61731                 
                                                         -.3812           
.56633                                                                    
__________________________________________________________________________
OPTIMAL SOLUTION                                                          
MINIMUM              ENTERS: COG3  BASIS X: 17 VARIABLES: 20              
PIVOTS: 25           LEAVES: COG2  BASIS S: 13 SLACKS: 16                 
LAST INV: 0            DELTA -.0039632                                    
                                    COST 4255.048                         
                                                CONSTRAINTS:              
__________________________________________________________________________
                                                30                        
BASIS S.14  S.18   S.22  STM4  OIL1   OIL2  OIL3  OIL4   BFG4             
COG4  S.27  S.12   S.13  STM1  S.15   BFG2  COG3  STM2   S.19             
BFG3  S.21  STM3   S.23  S.20  S.17   S.26  SW4   SW1    SW2              
SW3                                                                       
PRIMAL                                                                    
      .00000                                                              
            .00000 31.000                                                 
                         126.00                                           
                               6.5076 3.9188                              
                                            5.3318                        
                                                  .03577 70.00            
11.000                                                                    
      106.00                                                              
            40.000 6.000 90.000                                           
                               75.000 40.000                              
                                            4.0000                        
                                                  90.000 75.00            
40.000                                                                    
      7.00000                                                             
            129.00 109.00                                                 
                         30.000                                           
                               6.0000 34.000                              
                                            1.0000                        
                                                  1.0000 1.000            
1.0000                                                                    
DUAL  .00000                                                              
            -.02068                                                       
                   .00000                                                 
                         -.79668                                          
                               11.092 11.321                              
                                            11.674                        
                                                  11.928 -3.157           
-25.827                                                                   
      12.167                                                              
            .00000 .00000                                                 
                         .00000                                           
                               .00000 -.00052                             
                                            .00000                        
                                                  .00000 .00000           
.00000                                                                    
      .00000                                                              
            .00000 .00000                                                 
                         -.00545                                          
                               -.03776                                    
                                      .00000                              
                                            .00000                        
                                                  -1.0752                 
                                                         -.5063           
-.2 927                                                                   
__________________________________________________________________________
 
    
     ON-LINE OPTIMIZATION MATRIX 
     An examination of the available data indicates the essential linearity of equipment behavior, at least over restricted load ranges, and thus suggests that the linear programming technique for the optimization of closely-coupled systems would be appropriate and practical in this case. The flowsheet of FIG. 6 is the network to be implemented in the LP matrix, the matrix consisting of: 
     a. Variables 1 through 24 forming the columns of the matrix 
     b. Energy and Materials Balances, which form a large number of the rows of the matrix 
     c. The remaining rows, consisting of fixed or adjustable upper or lower constraints or bounds to appropriate variables. 
     d. The unit cost or revenue obtained from those resources (variables) which are either purchased or sold. 
     List of Variables 
     The variables shown in the columns of the matrix listed in Table 5.0, are defined below: 
     
