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Transcript of 1 MATHEMATICAL MODELLING OF INDUSTRIAL ENERGY SYSTEMS WITH OPTIMIZATION PROBLEMS OPTI ENERGY Andrzej...
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: [email protected]
Summer School on"Optimisation of Energy Systems and Processes"
Gliwice, June 24 - 27, 2003
2
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.
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 PLANTInterbranches flows
E lectr icen erg y
F eedw a ter
H ig h -p ressu re
stea m
L o wp ressu re
stea m
C o o 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
...
G
G
n
2
1
G
supplies, external of vector-n
D
...
D
D
n
2
1
D
,nconsumptioof
tscoefficientechnical ofmatrix
a...aa
............
a...aa
a...aa
nn2n1n
n22221
n11211
G
13
production final of vector-n
K
...
K
K
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.
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
Main product By -productionEnergycarrier
peakpart
basicpart
Energy subsystem Technologicalsubsystem
Externalsupply
. . .i
. . .
. . .G i
. . .
. . .P i
. . .
. . .
. . .
. . .. . .D
i. . .
jij
Pijj
Gij QP jfGf
kikkik QGf
. . .
. . .
18
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....-1
10
0....0
S
11
11
1
INPUT
Vii
i - relative losses of energy carriers
21
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.
22
ALGORITHM FOR THE CALCULATION OF EXERGY LOSSES IN AN ENERGY AND TECHNOLOGICAL SUBSYSTEM
e
T
e
DTT bB YGEFA
23
e
T
tt
T
u
TT
S
T
e
TTD
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.
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.
i
deii
ueii
Peii
Geii
ei HwDeLwPwG
w
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
eu
T
G
1D
eG
1
e
1DT
wQwS
w
GF
YXGA
T
G
T
G
T
G
e u
T
P
1
e P
1
e
T
P
1D wQwSw DT
P 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
P
D
PP
D
SPZ
T
P
D
P
U
T
G
D
GG
D
SGZ
T
G
D
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.
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.
31
minKK TsDssDGe DDDG
eK - variable operating costs,
G - row vector of the variable unit costsof operation,
D - row vector of the unit costs of basicpart 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.
32
Inequality constraints
NPG
Global constraints – balance equations
s
s
D
DD
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 INDUSTRY
Specific exergy of energy carriers, raw materials, technological main products and by-products (selected results of calculations)
Energy carrier Unit Specificexergy
Raw 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
dT
Tln
TT
T1Q
Q1
bp
h
ph
ot
h
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
GFAFAFAE
G
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 carriersBlast
High-purity oxygen
Compressed air for the
oxygen plant
Low-pressure steam
Medium-pressure steam
High-pressure steam
Demineralised w ater
Compressed air
Industrial w ater
Electric energy
Blast-furnace gas:
- production
- consumption
Coke-oven gas
Natural gas 1
Natural gas 2
Pow 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 cooling
Unit Majorprodu-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
ne
fech
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 P
PlnTRnE g
ngngcumEel
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
DDGFFAA
FAIBΔZ
(10)
Resources of waste energy:
elcumE
el
kcumE
k
kflchDZ
E
(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 CALCULATIONS
Results 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/Mg r.p.MJ/Mg r.p.
+86.3-41.8-9.8
-98.9-114.4-0.021
-7.1-6.4
--
+13.2-22.70.0
+83.3---
0.0--
-------
0.0-380.0
-4.0
66
Changes due to the recovery turbineEnergy carrier Unit Major
productionBy-
productionExternalsupply
Soft w aterDemineralised w aterLow-pressure steamMedium-pressure steamHigh-pressure steamCompressed airIndustrial w aterElectric energyPow er coalNatural gas
kg/Mg p.i.kg/Mg p.i.kg/Mg p.i.kg/Mg p.i.kg/Mg p.i.
kmol/Mg p.i.Mg/Mg 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 IES
C O EFFIC IEN TS O F C U M U LATIVEEN ER G Y C O N SU M PTIO N
IESPROCESS
OFWASTE
EN. REC.
C O EFFIC IEN TS O F C U M U LATIVEEN ER G Y C O N SU M PTIO N
LEVEL O F IN D U STR IALEN ER G Y SYSTEM
ELEM EN TS O F IN VER SE M ATR IXVER SU S
"IN PU T-O U TPU T" M ATR IXO F IN D U STR IAL EN ER G Y SYSTEM
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
kjij
jij
n
j
iik
P
kjij
jij
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
lkjij
jij
n
ljii
ii DUPUGUPGV
Decomposition of the calculations:
i
iii
i
ik
i
ikP
lki
iPHVYU
ZG:
11
1
0
GZGUG:
i
i
jij
jij
n
nii
i
1
2
11
76
.
ik
i
ikP
kjij
i
jijn
njii
iii
jij
jij
n
ji
n,......,ni
,UZ
PUPZ
PD
CHYPZGZ
21
1
2
11
1
1
1
1
11
11
jij
jij
jij
jij
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 TisDisiDiDiiPiiGi
n
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.
eK
e
Pi
Di
siD
siD
TK
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
jij
jij
jij
jij
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
YHEXXEGAPAGA
DEQQEGFPFPFPG
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,G
Relation 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
t
t
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 U 1
Medium -pressuresteam
0 Steam boilers fired w ith blast-furnace gas and oil U 2
Extraction turbine (steam extraction nozzle 0.8 MPa) U 3
Back-pressure turbine (exhaust pressure 0.8 MPa) U 4
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 U 6
Blast-furnace turboblow ers U 7
Blast 1 Electrically driven blast-furnace blow ers U 8
93
Energy carrier T b Equipment or engines SymbolHeat 1 Heat exchangers U 9
Soft 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 12
Compr essed air 1 Air compressors U 13
Pow 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 1
U 2
2 Low-pressure steam 1Electric energy
34
U 3 Λ U 5 Λ D 4
U 4 Λ U 5 Λ D 4
3 Low-pressure steam 2 5 U 6
4 Blast 67
U 7
U 8
5 Heat 8 U 9
6 Soft w ater 9 U 10
7 Boiler w ater 10 U 11
8 Industrial w ater 11 U 12
9 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
b
G
b
P
b
G
b
P
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
s
c
bb FAC
97
Intersection matrix
TCCW
98
Input-output binary matrix divided into blocks bb FA
99
001010
000011
010000
010001
000100
000010
FAC1
1s
s
c
bb
1
011110
000111
010011
010011
010101
000110
FAC2
1s
s
c
bb
2
100
011111
010111
010111
010111
010111
010111
FAC3
1s
s
c
bb
3
011111
010111
010111
010111
010111
010111
FAC4
1s
s
c
bb
4
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
000000
010111
000000
010111
010111
010111
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
IIII
D
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
jii
G
jii
P
jiZjUj
constantQGFHYEXGAk 2
T
Z
n
1iijUi
n
1ij
G
ijUi
n
1ij
P
ijUi
n
1iijZi
n
1ij
G
ijZi
n
1ij
P
ijZiRjj
minQkGfkPfk
XkGakPakK
108
After summation of the local objective functions including above mentioned equations we get:j
HYEXXEGAGAPAkK 21GP
T
ZR
n
1jj
minDEQQEGFGFPFGP 2GP
Because:
n
1jjL
we can write:
T
Zk
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
Equipment 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