1

MATHEMATICAL MODELLINGOF INDUSTRIAL ENERGY SYSTEMSWITH OPTIMIZATION PROBLEMS

OPTI ENERGY

Andrzej ZiebikInstitute 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

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Chapter 1

ENERGY MANAGEMENTOF 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.

 

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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 PLANTInterbranches flows

E lectr icen ergy

F eedw a ter

H igh -p ressu re

stea m

L o wp ressu re

stea m

C oo lin gw a ter

E lectr icen erg y

1 1 1 1 1

F eedw a ter

0 0 1 1 1

H ig h -p ressu re

stea m1 0 0 1 0

L o wp ressu re

stea m0 1 0 0 0

C o o lin gw a ter

1 0 0 0 0

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 and technological 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 negative ecological 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

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.

Application of input-output analysis

9

According to this assumption we can write:

(1.1)G ij = a ijG j

- 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.

G ij

a ij

G j

10

Table of interbranch flows according to Leontief

Interbranch flowsProduction

branch

G lobal

product

External

supply 1 2 ... n

Final

product

1 G 1 D 1 G 11 G 12 ... G 1n K 1

2 G 2 D 2 G 21 G 22 ... G 2n K 2

... ... ... ... ... ... ... ...

n G n D n G 1n G n2 ... G nn K n

11

The balance equation for the i-th production branch has the following form:  (1.2)i

n

1jjijii KGaDG

where:

iG

iD

iK

- global production of the of i-th branch,

- external supply of the i-th product,

- final production of the i-th branch.

The set of balance equations in matrix notation is as follows: 

(1.3) where:

KAGDG

12

production global of vector-n

G...GG

n

2

1

G

supplies, external of vector-n

D...DD

n

2

1

D

,nconsumptiooftscoefficientechnical ofmatrix

a...aa............

a...aaa...aa

nn2n1n

n22221

n11211

G

13

production final of vector-n

K...KK

n

2

1

K

If the vector G is looked for, equation (1.3) is transformed to: 

(1.4)  where:

- inverse matrix,

- unit matrix.

)DK()AE(G 1

1AE

E

In equation (1.4) the matrix must be a nonsingular matrix.)AE(

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.

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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.

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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).

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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

Main product By -productionEnergycarrier

peakpart

basicpart

Energy subsystem Technologicalsubsystem

Externalsupply

. . .i

. . .

. . .G i. . .

. . .P i

. . .

. . .

. . .

. . .. . .D

i. . .

jij

Pijj

Gij QP jfGf

kikkik QGf

. . .

. . .

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O u t p u t

branch. ogicalth technol-k of production - G

,production theoftly independen carriersenergy ofn consumptio - X,X

carriers,energy ofn consumptio theof tscoefficien - a,a,a

,production theoftly independen carriersenergy of production-by - Q,Q

carriers,energy of production-by theoft coefficien - f,f,f

k

ikij

ikPij

Gij

ikij

ikPij

Gij

Interbranch flowsEnergy carrier

Energy subsystem Technological subsystem

Generalneeds

Sale Losses

. . .i

. . .

. . .Y

i. . .

. . .H

i. . .

. . .V

i. . .

. . .

. . .

. . .

. . .

j

ijPijj

Gij XP jaGa

kikkik XGa

19

LINEAR MATHEMATICAL MODELAND ITS APLICATION

where:- column vectors of the peak and basic part of the production of

energy carriers, P,G

- matrices of the coefficients of the consumption of energy carriers in the energy and technological subsystem,

A,A

- matrices of the coefficients of the by-production of energy carriers in the energy and technological subsystem,

F,F

- column vector of external supply of energy carriers,D- matrices of the consumption and by-production of energy carriersX,X

THGASDGFG GG

PFSAESYEQXSEQSXGFAST PP21

20

independent of the production in the energy and technological subsystem,

Q,Q

- diagonal unit matrix and column vectors with unit elements.21 E,E,E

n

2

....00................

0....-110

0....0

S

11

11

1

INPUTVi

i

i - relative losses of energy carriers

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SIMULATION OF A LONG-TERM BALANCEOF THE ENERGY SYSTEM

DHTFSAEG 1GG

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.

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

eTe

DTT bB YGEFA

23

eTtt

T

uTT

S

T

eTTD

bb

bˆbbB

YEA

FUSFAG

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.

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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.

i

deii

ueii

Peii

Geii

ei HwDeLwPwGw

iiiii DLPGH

k

ikkikj

Pij

Gijjiji QGfQQGfL

26

C a l c u l a t i o n d i a g r a m c o n c e r n i n g t h e b a l a n c eo f c u m u l a t i v e e n e r g y c o n s u m p t i o n

euTG

1DeG

1

e

1DT

wQwS

w

GF

YXGA

TG

TG

TG

euT

P

1

eP1

eT

P

1D wQwSw DTP PYXP

"j"

iei

Pij

Gijjij wXXGa

i

eiPij

Gij wYY

Gejjj w1G

Pejjj w1P

i

Peii

Pij

Gijjij w1QQGf

27

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:

- 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.