         ______________________________________                                    
I.D. Name      Description          Units                                 
______________________________________                                    
 1.  GTGGAS    GTG - Fuel flow      MBTU/h                                
 2.  GTGPWR    GTG - Power due to fuel                                    
                                    MW                                    
 3.  NOXPWR    GTG - Incremental power from                               
                                    MW                                    
               Nox steam                                                  
 4.  TOTPWR    GTG - Total power generated                                
                                    MW                                    
 5.  NOXGT2    GTG - Nox due to Fuel                                      
                                    lb/h                                  
 6.  NINCG2    GTG - Nox increment due to steam                           
                                    lb/h                                  
 7.  TOTNX2    GTG - Total nox from GT2                                   
                                    lb/h                                  
 8.  PM10G2    PM.sub.10 from GTG2 fuel                                   
                                    lb/h                                  
 9.  HPSTM     HRSG - H.P. Steam generated                                
                                    Klb/h                                 
10.  LPSTM     HRSG - L.P. Steam generated                                
                                    Klb/h                                 
11.  THR       STG - Throttle flow  Klb/h                                 
12.  NOXEX     STG - Nox steam extraction flow                            
                                    Klb/h                                 
13.  PREX      STG - Process Steam extraction                             
                                    Klb/h                                 
               flow                                                       
14.  INJ       STG - Injection steam flow                                 
                                    Klb/h                                 
15.  COND      STG - Flow to condenser                                    
                                    Klb/h                                 
16.  STGPWR    STG - Generated power                                      
                                    MW                                    
17.  HPPRV     HP steam/NOX PRV flow                                      
                                    Klb/h                                 
18.  DUMPV     Steam dump valve flow to                                   
                                    Klb/h                                 
               condenser                                                  
19.  PRPRV     HP steam/process steam PRV flow                            
                                    Klb/h                                 
20.  TOTNX     Total Nox steam flow Klb/h                                 
21.  HTRST     Steam flow to heaters                                      
                                    Klb/h                                 
22.  SCE2      Power exported to switchyard                               
                                    MW                                    
23.  USBTIE    Power sold           MW                                    
24.  PRSTM     Process steam sold   Klb/h                                 
______________________________________                                    
 
    
     Costs and Prices 
     As shown in the first row of the matrix in Table 5.0, the following variables need to be assigned costs or prices: 
     
         ______________________________________                                    
GTGPWR      4.0            $/MMBTU                                        
SCE2        -50.00         $/MWh                                          
PRSTM       -3.5           $/Klb/h                                        
______________________________________                                    
 
    
     The cost associated with GTGPWR is the fuel cost in $/MMBTU. This will be inserted by the matrix preparation program and will represent the actual cost of whatever fuel is being fired in the gas turbogenerator. 
     The cost associated with SCE2 is actually a price and therefore carries a negative sign. It is the effective current revenue received per MWh of electricity exposed via the tie-line SEC2. It will be calculated by the schedule program every hour and stored in a reserved location, from which it will be recovered by the matrix preparation program. 
     Similarly, the cost of PRSTM also carriers a negative sign and will be calculated by the schedule program and stored in its own reserved location, from which it will be recovered by the matrix preparation program. 
     OPTIMIZATION MATRIX EQUATIONS 
     The equations included in the Cogeneration Plant optimization matrix are defined in Table 5.0 and described in the following: 
     Y.1 Gas Turbogenerator #2 Characteristic Equation 
     Westinghouse Curve No. CW251-91 shows an essentially linear relationship between fuel and power which varies as a function of temperature. For an ambient temperature of 59 deg.F., the relation ship is: 
     
         GTGGAS≦138.27+7.591* GTGPWR                         (Y.1) 
    
     In practice, the matrix preparation program will contain a model relation generated power to fuel flow as a function of ambient temperature. It will calculate the values of the coefficients needed by Y.1 from this model and load them into the appropriate elements of the matrix. 
     Y.2 Incremental Power due to Nox Steam 
     Westinghouse Curve No. CW251-95 shows an essentially linear relationship between incremental power due to Nox steam and the quantity of Nox steam injected. For an ambient temperature of 59 deg.F the relationship is: 
     
         NOXPWR=0.1455*TOTNX                                        (Y.2) 
    
     Again, in practice, the matrix preparation program will contain a model relating increase in generated power to N ox steam flow as a function of ambient temperature. It will calculate the values of the coefficient needed by Y.2 from this model and load it into the appropriate element of the matrix. 
     Y.3 Maximum Nox Steam Flow 
     As will be seen from Westinghouse Curve No. CW251-95, there is an upper limit to the amount of Nox steam which can be supplied, this amount varying with the ambient temperature. 
     In practice, the matrix preparation program will contain a model which relates the maximum Nox steam flow to ambient temperature. It will calculate the upper Nox flow limit from this model and load it into the appropriate RHS element of the matrix. 
     Y.4 Maximum Nox Steam/fuel Flow Heat Ratio 
     A model relating the maximum allowable Nox steam/Fuel Heat Flow ratio will exist in the matrix preparation program which will compute the ratio and load it into the appropriate element of the matrix. This model has yet to be defined. 
     Y.5 Total Power Generated by GTG2 
     The power generated by GTG2 is contributed mainly by the fuel fired and secondarily by the incremental power due to Nox steam. Thus total power: 
     
