# 9. NUMERICAL EXAMPLES METHODS OF ENERGY SYSTEMS OPTIMIZATION

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9. NUMERICAL EXAMPLES METHODS OF ENERGY SYSTEMS OPTIMIZATION
OPTI_ENERGY Summer School: Optimization of Energy Systems and Processes Gliwice, 24 – 27 June 2003 METHODS OF ENERGY SYSTEMS OPTIMIZATION 9. NUMERICAL EXAMPLES 9.1 Thermoeconomic Operation Optimization of a System

9.1.1 Description of the system
A combined cycle cogeneration system that covers the needs of a refinery in electricity and steam. Two-way interconnection with the utility grid. Main components: Two gas-turbine electricity generators of 17 MWe each. Two exhaust-gas boilers recovering heat from the gas turbine flue gases. One steam-turbine electricity generator of 16 MWe. Two steam boilers of 60 ton/h each. Two steam boilers of 30 ton/h each.

Fig. 9.1.1. Simplified diagram of the
combined-cycle cogeneration system.

Table 9.1.1. Steam grades used in the refinery.
Description of the system Table Steam grades used in the refinery.

9.1.2 Primary energy sources
Electricity supply from the utility grid. Fuel gas (FG): A by-product of the refinery process. The largest primary energy source. It consists of light hydrocarbons (methane to butane) and a small percentage of hydrogen (about 5% by volume). It is available at low pressure (LPFG) and high pressure (HPFG). It cannot be stored. If not used, it is burned in the flares. (continued)

9.1.2 Primary energy sources
(continued) Fuel oil (FO). Commercial industrial grade fuel oil (900 kg/m3, 370 cSt at 50°C max) of low sulfur content (0.7% by weight, maximum). The second largest primary energy source for the refinery. Propane. A sellable final product. Its use as a fuel in the refinery depends on propane storage availability and its selling price. There is actually a trade-off between FO and propane, and the use of one or the other depends on their selling price.

Energy conversion The various fuels are converted to heat, steam and electricity. Process heat needs are covered by fired heaters using FG and/or FO or by steam. Steam is produced by steam boilers, and by waste heat boilers in the process units as well as in the cogeneration system. Four grades of steam are produced. If the quantity of steam directly produced at a certain grade is not sufficient, then it is supplemented by desuperheating, which causes an exergy destruction and consequently must be avoided whenever possible.

9.1.4 The need for operation optimization
The energy needs of the refinery can be satisfied by several primary energy sources through various energy conversion systems. Important considerations: Electricity can be produced (within certain limits) either by the gas turbines or by the steam-turbine generator. The optimum load distribution is requested. Gas-turbine generators produce electricity and steam simultaneously. Thus, increased gas turbine level of electricity production results in an increase of steam availability, reducing the required production of steam by the steam boilers. Increasing the level of electricity production by the steam-turbine generator results in reduced steam availability, thus increasing the required production of steam boilers. (continued)

9.1.4 The need for operation optimization
Important considerations (continued): Electricity can be exported to the utility grid. The quantity of the exported electricity affects the operation of the gas turbines, steam turbine and boilers. Production and consumption of the various steam grades must be kept in balance to avoid degrading steam of higher levels to lower levels at a loss (i.e. without production of mechanical work). A heuristic approach or past experience only is not capable of determining the optimum mode of operation. The application of an optimization procedure is necessary.

9.1.5 The Optimization objective
Minimization of the capital and operating cost at any instant of time: (9.1.1) (9.1.2) Inequality constraints on the independent variables: (9.1.3)

9.1.5 The Optimization objective
(continued) Net electric power produced by the cogeneration system: (9.1.4) Total electric power supplied by the cogeneration system and the utility grid: (9.1.5) An analysis and simulation of the system including mathematical simulation of the main components and important auxiliary equipment has been performed.

9.1.6 Considerations on capital and operation expenses
The introduction of capital depreciation, maintenance and personnel costs in the objective function has an impact on the optimum point only if these costs can be expressed as functions of independent variables. The available information led to the following. Four main subsystems are considered: 1: fuel-oil boilers, 2: steam-turbine generator, 3: gas-turbine generator No. 1 with exhaust boiler, 4: gas-turbine generator No. 2 with exhaust boiler. (continued)

9.1.6 Considerations on capital and operation expenses
(continued) Capital cost: (9.1.6) Maintenance and personnel costs: (9.1.7) where (9.1.8)

9.1.7 Description of the computer program
The direct application of a mathematical programming algorithm has been used. The computer program consists of the following parts: Main program Optimization algorithm GRG2 Constraints subroutine GCOMP Objective function FZ Component simulation package File DSTEAM

Numerical results Results for typical load conditions Usual practice (example): Optimum mode of operation (for the same load conditions):

Example of Sensitivity Analysis
Fig Effect of unit cost of electricity purchased from the grid on the optimum operating point.