Pk

Gk

Uk

Dk

28

Unit cost loco consumer:  

Weighted average unit cost of an energy carrier

iDiDiUiUiGiGiPiPiZi Tkrkrkrkrk

Balance of costs for the energy branch “j”

Energy

branch

"j"

n

1iziijkZ

SGjK

GjjkG

n

1iUiijkU

29

ZU

U

T

PD

PPD

SPZ

T

PD

P

U

T

GD

GGD

SGZ

T

GD

G

kk

kQPFkPKkXPA

kQGFkGKkXGA

where:i - weights,

iT - costs of transport and distribution,

SPSGK,K - fixed and operating costs (without the costs of energy carriers,

, - coefficients which follow from the method of dividing costs in combined process.

The test of the correctness of calculated results of theunit costs of energy carriers by means of the balanceequation of the costs of all the energy management istaken into account.

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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.

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minKK TsDssDGe DDDG

eK - variable operating costs,G - row vector of the variable unit costs

of operation,D - row vector of the unit costs of basic

part of external supplies of energycarriers,

Ds - row vector of the unit costs of thepeak part of external supplies ofenergy carriers,

sD - column vector of the peak part ofexternal energy supplies,

TK - cost of losses in the technologicalsubsystem because of the deficiency ofenergy carriers.

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Inequality constraints

NPG

Global constraints – balance equations

s

s

DDD

where:N - column vector of the production

capacities of the energy branches, - column vector of the limits of the

basic part of the external supply ofenergy carriers,

s- column vector of the limits of the peakpart of the external supply of energycarriers.

33

After transformations

minKK TsDGGGDe DGEFA DsS

sGG DTEFA SHGS

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.

35

36

37

38

EXAMPLE OF THE APPLICATIONS OF EXERGY ANALYSIS TO ENERGY AND TECHNOLOGICAL SUBSYSTEMS

OF THE STEEL INDUSTRYSpecific exergy of energy carriers, raw materials, technological main products

and by-products (selected results of calculations)Energy carrier Unit Specific

exergyRaw materialsTechnological

products

SpecificexergyGJ/Mg

Heat:- basic part- peak partBlastCompressed oxygenCompresses air foroxygen plantOxygenLow -pressure steamMedium -presure steamHigh -pressure steamBlast -fur nace gasConverter gasCoke -oven gasNatural gasPower coalCoke

GJ/GJGJ/GJ

GJ/MmolGJ/MmolGJ/Mmol

GJ/MmolGJ/MgGJ/MgGJ/MgGJ/GJGJ/GJGJ/GJGJ/GJGJ/GJGJ/Mg

0.1710.2524.091

10.1084.187

3.5570.6631.0451.6170.9900.9971.0131.0411.085

30.153

Iron oreLomestoneBurnt limeSinterPig iron:- chemical exergy- physical exergySteel:- chemical exergy- physical exergyRolled productsBlast -furnace slagConverter slagBlast -furnace dustFire scale

0.3010.0462.0490.338

8.1550.930

6.9190.3106.9190.55 41.315

11.8881.324

39Duration function of heat production

40

Duration function of exergy of heat-production

41

dTTln

TTT1Q

Q1b

p

h

ph

oth

0a

q

Comparison of the effects of applying conti-casting

Without conti-casting With conti-castingRelative exergy losses0.693 0.690

Relative decrease of the inputexergy

0.0065

Relative decrease of exergylosses

0.0108

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

S1SS2SSS GFAFAT

where:

  - column vectors concerning pig iron,1,2 - state before and after the change of the blast

furnace parameters,- production of pig iron.

SS ,FA

SG

1S2SS

1

GG

S

S

GFAFAFAEG

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

Energy characteristic UnitBefore

rationa-lisation

Afterrationa-lisation

Specific consumptionof coke

kg/Mg p.i. 503.9 479.5

Specific consumptionof blast

kmol/Mg p.i. 54.4 52.6

Specific production ofchemical energy oftop-gas

GJ/Mg p.i. 7.978 8.147

Specific consumptionof chemical energy oftop-gas in Cow perstoves

GJ/Mg p.i. 2.552 2.342

49

RESULTS OF THE FORECAST OF ENERGY CHARACTERISTICS OF BLAST FURNACE PROCESSES

50

51

Results of the system analysis

Energy carrier UnitChanges of

consumption or by -production

of energy carriersBlastHigh-purity oxygenCompressed air for theoxygen plantLow-pressure steamMedium -pressure steamHigh-pressure steamDemineralised w aterCompressed airIndustrial w aterElectric energyBlast-furnace gas:- production- consumptionCoke-oven gasNatural gas 1Natural gas 2Pow er coal

kmol/Mg p.i.kmol/Mg p.i.kmol/Mg p.i.

kg/Mg p.i.kg/Mg p.i.kg/Mg p.i.kg/Mg p.i.

kmol/Mg p.i.kg/Mg p.i.

kW h/Mg p.i.