         TOTPWR=GTGPWR+NOXPWR                                       (Y.5) 
    
     Y.6 Maximum Generated Power 
     The maximum power which can be generated by GTG2 is 42.5MW, but this will vary with the power factor and reactive capability curve for the generator. The power limit will be calculated by the matrix preparation program from the reactive capability curve and the result stored in the RHS element of equation Y.6. 
     The reactive capability curves are contained in Data Sheets H.E.P. 4959A and H.E.P. 4960 dated Aug. 20, 1986 and issued by Brush Electrical Machines Ltd. 
     Y.7 Minimum Generated Power 
     There is a minimum allowable amount of power which can be generated once the machine is synchronized to the system and this is at present assumed to be 2MW. 
     Y.8 GTG2 Nox Emission 
     Data contained on page 194 of Appendix X of The California Energy Commission Staff Report contains the PM 10  emission rate for the project when firing gas as 0.0289 lb/MMBTU. Thus the relationship is: 
     
         NOXTG2=0.0289*GTGGAS                                       (Y.8) 
    
     Y.9 Nox Decrease due to Nox Steam 
     Data from a Westinghouse curve dated Feb. 2, 1988 (Mumford), showing the relationship between Nox reduction and Nox steam flow, has been used to determine the following linear relationship: 
     
         NINCG2=A.sub.1 *TOTNX                                      (Y.9) 
    
     The value of A 1  will be calculated from an analysis of the curve referred to above. 
     Y.10 Total Nox emission 
     The total Nox emission from GTG2 is given by: 
     
         TOTNX2=NOXGT2-NINCG2                                       (Y.10) 
    
     Y.11 Particulate Emission--GTG2 
     From page 194 of the CEC Report, the PM 10  emission is given as 0.0159 lb/MMBTU. The equation relation PM 10  emission to fuel firing rate is therefore: 
     
         PM10G2=0.0159*GTGGAS                                       (Y.11) 
    
     Y.12 HRSG H.P. Stem Flow vs. Generated Power 
     The relationship between the flow of the H.P. steam generated in the HRSG as a function of the power generated by GTG2 is derived from data in the Nooter-Eriksen data sheet dated Feb. 7, 1989. The data with gas at 73 deg.F. has been plotted and an approximation to the curve at that temperature is given by: 
     
         HPSTM=95+1.388*GRGPWR                                      (Y.12) 
    
     This model will need to be developed further 
     Y.12 HRSG l.P. Steam Flow vs. Generated Power 
     The relationship between the flow of L.P. steam generated in the HRSG as a function of the power generated by GTG2 is derived from data in the Nooter-Eriksen data sheet dated Feb. 7, 1989. The data with gas at 73 deg.F. has been plotted and an approximation to the curve at that temperature is given by: 
     
         LPSTM=0.576*GTGPWR                                         (Y.12) 
    
     This model will need to be developed further 
     Y.14 Steam consumed by Deaerator, etc. 
     Steam consumed as a function of HRSG H.P. and L.P. steam flows is approximately 0.054 lb/lb HRSG steam. Thus: 
     
         HTRST=0.054*HPSTM+0.054*LPSTEM                             (Y.14) 
    
     The values of these two coefficients will need to be further defined and an appropriate model generated if necessary. 
     Y.15 Steam Turbogenerator Throttle Flow Equation 
     From the heat balances given in Appendix A of the Westinghouse Contract for the Project, steam flow and power data were extracted for presentation to a regression analysis program. The coefficients in the following model give a 0.5% worst error in predicted throttle flow: 
     