Example of Sensitivity Analysis
Fig Effect of unit cost of fuel oil on the optimum operating point.

9.1.9 Conclusions on the example
The application of an optimization procedure to a complex system is very beneficial: if the common practice is replaced by the optimization procedure, a very significant reduction in operating expenses can be achieved with no need of additional investment. The simplifying assumptions leave much room for further development and improvement of the procedure and the software. In a further development, the limits of the system under optimization may be extended to include the refinery processes. Off-line optimization has been applied, which is satisfactory when the plant operates at nearly constant conditions for relatively long periods of time. For frequent changes of conditions however, on-line optimization is necessary. On-line optimization requires fast simulation and optimization software.

9. NUMERICAL EXAMPLES METHODS OF ENERGY SYSTEMS OPTIMIZATION
9.2 Thermoeconomic Design Optimization of a System

9.2.1 Description of the system and main assumptions
The system consists of a gas-turbine unit with regenerative air preheater, and a heat recovery steam generator (HRSG). Main Assumptions: a. The air and combustion gases behave as ideal gases with constant specific heats. b. For combustion calculations, the fuel is considered as methane. c. All components, except the combustion chamber, are adiabatic. d. Pressure and temperature losses in the ducts connecting the components are neglected. However, a pressure drop due to friction is taken into consideration in the air preheater (both streams), combustion chamber and the HRSG. e. Mechanical losses in the compressor and turbine are negligible.

Fig. 9.2.1. Flow diagram of the gas-turbine cogeneration system.

Table 9.2.1. Thermodynamic parameters for the system.
Net shaft power: Steam flow rate: Steam condition: p9 = 20 bar, saturated Feedwater conditions: p8 = 20 bar T8 = 25°C Temperature difference: Fuel properties (CH4) Molar mass: Mf = kg/kmol Lower heating value: Hu = kJ/kg Specific chemical exergy: Conditions at the combustor inlet: T10 = 25°C Reference environment: p0 = bar T0 = 25°C (continued)

Table 9.2.1. Thermodynamic parameters for the system.
(continued) Other pressures and temperatures p1 = bar T1 = 25°C p7 = bar T7min = 25°C Overall heat transfer coefficient in the air preheater: U = kW/m2K Properties of air and exhaust gas for compression and expansion calculations (ideal gas model): cpa = kJ/kg·K Ra = kJ/kg·K cpg = kJ/kg·K Rg = kJ/kg·K Efficiency of the combustor : (i.e. thermal losses 2%) Exit/inlet pressure ratios in components due to friction Air preheater – air side: rAa = 0.95 Air preheater – exhaust gas side: rAg = 0.97 Combustor and HRSG: rB = rR = 0.95

9.2.2 Preliminary Calculations
Steam temperature: T9 = Tsat(20 bar) = °C Preheated water temperature: Useful heat rate (product of the system): Useful heat rate of the economizer: Useful heat rate of the evaporator:

9.2.3 Thermodynamic Model of the System
It consists of 21 equations including 47 quantities (pressures, temperatures, mass flow rates, heat transfer area, etc.). Examples:

9.2.3 Thermodynamic Model of the System
(continued) Quantities involved: Parameters given or already calculated: 21 Number of equations available: Number of unknown quantities (independent variables): 5 Selected independent variables:

Installed capital cost functions of components
Economic model of the system Installed capital cost functions of components Compressor: Air preheater: Combustor: Turbine: HRSG:

9.2.4 Economic model of the system
Annualized capital cost of a component including depreciation and maintenance: (9.2.4) Total annual cost of the system: (9.2.5) where Cr installed capital cost of component r, FCR annual fixed charge rate, maintenance factor, cf cost of fuel per unit of energy, t time period of operation during a year.

Fig. 9.2.2. Functional diagram of the system.
Thermoeconomic Functional Analysis of the system Fig Functional diagram of the system.

Functions (products) of the units
Compressor: Air preheater: Combustor: Turbine: HRSG: Junction:

Additional functions Function from the environment: Functions to the environment: Distribution of mechanical exergy (due to pressure difference from the environment): Shaft power from the turbine to the compressor:

Additional functions (continued) Thermal exergy due to temperature increase in the compressor: Thermal exergy from exhaust gases: Product of the air preheater given to the junction: Combustion function given to the junction: Thermal exergy from the junction to the turbine: Thermal exergy from the junction to the HRSG:

9.2.5 Thermoeconomic Functional Analysis of the system
(continued) Cost balance for each unit considering a break-even operation (physical or monetary costs): (6.2.27) The system of equations is solved for the unit product costs, cn. The costs are distributed to the units and to the final products by the function distribution network.

9.2.6 Statement of the optimization problem
Optimization objective function (minimization of the total cost rate of the system): (9.2.28) Equality constraints: the thermodynamic and economic model of the system. Inequality constraints: (9.2.29)

Basic procedure for solution of the optimization problem
by the Functional Approach 1. Select an initial set of values for x. 2. Determine the values of y by the system of equality constraints. 3. Evaluate the Lagrange multipliers. 4. Check the necessary conditions. If they are satisfied to an acceptable degree of approximation, then stop. Otherwise, select a new set of values for x and repeat steps 2-4.