MJ/Mg p.i.MJ/Mg p.i.MJ/Mg p.i.MJ/Mg p.i.GJ/Mg p.i.MJ/Mg p.i.

- 2.23+ 1.424+ 8.55

+ 10.6+ 14.3+ 61.9+ 18.9

+ 0.012+ 4.3+ 1.2

+ 189.- 210.- 14.4- 43.+ 1.

+ 103.

52

SYSTEM ANALYSIS OF EVAPORATIVE COOLING IN A HEATING FURNACE

Coefficients of consumption and by-production of energy carriers before and after the installation of evaporative cooling

Coefficients of consumption and by-production of energy carriers

Beforerationali-

sation

Afterrationali-

sationCoefficient of industrial w aterconsumption (0 = 10 K),Mg/mg r.p.

5.658 0.258

Coefficient of soft w aterconsumption,Mg/Mg r.p.

0 0.098

Coefficient of by-production ofmedium -pressure steam,Mg/Mg r.p.

0 0.0833

53

54

55

Results of the system analysis of evaporative cooling

Energycarrier

Changes of major production, by-production and external supplies due to

evaporative coolingUnit Major

produ-ction

By-produ-ction

Externalsupply

Soft w ater

Demineralizedw ater

Low-pressuresteam

Medium-pressure steam

High-pressuresteam

Compressed air

Industrial w ater

Electric energy

Pow er coal

Natural gas

kg/Mg r.p.

kg/Mg r.p.

kg/Mg r.p

kg/Mg r.p.

kg/Mg r.p.

kmol Mg/r.p.

Mg/Mg r.p.

kW h/Mg r.p.

MJ/Mg r.p.

MJ/Mg r.p.

+ 86.3

- 41.8

- 9.8

- 98.9

- 114.4

- 0.021

- 7.1

- 6.4

-

-

+ 13.2

- 22.7

0

+ 83.3

-

-

-

0

-

-

-

-

-

-

-

-

-

0

- 380.0

- 4.0

56

CONCLUSIONS  In the process-method of the assessment of the effects of rationalization of energy management of industrial plants the energy effects of the relations between energy processes have been neglected.  The application of the mathematical model of energy management of an industrial plant in the system approach of the evaluation of the effects of rationalization provides possibilities to take into account all the interdependencies between energy processes and to obtain accurate results.  The energy effects of the rationalization of industrial energy management are determined at the boundary of the balance shield of industrial plant. The final results of this calculations is a decrease of external supplies of energy carriers.

57

A SYSTEM APPROACH TO THE ASSESSMENTOF INDUSTRIAL WASTE ENERGY RESOURCES

Waste energy resources – amount of the chemical energy of fundamental fuels, which can be saved in result of waste energy utilization. Interior utilization – preheating the process substrates.Exterior utilization – production of secondary energy carriers.

INFLUENCE OF WASTE ENEGY UTILIZATION UPON THE ENERGY MANAGEMENT OF AN INDUSTRIAL PLANT

Interior utilization → saving fuel in a process → fuel management → external supply of fuels.

58

S i m p l e r e l a t i o n s 

Exterior utilization → production of steam, hot water, electric energy → heat and power management → external supply of fuels.  

C o m p l i c a t e d r e l a t i o n s 

among othersdependences of feedback character

59

PROCESS-METHOD OF THE ASSESSMENT OF WASTE ENERGY RESOURCES AND ITS DRAWBACKS

Recuperation

nrrr

cumEf

frch QTSPE

11

1 (1)

Equation (1) gives accurate results if the saved fuel is supplied from outside.Otherwise (saving of own fuels, e.g. coke oven gas) – inaccurate result. Waste - heat boiler and evaporative cooling.

n

cumEl

ra

cumEh

0fgfg

fbch

NNinE

1

(2)

cumEh

nefech

QE

(3)

60

In the equations (2) and (3) the indirect relations between energy carriers in a heat and power generating plants have not been taken into account. The problem of dividing the production cost in cogeneration processes must be solved. Recovery turbine

0

0 PP

lnTRnE gngng

cumEel

Btftch

(4)

The production of electric energy substitutes:a) the external supply – nearly accurate result,b) the own production of electric energy – in equation (4) the interdependences existing in the energy management have not

been taken into account.