         THR=8.5378+1.893*NOXEX+0.707*PREX-1.077*INJ+7.598*STGPWR   (Y.15) 
    
     Y.16 STG Steam Balance 
     The sum of the throttle flow plus injection steam flow must equal the sum of Nox and Process Steam extraction flows and the flow to the condenser. Thus: 
     
         THR+INJ=NOXEX+PREX+COND                                    (Y.16) 
    
     Y.17 Maximum Throttle Flow 
     The maximum throttle flow is assumed to be 170 Klb/h 
     Y.18 Maximum Nox Steam Flow 
     The maximum Nox steam flow is assumed to be 37.5 Klb/n 
     Y.19 Maximum Process Extraction Steam Flow 
     The maximum Process Steam Extraction flow is assumed to be 109 Klb/h. 
     Y.20 Minimum Condenser Flow 
     The flow to the condenser is normally the exhaust from the steam turbogenerator but this may be augmented or replaced by live H.P. steam dumped to the condenser. Thus: 
     
         (COND+DUMPV)≧10Klb/h                                (Y.20) 
    
     Y.21 Maximum Condenser Flow 
     The maximum condenser flow may be defined as: 
     
         (COND+DUMPV)≦152 Klb/h                              (Y.21) 
    
     Y.22 Maximum STG Generated Power 
     The maximum allowable power which can be generated by the steam turbogenerator is subject to the power factor and the reactive capability curve for the generator. 
     Y.23 Minimum Allowable STG Power 
     There is a minimum allowable amount of power which can be generated once this machine is synchronized to the system and this is at present assumed to be 2MW. 
     Y.24 Process Steam Header Flow Balance 
     Prior to the process steam leaving the plant, steam can enter the process steam header from the process steam extraction port on the STG and via the pressure reducing valve connecting this header to the H.P. Steam supply. It is assumed that 15% water is added at the desuperheater provided for each of these process steam sources. The balance in the process steam header may thus be defined as: 
     
         PRSTM=1.15*PREX+1.15*PRPRV                                 (Y.24) 
    
     The desuperheating water addition flow ratios given in Y.24 will need to be calculated with greater precision. 
     Y.25 L.P. Steam Header Balance 
     The L.P. steam generated by the HRSG is used to supply the deaerator steam, the balance being conducted to the steam injection port of the STG. Thus: 
     
         LPSTM=INJ+HTRST                                            (Y.25) 
    
     Y.26 H.P. Steam Header Balance 
     The H.P. steam leaving the HRSG supplies: 
     Steam turbogenerator throttle 
     Dump valve 
     PRV between H.P. header and Nox steam header 
     PRV between H.P. header and Process Steam header 
     The H.P. steam header balance may therefore be defined as: 
     
         HPSTEM=THR+HPPRV+DUMPV+PRPRV                               (Y.26) 
    
     Y.27 Nox Steam Header Balance 
     The Nox steam header receives steam from the Nox steam extraction port of the STG as well as from the bypass pressure reducing valve. Thus the header balance may be defined as: 
     
         TOTNX=1.109*NOXEX+1.17*HPPRV                               (Y.27) 
    
     where a desuperheating water addition rate of 17% is assumed. 
     Y.28 Power balance 
     The sum of the generated powers must equal the power dispatched 
     
         TOTPWR+STGPWER=SCER2+USBTIE                                (Y.28) 
    
     Y.29 Power to be Despatched to 
     The power to be dispatched to electrical utility will be calculated in a DPU as a function of the time of day and period of the year, these data being identified in the VAX schedule program and passed to the DPU over the data highway. The power to be dispatched to SCE (MWMAX) will be received from the DPU over the data highway, so that: 
     
         SCE2&lt;=MWMAX                                                (Y.29) 
    
