MATHEMATICAL MODELLING OF INDUSTRIAL ENERGY SYSTEMS WITH OPTIMIZATION PROBLEMS

1. OPTI ENERGY MATHEMATICAL MODELLING OF INDUSTRIAL ENERGY SYSTEMS WITH OPTIMIZATION PROBLEMS Andrzej Ziebik Institute of Thermal Technology, Technical University of Silesia ul. Konarskiego 22, 44-101 Gliwice, POLAND Tel. +(48 32) 237 16 61, Fax +(48 32) 237 28 72 email: itc@itc.ise.polsl.gliwice.pl Summer School on "Optimisation of Energy Systems and Processes" Gliwice, June 24 - 27, 2003

2. Chapter 1 ENERGY MANAGEMENT OF AN INDUSTRIAL PLANT AS A SYSTEM Technological and energy subsystem of an industrial plant Every industrial process can be divided into a technological subsystem (the assembly of technological branches) and an energy subsystem (energy management). In an industrial plant the production of energy carriers is meant, first of all, for the technological subsystem. A part of the produced energy carriers is used up in the energy subsystem itself. Due to the complexity of relations between the energy branches (some of these relations are of feedback character), the whole energy management is more than the sum of its parts (meant as separate energy branches considered individually). The last conclusion (with no energy terminology used) is the oldest definition of any system, originally formulated by Aristotle.

3. Thus, the energy management of an industrial plant is a system defined as a set of energy equipment and engines and the inner relations between them and external relations between the energy management and environment, the aim of which is the production, conversion, transmission and distribution of energy carriers consumed in industrial plants. Due to these relations the energy management, treated as a complex, has attributes which its parts (energy branches) do not possess. The energy management of an industrial plant, treated as a large-scale energy system belongs to artificial systems, continually developing and having a hierarchical structure. In this system people belong to its controlling or controlled part. The energy subsystem of an industrial plant can be considered as a cybernetic-type engineering system having attributes of a socio-economic system. The participation of people in the controlling as well as in the controlled part of the system decides about the attributes of the socio-economic system.

4. Input-output relations in the energy subsystem of an industrial plant As a simple example for an energy system one may consider a heat-and-power generating plant (Fig. 1.1). Another form of presenting such an energy system besides a schematic diagram, is the binary input-output matrix (Table 1.1). Some relations situated under the main diagonal have a feedback character. The existence of feedback relations is responsible for the fact, that the partial balances of energy carriers lead to an agreement of the balance by means of subsequent approximations. Therefore, a mathematical model of the balance of energy systems of industrial plants has been prepared.

5. HEAT AND POWER GENERATING PLANT Interbranches flows

6. External relations of an industrial energy subsystem The energy subsystem of an industrial plant is characterised by its large-scale as well as compactness, the latter being due to the relations of nets and pipe-lines. The energy subsystem has a great influence on the efficiency of the technological subsystem, in spite of its auxiliary role in relation to the technological subsystem. The energy subsystem of an industrial plant is a goal-seeking system which has a hierarchical structure. This means, that the particular elements of the subsystem (energy branches) are low-level subsystems while the whole energy management is a high-level system. Next, the energy management of the industrial plant, treated as a complex, is a subsystem in the large-scale national energy system. The hierarchical attribute of the energy subsystem is employed to decompose the global optimisation problem (if we apply this model as an optimisation model).

7. The energy subsystem of an industrial plant is an open system which exchanges matter, energy and information with the environment. The relations with the environment are external ones. These are relations with other systems, that are on a higher or on the same level. There are the following external relations: - input-output relations in industrial plants between the energy andtechnological subsystem, - relations between the energy subsystem of an industrial plant and the national energy system, - restrictions in the outlay and supply of engines, materials, fuels and energy, • relations to the natural environment creating mainly negativeecological effects. Scientific researches of the energy management of an industrial plant should be characterised by a system approach. It is connected with mathematical modelling of the energy balance of the industrial plant. The mathematical model of a long-term balance plan of energy carriers and the mathematical model of the energy subsystem for production control are considered.

8. Application of input-output analysis • The basis of linear mathematical models in system investigations is Leontief’s “input – output analysis”. The structure of the table of interbranch flows in Leontief’s theory bases on the following assumptions: • the manufacturing process is divided into “n” branches, • in each branch only one product is produced, • the global production of each branch is partially consumed by other branches including own consumption; the remaining part of global production is a final product, • the consumption of the i-th product in j-th branch is directly proportional to the global production of this branch, • the values in the table of the interbranch flows do not depend on time;Leontief’s model is a static one, • the values in Leontief’s table may be expressed by natural or monetary units.