61

SYSTEM-METHOD OF THE ASSESSMENT OF WASTE ENERGY RESOURCES

Base – linear mathematical model of the energy management of an industrial plant (simple model).

HYGAAGDGFFGG

GBBGZ

(5)

(6)

Principle- the results of waste energy utilization are calculated at the boundary balance shield of the industrial plant.

  The decrease of supplies of external energy carriers obtained as a result of waste energy recovering is a direct saving.

62

General formula

GBBDYGFAFAIB

DYGFAFAIBΔZ

212221

222

1111

111

(7)

Interior utilization (recuperation) GBBZΔ 21r (8)

GBBAAFAIBZΔ 21211

r (9)

Exterior utilization

212121

1ex

DDGFFAAFAIBΔZ

(10)

Resources of waste energy:

elcumE

el

kcumE

kkflch

DZE

(11)

63

EXAMPLES OF APPLICATIONS OF THE SYSTEM-METHODOF THE ASSESSMENT OF INDUSTRIAL

WASTE ENERGY RESOURCES

A. The utlization of the exergy of blast-furnace gas due to increased pressure.

Volume of blast furnace: 3200 m3

Thermodynamics parameters of the blast:temperature 1100 o C,oxygen enrichment 26%,injection of natural gas 3GJ/Mg p.i.,top-gas pressure 0,3 Mpa,

Coefficient of electric energy productionIn a recovery turbine – 31.4 kWh/Mg p.i.

Other data:ηBt = 0.6 ; ηE cum el =0.261;ng =83.1 kmol/Mg p.i.; ζg = 0.85; T0 = 281K.

64

Assumption: By-production of electric energy in the recovery turbine substitutes the own production in an industrial heat and power generating plant.

B. Applications of evaporative cooling of the heating furnace.

Technical coefficient Before AfterConsumption of industrialw ater

Consumption of softw ater

By-production of medium -pressure steam

658.5a 'iw 258.0a ''

iw

0.0a 'sw 098.0a ''

sw

0.0f 'ms 0833.0f ''

ms

Other data: 

194.6 MJ/Mg r.p.  1.2 1

eQ cumEh

65

RESULTS OF CALCULATIONSResults of the system analysis of industrial

waste energy utilisation

Changes due to evaporative coolingEnergy carrier

UnitMajor

productionBy-

productionExternalsupply

Soft w aterDemineralised w aterLow-pressure steamMedium-pressure steamHigh-pressure steamCompressed airIndustrial w aterElectric energyPow er coalNatural gas

kg/Mg r.p.kg/Mg r.p.kg/Mg r.p.kg/Mg r.p.kg/Mg r.p.

kmol/Mg r.p.Mg/Mg r.p.

KWh/Mg r.p.MJ/M g r.p.MJ/M g r.p.

+86.3-41.8-9.8

-98.9-114.4-0.021

-7.1-6.4

--

+13.2-22.70.0

+83.3---

0.0--

-------

0.0-380.0

-4.0

66

Changes due to the recovery turbineEnergy carrier Unit M ajor

productionBy-

productionExternalsupply

Soft w aterDemineralised w aterLow-pressure steamMedium-pressure steamHigh-pressure steamCompressed airIndustrial w aterElectric energyPow er coalNatural gas

kg/M g p.i.kg/M g p.i.kg/M g p.i.kg/M g p.i.kg/M g p.i.

kmol/M g p.i.M g/M g p.i.

KWh/Mg p.i.MJ/Mg p.i.MJ/Mg p.i.

0.0-69.3-38.9-48.3

-229.0-0.043-10.8-40.4

--

0.0-37.70.00.0

---

+31.4--

--------

-761.8-8.1

67

Comparison of the results of calculations of waste energy resources by means

of the system method and process method

W aste energyresourcesKind of w aste energy and

the w ay of its utilisation Systemmethod

Processmethod

Cooling heat of the heatingfurnace – evaporativecooling,MJ/Mg rolled products

408.3 172.7

Exergy of blast-furnace gas-recovery turbine,MJ/Mg p.i.

815.5 415.7

68

The cause of inaccuracy of the process - method

n

1

i

...

...PRO CESSO F W ASTEEN. REC.

NATIONAL ENERGY SYSTEM

INDUSTRIAL ENERGY SYSTEM

NES IESCO E FFIC IEN TS O F C UM U LATIV E

EN ER G Y CO N SUM PTIO N

IESPROCESS

OFWASTE

EN. REC.