     Y.30 Power to be Despatched to Industry 
     The power to be despatched to industry will normally be zero but at times may rise to a limited value when there own generating equipment is down, in order to reduce the maximum demand limit. Thus this relationship is defined as an inequality of the form: 
     
         ______________________________________                                    
IF (SCE/USB TIE OPEN)                                                     
       THEN CONST=ZERO                                                    
         ELSE CONST=2000 (Large number)                                   
END IF                                                                    
USBTIE &lt;= CONST                                                           
______________________________________                                    
 
    
     Y.31 Process Steam to be Supplied to Industry 
     This equality determines the quantity of steam to be supplied to industry and would be loaded by the matrix preparation program in accordance with the amount presently being consumed 
     
         __________________________________________________________________________
Optimization Matrix                                                       
MOJAVE       OBJECTIVE: MIN                                               
                        VARIABLES: 24                                     
                                 DATE 12-14-1989                          
BASIS: MOJAVE                                                             
             CONSTRAINTS: 30                                              
                        SLACKS: 12                                        
                                 TIME 16:09:57                            
__________________________________________________________________________
    GTGGAS                                                                
          GTGPWR                                                          
                NOXPWR                                                    
                      TOTPWR                                              
                            NOXGT2                                        
                                  NINCG2                                  
                                       TOTNX2                             
COST                                                                      
    4.0000                                      COST                      
Y.1 1.0000                                                                
          -7.5910                               Y.1                       
Y.2             1.0000                          Y.2                       
Y.3                                             Y.3                       
Y.4 .04880                                      Y.4                       
Y.5       1.0000                                                          
                1.0000                                                    
                      -1.0000                   Y.5                       
Y.6                   1.0000                    Y.6                       
Y.7                   1.0000                    Y.7                       
Y.8 .02890                  -1.0000             Y.8                       
Y.9                               -1.0000       Y.9                       
Y.10                        1.0000                                        
                                  -1.0000                                 
                                       -1.0000  Y.10                      
Y.11                                                                      
    .01590                                      Y.11                      
Y.12      -1.3880                               Y.12                      
Y.13      -.57600                               Y.13                      
Y.14                                            Y.14                      
Y.15                                            Y.15                      
Y.16                                            Y.16                      
Y.17                                            Y.17                      
Y.18                                            Y.18                      
Y.19                                            Y.19                      
Y.20                                            Y.20                      
Y.21                                            Y.21                      
Y.22                                            Y.22                      
Y.23                                            Y.23                      
Y.24                                            Y.24                      
Y.25                                            Y.25                      
Y.26                                            Y.26                      
Y.27                                            Y.27                      
Y.28                  1.0000                    Y.28                      
Y.29                                            Y.29                      
Y.30                                            Y.30                      
Y.31                                            Y.31                      
    GTGGAS                                                                
          GTGPWR                                                          
                NOXPWR                                                    
                      TOTPWR                                              
                            NOXGT2                                        
                                  NINCG2                                  
                                       TOTNX2                             
__________________________________________________________________________
    PM10G2                                                                
          HPSTM LPSTM THR   NOXEX PREX INJ                                
COST                                            COST                      
Y.1                                             Y.1                       
Y.2                                             Y.2                       
Y.3                                             Y.3                       
Y.4                                             Y.4                       
Y.5                                             Y.5                       
Y.6                                             Y.6                       
Y.7                                             Y.7                       
Y.8                                             Y.8                       
Y.9                                             Y.9                       
Y.10                                            Y.10                      
Y.11                                                                      
    -1.0000                                     Y.11                      
Y.12      1.0000                                Y.12                      
Y.13            1.0000                          Y.13                      
Y.14      .05400                                                          
                .05400                          Y.14                      
Y.15                  1.0000                                              
                            -1.8930                                       
                                  -.70700                                 
                                       1.0770   Y.15                      
Y.16                  1.0000                                              
                            -1.0000                                       
                                  -1.0000                                 
                                       1.0000   Y.16                      
Y.17                  1.0000                    Y.17                      
Y.18                        1.0000              Y.18                      
Y.19                              1.0000        Y.19                      
Y.20                                            Y.20                      
Y.21                                            Y.21                      
Y.22                                            Y.22                      
Y.23                                            Y.23                      
Y.24                              1.1500        Y.24                      
Y.25            1.0000                 -1.0000  Y.25                      
Y.26      1.0000      -1.0000                   Y.26                      
Y.27                        1.1090              Y.27                      
Y.28                                            Y.28                      
Y.29                                            Y.29                      
Y.30                                            Y.30                      