9. According to this assumption we can write: (1.1) - consumption of the i-th product in the j-th production branch, - technical coefficient of the consumption of the i-th product per unit of production of the j-th branch, - global production of the j-th branch.

10. Table of interbranch flows according to Leontief

11. The set of balance equations in matrix notation is as follows: (1.3) where: The balance equation for the i-th production branch has the following form: (1.2) where: - global production of the of i-th branch, - external supply of the i-th product, - final production of the i-th branch.

13. If the vector G is looked for, equation (1.3) is transformed to: (1.4) where: - inverse matrix, - unit matrix. In equation (1.4) the matrix must be a nonsingular matrix.

14. Input – output analysis may be applied in mathematical modelling of various economy systems (national economy, regional economy, economy of an industrial plant). The theory of „input-output” was published by Leontief in 1936 in the USA. It was first applied in the USA in 1941 when the USA joined war operations. Then the problem arose to transform the American economy into war production and to balance it. Leontief’s model of USA’s economy proved to be adequate. In the 1980’s Leontief was awarded the Nobel prize.

15. Chapter 2 LINEAR MATHEMATICAL MODEL OF THE ENERGY BALANCE OF AN INDUSTRIAL PLANT The linear mathematical model of energy balance of an industrial plant comprises the system of interdependences existing in a real plant between the technological and energy subsystems and between the energy branches in the form of matrix equations. The matrix equations result from the balances of energy and fuels. This model is a development of Leontief’s “input-output” theory. The aim of the model is to replace the existing traditional method of partial balances by a computer-aided system method. The computer program is based on typical software of a microcomputer. A month is the shortest balance period to which the elaborated linear model can be applied.

16. EQUATIONS OF A LINEAR MATHEMATICAL MODEL OF ENERGY BALANCE The production process of an industrial plant is divided into two groups of productive branches: the technological and energy branches. The productive branch is a technological or energy process producing one given major product as well as an optional number of by-products. The by-products can exist only near the major product. If there is more than one source of energy carrier as the major product, the production must be divided into the basic part and the variable (peak) part (for example the steam extraction nozzle and the steam from the pressure-reducing valve). Energy carriers can be produced as by-products in the energy and technological subsystem. If a given energy carrier is the major product in one branch and the by-product in another, it should be treated as a whole in the balance equations of the major product (for example steam from evaporative cooling or the waste-heat boiler). In another case the energy carriers produced as a by-product can provide fuel (e.g. blast-furnace gas or coke-oven gas in ironworks).

17. In some cases the own production of energy carriers must be supplemented by external supplies (e.g. electric energy). Some energy carriers are only brought from outside (e.g. mainly fuels). Sometimes part of the production of energy carriers is sold to external consumers (e.g. heat, hot water and electric energy from the heat and power generating plant). Balance sheet of energy carriers I n p u t

18. O u t p u t

19. -matrices of the coefficients of the consumption of energy carriers in the energy and technological subsystem, -matrices of the coefficients of the by-production of energy carriers in the energy and technological subsystem, -column vector of external supply of energy carriers, - matrices of the consumption and by-production of energy carriers LINEAR MATHEMATICAL MODEL AND ITS APLICATION where: -column vectors of the peak and basic part of the production of energy carriers,

20. independent of the production in the energy and technological subsystem, - diagonal unit matrix and column vectors with unit elements. - relative losses of energy carriers

21. SIMULATION OF A LONG-TERM BALANCE OF THE ENERGY SYSTEM This equation may be applied to calculate various balances of the energy system concerning a number of variants of the production of technological branches (than the vector T is changed). This equation may also be used to analyse the influence of thermal parameters of energy carriers and of the introduction of new processes, and the modernisation of old ones upon the energy balance of industrial plants.

22. ALGORITHM FOR THE CALCULATION OF EXERGY LOSSES IN AN ENERGY AND TECHNOLOGICAL SUBSYSTEM

24. SYSTEM METHOD OF DETERMINATION OF CUMULATIVE ENERGY CONSUMPTION Technological and energy products manufactured in an industrial plant are interconnected due to the existence of network of mutual technological and energy connections. The direct consumption of energy does not comprise all the energy required to produce some given useful product, because the raw materials, energy carriers, materials and semi-products used for its production also required energy. Thus, every product results not only from direct but also indirect consumption of energy in numerous previous technological and energy processes. The total consumption of energy charging all the processes of production and transport leading to the final product have been called cumulative energy consumption.