CO E FFIC IEN TS O F C UM U LATIV EEN ER G Y CO N SUM PTIO N

LEVEL O F IN DU ST RIA LENE RG Y SYSTEM

ELEM ENTS O F IN VER SE M AT RIXVE RS US

"INPU T-O UT PU T" M ATRIXO F IND USTRIAL EN ER G Y SY STEM

69

Conclusions

In the process method of the assessment of waste energy resources the energy effects of the relations between energy processes have been omitted. In cases of interior waste energy utilisation there are only weak interdependences. Therefore, then the process and system methods yield similar results.

But in cases of exterior waste energy utilisation the process method gives lower results, because strong interdependences are neglected.

The application of the mathematical model of energy management of an industrial plant in the system method of the assessment of waste energy resources provides possibilities to take into account all the interdependencies between energy processes and to obtain accurate results.

70

Chapter 3 

NONLINEAR MATHEMATICAL MODEL OF A SHORT-TERM BALANCE OF THE ENERGY SUBSYSTEM OF AN INDUSTRIAL

PLANT 

ASSUMPTIONS OF THE MODEL

-        the balances of energy carriers are set up for time intervals of one hour; the balances for a shift and twenty-four hours are assembled by means of one-hour balances,

-   the time-tables of work and repair idle-time for energy and energy-technological equipment are known; the planned repairs based on a long-term plan of energy balance are determined; it results from the connection of the model of long-term energy balance with the model of short-term balance,

71

-       the characteristics of engines or a complex of these are given; these may be non-linear or piecewise linear functions and sometimes linear dependences,

- the dependences of the consumption of energy carriers on the parameters of energo-technological processes are taken into account (e.g. the influence of the blast parameters and of the injection of auxiliary fuels on the energy characteristics of the blast-furnace),

-     the storage volume of energy carriers (gasholders, steam-storage tanks and hot-water accumulators) has been taken into account,

-  short-time fluctuations between the production and consumption of energy carriers existing in time intervals of one hour are covered by the ability to accumulate the heat and gas distribution network.

-  forecasts of hour-diagrams of the demand for energy carriers in a technological subsystem and the general needs of the plant and external consumers are known; the hour-diagrams show the average demands for energy carriers in particular hours of the considered shift or twenty-four hours,

72

MATHEMATICAL SIMULATION MODEL OF SHORT-TERM BALANCE OF AN ENERGY SYSTEM

The main aims of the model:

-       forecast of the energy balance of an industrial plant for a work-shift and twenty-four hours for the purpose of production control,

-       hour by hour correction of the forecast of the energy balance,

-  preparation of the energy balance of the industrial plant in case of failure.

Input data of the model:

-   forecast of hour-diagrams of the demands for energy carriers by a technological subsystem,

-       forecast of hour-diagrams of the by-production of energy carriers in the technological subsystem,

-       forecast of hour-diagrams of the consumption of energy carriers for the general needs of an industrial plant,

73

-  forecast of hour-diagrams of the demands for energy carriers by external consumers (sale),

-      forecast of hour-diagrams of supplementary external supplies which are known a priori,

-       energy characteristics of engine assemblies; at the same time the time-tables of work and repair idle-time are taken into account,

-       initial amount of energy carriers in the energy storage system,

-       average hour values of the input or output flux of energy carriers for the energy storage system.

iiiiik

P

kjijjij

n

j

iik

P

kjijjij

n

jii

i

VHCYZPZGZ

DUPUGUPG:

11

11

74

where:

- average hour flux of the variable (peak) part of the production of energy carriers,

- average hour flux of the basic part of the production of energy carriers,

ji G,G

ji P,P

- average hour flux of the by-production of energy carriers in energy and technological branch, respectively,

- average hour flux of the external supply of energy carriers,

- average hour flux of the consumption of energy carriers in the energy and technological branch, respectively,

- average hour flux of the consumption of energy carriers for general needs of a plant,

- increase of the energy carrier in the energy storage system,

- average hour flux of energy carriers for sale,- average hour flux of losses of energy carriers,

i, j= 1,2,.....,n, k=1,2,......,p.

ikij U,U

iD

ikij Z,Z

iY

iC

iH

iV

75

iik

P

lkjijjij

n

ljiiii DUPUGUPGV

Decomposition of the calculations:

iiii

i

ik

i

ikP

lki

iPHVYUZG:

11

1

0

GZGUG: i

i

jij

jij

n

nii

i

12

11

76

.

ik

i

ikP

kjij

i

jijn

njii

iiijijjij

n

ji

n,......,ni

,UZPUPZ

PD

CHYPZGZ

21

1

2

11

1

11

1

11

11

jijjijjijjij

n

j

iikik

P

kiiii

i

PUGUPZGZ

YUZSHVD:

2

1

1

77

OPTIMIZATION MODEL OF THE SHORT-TERM BALANCE OF AN ENERGY SUBSYSTEM

Objective function:

minKDDPGK TisDisiDiDiiPiiGin

1ie

where:

- variable operating costs of energy management,- variable operating unit costs of peak energy equipment

(without the costs of external energy carriers),- variable operating unit costs of basic energy equipment

(without the costs of external energy carriers),- unit cost of the basic part of the external supply of an

energy carrier,- unit cost of the peak part of the external supply of an

energy carrier,- peak part of the external supply of an energy carrier,- losses in the technological subsystem due to the deficiency

of energy carriers.

eKe

Pi

DisiDsiDTK

78

Global constraints – set of balance equations of energy carriers. 