Y.31                                            Y.31                      
    PM10G2                                                                
          HPSTM LPSTM THR   NOXEX PREX INJ                                
__________________________________________________________________________
    COND  STGPWR                                                          
                HPPRV DUMPV PRPRV TOTNX                                   
                                       HTRST                              
COST                                            COST                      
Y.1                                             Y.1                       
Y.2                               -.14550       Y.2                       
Y.3                               1.0000        Y.3                       
Y.4                               -1.0000       Y.4                       
Y.5                                             Y.5                       
Y.6                                             Y.6                       
Y.7                                             Y.7                       
Y.8                                             Y.8                       
Y.9                               .14440        Y.9                       
Y.10                                            Y.10                      
Y.11                                            Y.11                      
Y.12                                            Y.12                      
Y.13                                            Y.13                      
Y.14                                   -1.0000  Y.14                      
Y.15      -7.5980                               Y.15                      
Y.16                                                                      
    -1.0000                                     Y.16                      
Y.17                                            Y.17                      
Y.18                                            Y.18                      
Y.19                                            Y.19                      
Y.20                                                                      
    1.0000            1.0000                    Y.20                      
Y.21                                                                      
    1.0000            1.0000                    Y.21                      
Y.22      1.0000                                Y.22                      
Y.23      1.0000                                Y.23                      
Y.24                        1.1500              Y.24                      
Y.25                                   -1.0000  Y.25                      
Y.26            -1.0000                                                   
                      -1.0000                                             
                            -1.0000             Y.26                      
Y.27            1.1770            -1.0000       Y.27                      
Y.28      1.0000                                Y.28                      
Y.29                                            Y.29                      
Y.30                                            Y.30                      
Y.31                                            Y.31                      
    COND  STGPWR                                                          
                HPPRV DUMPV PRPRV TOTNX                                   
                                       HTRST                              
__________________________________________________________________________
    SCE2  USBTIE                                                          
                PRSTM                  RHS                                
COST                                                                      
    -50.000     -3.5000                .0000000 COST                      
Y.1                                    = 138.2700                         
                                                Y.1                       
Y.2                                    = .O000000                         
                                                Y.2                       
Y.3                                    &lt;= 50.40000                        
                                                Y.3                       
Y.4                                    &gt;= .0000000                        
                                                Y.4                       
Y.5                                    = .0000000                         
                                                Y.5                       
Y.6                                    &lt;= 42.50000                        
                                                Y.6                       
Y.7                                    &gt;= 2.000000                        
                                                Y.7                       
Y.8                                    = .0000000                         
                                                Y.8                       
Y.9                                    = .0000000                         
                                                Y.9                       
Y.10                                   = .0000000                         
                                                Y.10                      
Y.11                                   = .0000000                         
                                                Y.11                      
Y.12                                   = 95.00000                         
                                                Y.12                      
Y.13                                   = .0000000                         
                                                Y.13                      
Y.14                                   =  .0000000                        
                                                Y.14                      
Y.15                                   = 8.537800                         
                                                Y.15                      
Y.16                                   = .0000000                         
                                                Y.16                      
Y.17                                   &lt;= 170.0000                        
                                                Y.17                      
Y.18                                   &lt;= 37.50000                        
                                                Y.18                      
Y.19                                   &lt;= 109.0000                        
                                                Y.19                      
Y.20                                   &gt;= 10.00000                        
                                                Y.20                      
Y.21                                   &lt;= 152.0000                        
                                                Y.21                      
Y.22                                   &lt;= 14.90000                        
                                                Y.22                      
Y.23                                   &gt;= 2.000000                        
                                                Y.23                      
Y.24      -1.0000                      = .0000000                         
                                                Y.24                      
Y.25                                   = .0000000                         
                                                Y.25                      
Y.26                                   = .0000000                         
                                                Y.26                      
Y.27                                   = .00000000                        
                                                Y.27                      
Y.28                                                                      
    -1.0000                                                               
          -1.0000                      = .0000000                         
                                                Y.28                      
Y.29                                                                      
    1.0000                             = 44.00000                         
                                                Y.29                      
Y.30            1.0000                 &lt;= .0000000                        
                                                Y.30                      
Y.31            1.0000                 = 130.00000                        
                                                Y.31                      
    SCE2  USBTIE                                                          
                PRSTM                  RHS                                
__________________________________________________________________________
 