9.2.7 Application of the modular approach
Module 1: Compressor Parameters and variables: Simulation model: Eqs. (A.1), (A.2), Appendix A in the text. Module 2: Combustor and turbine Parameters and variables: Simulation model: Eqs. (A.7) – (A.9) and (A.11) – (A.13).

Module 4: Heat recovery steam generator
Application of the modular approach Module 3: Air preheater Parameters and variables: Simulation model: Eqs. (A.10), (A.18) (A.19). Module 4: Heat recovery steam generator Parameters and variables: Simulation model: Eqs. (A.14), (A.15) (A.20), (A.21).

Numerical results Table Optimization results for the nominal set of parameter values.

Table 9.2.3. TFA: values of functions at the optimum point (in kW).
Numerical results Table TFA: values of functions at the optimum point (in kW). Table TFA: values of Lagrange multipliers and unit product costs at the optimum point (in \$/106 kJ).

Sensitivity analysis Table Sensitivity of the optimal solution to the fuel price and capital cost.

Sensitivity analysis Table Sensitivity of the objective function to the independent variables: , %.

compressor pressure ratio.
Sensitivity analysis Fig a. Effect of fuel price and capital cost on the optimum value of compressor pressure ratio.

compressor isentropic efficiency.
Sensitivity analysis Fig b. Effect of fuel price and capital cost on the optimum value of compressor isentropic efficiency.

preheated air temperature.
Sensitivity analysis Fig c. Effect of fuel price and capital cost on the optimum value of preheated air temperature.

the objective function.
Sensitivity analysis Fig c. Effect of fuel price and capital cost on the optimum value of the objective function.

9.2.10 General comments derived from the example
The application of three methods for the optimization of thermal systems has been demonstrated through this example. All three approaches have been successful in the particular application. The direct use of an optimization algorithm is the simplest way, because it requires the least effort in system analysis, but it gives no information about the internal economy of the system (physical and economic relationships among the components). Scaling of the variables and of the objective function is usually required in order to achieve convergence to the optimum point. Since no method can guarantee convergence to the global optimum, there is need to start the search from different initial points. If the same final point is reached, then we are more or less confident that this is the true optimum.

9. NUMERICAL EXAMPLES METHODS OF ENERGY SYSTEMS OPTIMIZATION
9.3 Environomic Analysis and Optimization of a System

9.3.1 Description of the system and main assumptions
Main Characteristics of the System Fuel oil is considered in this example, because it is more polluting than the natural gas. The system produces a specified amount of electric power. The system is equipped with a flue gas desulfurization (FGD) unit for SO2 abatement. Its operation requires electricity, water and limestone. The size and the capital cost of the FGD unit depend largely on the exhaust gas flow rate. Therefore, it is less expensive to desulfurize a partial flow at the maximum possible degree than the total flow at a lower degree.

Fig. 9.3.1. Gas-turbine system with flue gas desulfurization unit.

9.3.1 Description of the system and main assumptions
Mass and volume flow rates through the FGD unit: (9.3.1) Degree of SO2 abatement: (9.3.2) where desirable degree of SO2 abatement, mass, volume flow rate of exhaust gases through the FGD unit, total mass, volume flow rate of exhaust gases, initial mass flow rate of SO2 : (9.3.3) final mass flow rate of SO2 (after abatement).

Two thermodynamic objectives
Statement of the optimization problems Two thermodynamic objectives Maximization of the cycle efficiency: (9.3.4) Maximization of the net power density, defined as: (9.3.5) where (9.3.6) Independent variable: (9.3.7) Comment: and w increase continuously with and

9.3.2 Statement of the optimization problems
Thermoeconomic objective is the minimization of the annual cost of owning and operating the system: (9.3.8) Independent variables: (9.3.9) Environomic objective: (9.3.10) Independent variables: (9.3.15)

9.3.2 Statement of the optimization problems
Capital cost of the FGD unit: (9.3.11) Cost of resources for the first year: (9.3.12) (9.3.13) First year penalty for emitted SO2: (9.3.14)

Table 9.3.1. Parameter values for optimization of the system.
Numerical results and comments Table Parameter values for optimization of the system.

Table 9.3.2. Optimization results.
Numerical results and comments Table Optimization results. Variable O b j e c t i v e rC 25.83 28.1 10.39 10.82 15.22 16.14 * # 0.8460 0.8555 rB 0.9820 0.9839 T3 (K) 1467.4 1478.6 0.8947 0.8993 __ 0.9500 0.4056 0.4202 0.3636 0.3750 0.3900 0.4034 * Equal to the thermoeconomic optimum value. # Equal to the environomic optimum value.