Local inequality constraints:

Gii NG

Pii NP

isii DD

sisiD

where:

- maximum capacity of peak energy equipment,

- maximum capacity of basic energy equipment,

- limit of the basic part of the external supply of an energy carrier,

- limit of the peak part of the external supply of an energy carrier.

GiN

PiN

i

si

79

Decomposition of the global optimization problem.

Method of Lagrange’s multipliers.Matrix method of calculating the unit costs of energy carriers –

coordination procedure. Lagrange multipliers = unit costs of energy carriers.

Lagrangian function:

min

DPGC

PUGUPZGZkKL

iiii

jijjijjijjij

n

j

i

n

ie

1

1

Objective function on the level of local optimization:

min

kPUGU

kPZGZ

KL

Uijijjij

ijijjij

n

iejj

1

80

Results of local optimization

-       optimal load distribution among the engines,-       optimal distribution of the demand for energy carriers between own production and external supplementary supply,-       optimization of the amount of energy carriers from the energy storage system,-  optimization of the fuel-feeding system of the industrial plant; the substitution of fuels has been taken into account.

The decomposition algorithm is solved by means of an iterative method. The unit costs are fixed in each successive iteration on the level of optimization.

After the determination of the optimal values of the decision variables in successive iteration follows a return to the level of coordination. The detailed information of this method is gathered in Chapter 4.

81

EXAMPLE OF THE APPLICATION OF A NONLINEAR MODEL FOR SIMULATION OF A TWENTY-FOUR HOURS BALANCE OF AN

INDUSTRIAL ENERGY SYSTEM

1. FORECAST OF THE HOUR DIAGRAM OF EXTERNAL AIR TEMPERATURE FOR THE CONSIDERED TWENTY-

FOUR HOURS

82

2. HOUR DIAGRAM OF THE BASIC PART PRODUCTION OF HEAT

3. HOUR DIAGRAM OF THE PEAK PART PRODUCTION OF HEAT

83

4. HOUR DIAGRAM OF THE PEAK PART PRODUCTION OF LOW-PRESSURE STEAM

5. HOUR DIAGRAM OF THE PEAK PART PRODUCTION OF MEDIUM-PRESSURE STEAM

84

6. HOUR DIAGRAM OF THE PEAK PART PRODUCTION OF HIGH-PRESSURE STEAM

7. THE DEPENDENCE OF CAPACITY OF A DOUBLE-FUEL BOILER ON FRACTION OF CHEMICAL ENERGY

OF BLAST FURNACE GAS IN THE FUEL MIXTURE

85

8. HOUR DIAGRAM OF THE LOSSES OF BLAST-FURNACE GAS

9. HOUR DIAGRAM OF THE EXTERNAL SUPPLY OF POWER COAL

86

Chapter 4  

MATHEMATICAL OPTIMIZATION MODEL FOR THE PRELIMINARY DESIGN OF INDUSTRIAL ENERGY

SYSTEMS 

AIM OF PRELIMINARY DESIGN

„to choose an optimal variantof an industrial energy system structure

from among a numerous setof possible variants”

STRUCTURE OF THE INDUSTRIAL ENERGY SYSTEM: 

“the set of the main energy equipmentand engines, determined by power ratings

and numbers as well as the relations between them”

87

YHEXXEGAPAGADEQQEGFPFPFPG

21PG

21PG

Data from a brief foredesign: 

- consumption of energy carriers in a technological subsystem,

- by-production of energy carriers in a technological subsystem,

- vectors of the sale and consumption for the

general needs of an industrial plant.

2EXGA

2EQGF

Y,H

APPLICATION OF THE MATHEMATICAL MODEL OF ENERGY MANAGEMENT BALANCE

88

UNKNOWN VALUES

Input-output matrices 

Vector of the production and supplies

Q,X,F,F,A,A PGPG

D,P,GRelation between production and power

rating

dDDD

dPPP

dGGG

0

0

0

0inini

0inini

0inini

D,P,G - duration functions

89

niD

iD

niG

iG

niPiP

0

maxiΩ

90

CONTENTS OF THE ALGORITHM

Elaboration of the set of variants. Determination of the structure of binary input-output matrices and structural analysis. Determination of duration functions of the demand for energy carriers. Determination of the elements of the input-output matrices. Determination of the optimal power-rating and capacity of the engine and energy equipment.