    
     
         __________________________________________________________________________
44 mw AND 130 k1B/H                                                       
       SOLUTION IS MINIMUM                                                
                         COST -1051.761436                                
       PRIMAL PROBLEM SOLUTION                                            
VARIABLE                                                                  
       STATUS VALUE    COST/UNIT                                          
                                VALUE/UNIT                                
                                         NET COST                         
__________________________________________________________________________
GTGGAS BASIS  400.80964                                                   
                       4.00000000                                         
                                4.0000000                                 
                                         .00000000                        
GTGPWR BASIS  34.585646                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
NOXPWR BASIS  2.8459088                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
TOTPWR BASIS  37.431555                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
NOXGT2 BASIS  12.264775                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
NINCG2 BASIS  2.8243933                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
TOTNX2 BASIS  15.089168                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
PM10G2 BASIS  6753.6425                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
HPSTM  BASIS  143.00488                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
LPSTM  BASIS  19.921332                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
THR    BASIS  126.38677                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
NOXEX  NONBASIS                                                           
              .00000000                                                   
                       .00000000                                          
                                -3.0113211                                
                                         3.01132                          
PREX   BASIS  113.04348                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
INJ    BASIS  11.123317                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
COND   BASIS  24.46610 .00000000                                          
                                .00000000                                 
                                         .00000000                        
STGPWR BASIS  6.5684449                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
HPPRV  BASIS  16.618106                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
DUMPV  NONBASIS                                                           
              .00000000                                                   
                       .00000000                                          
                                -3.1672313                                
                                         3.16723                          
PRPRV  NONBASIS                                                           
              .00000000                                                   
                       .00000000                                          
                                -.92799876                                
                                         .927998                          
TOTNX  BASIS  19.559510                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
HTRST  BASIS  8.7980153                                                   
                       -50.000000                                         
                                -50.00000                                 
                                         .00000000                        
SCE2   BASIS  44.000000                                                   
                       -50.000000                                         
                                -50.00000                                 
                                         .0000000                         
USBTIE NONBASIS                                                           
              .00000000                                                   
                       .00000000                                          
                                -24.064623                                
                                         24.0646                          
PRSTM  BASIS  130.00000                                                   
                       -3.5000000                                         
                                -3.500000                                 
                                         .00000000                        
S.3    BASIS  30.840490                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.4    NONBASIS                                                           
              .00000000                                                   
                       .00000000                                          
                                -8.1046702                                
                                         .810467                          
S.6    BASIS  5.0684449                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.7    BASIS  35.431555                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.17   BASIS  43.613229                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.18   BASIS  110.00000                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.19   BASIS  16.9566522                                                  
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.20   BASIS  19.466610                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.21   BASIS  135.53339                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.22   BASIS  9.0515551                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.23   BASIS  5.5684449                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
S.30   BASIS  .00000000                                                   
                       .00000000                                          
                                .00000000                                 
                                         .00000000                        
__________________________________________________________________________