25. In order to determine the relations concerning the indices of cumulative energy consumption a mathematical model of the energy balance of the industrial plant may be applied. An energy carrier may be produces in a basic or peak installation; it can also be a by-product or be supplied from outside. For this reason the average index of cumulative energy consumption has been introduced.

27. -unit cost of the basic part of the production, -unit cost of the peak part of the production, -unit cost of the by-production, -unit cost of external supplies. MATRIX METHOD OF CALCULATING THE UNIT COSTS OF ENERGY CARRIERS Application of the principle of the linear mathematical model of an industrial energy systems Unit costs loco production process:

28. Unit cost loco consumer: Weighted average unit cost of an energy carrier Balance of costs for the energy branch “j”

30. OPTIMISATION OF THE BALANCE PLAN • OF ENERGY MANAGEMENT • Assumptions • steady state of investment of energy management, • structure for the feed with the energy carriers is fixed, • the sale of energy carriers (vector H) is known, • the basic part of the production of energy carriers (vector P) is also known.

32. Global constraints – balance equations • Inequality constraints

33. After transformations Due to the linear form of the aim function and constraints this optimisation problem is solved by linear programming.

34. EXAMPLE OF LONG-TERM MATHEMATICAL MODEL OF ENERGY BALANCE OF IRONWORKS This plant is the most modern one in Poland. It is equipped with three blast furnaces (3200 m3 each) and three steel converters. The structure of production comprises: - 33 energy carriers, - 7 technological branches: - sinter plant, - blast-furnace plant, - converter plant, - four rolling mills.

38. EXAMPLE OF THE APPLICATIONS OF EXERGY ANALYSIS TOENERGY AND TECHNOLOGICAL SUBSYSTEMS OF THE STEEL INDUSTRY Specific exergy of energy carriers, raw materials, technological main productsand by-products(selected results of calculations)

39. Duration function of heat production

40. Duration function of exergy of heat-production

41. Comparison of the effects of applying conti-casting

42. RELATIVE EXERGY LOSSES IN THE ENERGY SUBSYSTEM 1 - high pressure steam (boiler house); 2 - low-pressure steam and electric energy (extraction turbine and pressure reducing valve); 3 - heat (heat exchangers and water heater); 4 - blast (turboblowers); 5 - compressed air for oxygen plant (turbocompressors), 6 - oxygen plant

43. RELATIVE EXERGY LOSSES IN THE TECHNOLOGICAL SUBSYSTEM 1 - sinter plant; 2 - blast furnace plant; 3 - converter plant; 4 - slabbing mill; 5 - steel conti-casting; 6 - heavy-section and medium-section mills

44. CONCLUSIONS We have investigated the influence of energy and technological changes on the exergy losses in one process. The effects of the applied energy-technological changes on the exergy losses in other processes must also be investigated. For this purpose the mathematical model of the material and energy balance of an industrial plant can be applied, in which all the quantities may be implemented by means of exergy. Such a model of the exergy balance may be applied in order to determine the effects of thermal improvements upon the exergy losses in the whole network of correlated energy and technological processes in industrial plants.

45. SYSTEM ANALYSIS OF RATIONALISATION OF ENERGY MANAGEMENT OF INDUSTRIAL PLANT EVALUATION OF ENERGY RATIONALISATION EFFECTS PROCESS METHOD The process method of evaluation of energy rationalisation effects does not take into account the interdependences existing between energy processes. Therefore this method gives incomplete energy effects of energy rationalisation. SYSTEM METHOD The energy rationalisation effects in the energy or technological subsystem should be determined at the boundary of the balance shields of an industrial plant. In this way the direct and indirect connections between considered process, in which the energy rationalisation is carried out and other processes will be taken into account.

46. where: - column vectors concerning pig iron, 1,2 - state before and after the change of the blast furnace parameters, - production of pig iron.

47. EXAMPLES OF APPLICATION OF SYSTEM ANALYSIS System analysis of intensification of blast-furnace process Blast-furnace 3 200 m3 ·blast temperature 1 100 ºC ·top-gas pressure 0.3 MPa ·oxygen enrichment of blast 26 % ·amount of auxiliary fuel 3 GJ/Mg p.i.

48. Energy characteristic of blast-furnace plant before and after rationalisation

49. RESULTS OF THE FORECAST OF ENERGY CHARACTERISTICS OF BLAST FURNACE PROCESSES