91

CHOICE OF THE STRUCTURE OF THE INPUT-OUTPUT MATRIX AND ITS STRUCTURAL ANALYSIS

Scenario of energy management

General list of energy carriers.

Energy carrier – major products. 

Project. 

Subset of designs. 

Binary input-output design matrix.

t

tt

t mmandmN

Elaboration of a set of variants of the designed energy management of an industrial plant

92

General list of energy carriers, structure vector Tb of the demand of energy carriers for the technological subsystem and set of energy equipment and engines

(Example concerning ironworks)

Energy carrier Tb Equipment or engines SymbolSteam boilers fired w ith blast-furnace gas and coal U1

Medium -pressuresteam

0 Steam boilers fired w ith blast-furnace gas and oil U2

Extraction turbine (steam extraction nozzle 0.8 MPa) U3

Back-pressure turbine (exhaust pressure 0.8 MPa) U4

Low-pressuresteam 1

Electricenergy

1

1Pressure-reducing valve 3.7/0.8 MPa U5

Low-pressuresteam 2

0 Pressure-reducing valve 0.8/0.12 MPa U6

Blast-furnace turboblow ers U7Blast 1 Electrically driven blast-furnace blow ers U8

93

Energy carrier Tb Equipment or engines SymbolHeat 1 Heat exchangers U 9Soft w ater 0 W ater - softening plant U 10

Boilers w ater 0Deaerating heater and pumping station of boiler

w aterU 11

Industrial w ater 1 Pumping station of industrial w ater U 12Compr essed air 1 Air com pressors U 13Pow er coal 0Fuel oil 0Natural gas 1Blast- furnacegas

1

The technological subsystem consists two branches: blast furnaces, converters and electric furnaces.

The binary vector Tb describe the structure of the demand for energy carriers by the technology subsystem. Zero elements of this vector concern energy carriers which are consumed only in the energy subsystem.

On the basis of vector Tb a general list of energy carriers has been set up.

94

Projects and designs

t Project p Design1 Medium -pressure steam 1

2U 1U 2

2 Low-pressure steam 1Electric energy

34

U 3 Λ U 5 Λ D 4U 4 Λ U 5 Λ D 4

3 Low-pressure steam 2 5 U 64 Blast 6

7U 7U 8

5 Heat 8 U 96 Soft w ater 9 U 107 Boiler w ater 10 U 118 Industrial w ater 11 U 129 Compressed air 12 U 13

In the considered example eight variants of the energy system have been created

 For example:

131211109764541 U,U,U,U,U,U,U,DUU,U

95SCHEMATIC DIAGRAM OF THE ENERGY SYSTEM OF THE CONSIDERED IRONWORKS

96

For each variant 

The input-output matrix is set up

bG

bP

bG

bP

bb FFAAFA

Structural analysis 

Its aim “to obtain a structure near the upper triangular matrix” 

Three groups of energy carriers: 

“input-type”“centre-type”“output-type”

 Separation of strongly coherent subsystems

in the “centre-submatrix”

r

1s

sc

bb FAC

97

Intersection matrix

TCCW

98

Input-output binary matrix divided into blocks bb FA

99

001010000011010000010001000100000010

FAC1

1s

sc

bb1

011110000111010011010011010101000110

FAC2

1s

sc

bb2

100

011111010111010111010111010111010111

FAC3

1s

sc

bb3

011111010111010111010111010111010111

FAC4

1s

sc

bb4

101

Because:43 CC

we can write:

43 CCC

The matrix intersection W is deduced from the following equation:

TCCW

Matrix W has non-zero elements only in those places, where matrix C has non-zero elements as well as matrix . In the considered case matrix W takes the following form:

TC

000000010111000000010111010111010111

W

102

Input-output binary matrix transformed to a block-triangular matrix with a minimal number of elements of feedback character

bb FA

103

OBJECTIVE FUNCTION AND CONSTRAINTS

minKDGPDG

PIIIK

TDGPnDnG

nPDDDGGGPPPR

Global constraint 

BALANCE EQUATION OF ENERGY MANAGEMENT(matrix equation)

 Local constraint

iDiGiPi

ii

maxini

maxininini

IIIID

D

DGP

104

DECOMPOSITION OF THE GLOBAL OPTIMISATION PROBLEM

GPXEGAPAKL 1GPR

minDQEGFPF 1GP

In order to determine the vector the procedure of coordination must be known.

It is possible to prove mathematically that in the considered case Lagrange’s multipliers are equal to the unit costs of energy carriers.

Lagrange’s decomposition method is commonly known as the method of prices.

105

MATRIX METHOD OF CALCULATING THE UNIT COSTS AS A CO-ORDINATION PROCEDURE

IN THE OPTIMISATION PROBLEM

Optimisation problem 

PRELIMINARY DESIGN OF AN INDUSTRIAL ENERGY SYSTEM 

Lagrangian function – global objective function

PHYEXXEGAPAGAKL 21PGR

minDEQQEGFGFPF 21GP

106

T j

AUXILIARY DIAGRAM USED FOR THE FORMULATION OF THE OBJECTIVE FUNCTION OF THE ENERGY BRANCH

107

Objective function for the energy carrier “j” – local objective function

On the level of optimisation in successive iteration we can write:

constantDGPk jjjZj

constantQGfPfkkn

ji1i

n

ji1i

n

ji1i

jiiGjii

PjiZjUj

constantQGFHYEXGAk 2TZ

n

1iijUi

n

1ij

GijUi

n

1ij

PijUi

n

1iijZi

n

1ij

GijZi

n

1ij

PijZiRjj

minQkGfkPfk

XkGakPakK

108

After summation of the local objective functions including above mentioned equations we get:j

HYEXXEGAGAPAkK 21GPTZR

n

1jj

minDEQQEGFGFPFGP 2GP

Because:

n

1jjL

we can write:

TZk

109

CONCLUSION

The matrix method of calculating the unit costs of energy carriers is a co-ordination procedure in the decomposition algorithm of the mathematical optimisation model for the preliminary design of industrial energy systems

In the first approximation of the iterative procedure the vector kZ of the unit costs of energy carriers is assumed. Technical coefficients with feedback characteristics are assumed, too. Problems of the optimization of the particular energy branches are solved according to the sequence of energy carriers in the upper triangular “input-output” matrix. In strongly coherent subsystems (i.e. subsystems with feedback-type relations) the inner iterative loops are solved. The accuracy of calculating the technical coefficients, which have feedback characteristics, determines the end of iterations in the inner iterative loops.

Short description of algorithm

110

The determination of the optimum values of all the decision variablesin successive iteration is followed by a return to the level of coordination.

Then a corrected balance of energy carriers is set up and the corrected values of the unit costs of energy carriers are calculated by means ofthe matrix method.

In the next iteration the corrected vector of the unit costs of energy carriers is applied on the level of optimization of the particular energy carriers.

The accuracy of calculating the unit costs of energy carriers determines the end of iterations in the external iterative loop.

111

Procedure of co-ordination “Matrix method of calculating the unit cost

of energy carriers”Tk

LEVEL OF COORDINATION

Complex of energy managementSetting-up the energy balance

Calculating the unit cost of energycarriers

LEVEL OF OPTIMIZATION

Energycarrier 1

Energycarrier 1

Energycarrier 1......

11u k,k

22u k,k

. . . . . .

nun k,k

jjn P,P

jjn G,G

jjn D,D

112

Some selective results of determining the industrial energy structure of ironworks

in preliminary design(application of Lagrange’s decomposition method)

i Energy carrierRatio of unit

costs in the lastand first iteration

123456789

10111213

Electric energyPow er coalBlast-furnace gasIndustrial w aterSoft w aterLow-pressure steam 2Boiler w aterMedium-pressure steamLow-pressure steam 1BlastHeatCompressed airNatural gas

1111

1.362.962.262.892.761.812.593.81

1

113

RESULTS OF CALCULATIONS

Equipm ent or engines Power rating ornominal capacity

Steam boilers fired withblast-furnace gas andcoal

3 x 40 Mg/h

Back pressure turbine 4 MWPressure-reducingvalve 3.7/0.8 MPa

3 x 35 Mg/h

Pressure-reducingvalve 0.8/0.12 MPa

15 Mg/h

Blast-furnaceturboblowers

3.480 kmol/h4.25 MW

Boiler-water pumpingstation

4 x 45 Mg/h

Air compressors 2 x 446 kmol/h

114

i Power rating andcapacity

Annual productionand external

supplies123456789

10111213

4 MW

6 000 Mg/h40 Mg/h15 Mg/h

135 Mg/h120 Mg/h35 Mg/h

6 950 kmol/h

890 kmol/h

65 2851) MW h39 600 GJ4 819 TJ

47 800 Gg231 900 Mg53 300 Mg

561 300 Mg527 150 Mg

328 0502) Mg40 300 Mmol180 000 GJ2 940 Mmol376 300 GJ

1) - own production – 25 800 MWh2) - basic part of the production (back-pressure turbine -

306 000 Mg)

115