PRODUCTION AND ENERGY OPTIMIZATION IN AN INDUSTRIAL COMPLEX: A GENETIC ALGORITHM APPROACH A. Santosõ õõ, A. Douradoõõ Instituto Superior de Engenharia do Instituto Politécnico de Coimbra Quinta da Nora, apartado 4065, 3030 Coimbra, Portugal õõ Departamento de Engenharia Informática da Universidade de Coimbra Pólo II, Pinhal de Marrocos, 3030 Coimbra, Portugal e-mail:
[email protected] õ
Keywords: Production Scheduling; Genetic Algorithms; Pulp and Paper ABSTRACT The pulp and paper industry exhibits nowadays an increasing need for efficient management of all those factors which may provide a reduction in operation costs. This leads to the necessity of an adequate optimization system which enables the generation of one or more optimum strategies with several objectives that fulfil the required restrictions. Herein we propose a system that looks for the optimum assignment for all the production sections in a particular mill of the kraft pulp and paper industry in order to optimize energy costs and the production rate changes using a genetic algorithm for the optimization task. This system is intended to fulfil all programmed or forced maintenance shutdowns as well as all the imposed reductions in production rates. INTRODUCTION A major number of continuos production industries can be described by a group of departments responsible for some specific operations and separated by intermediate buffers. The kraft pulp and paper is one of these industries. Consider the notation of fig. 1, suggested in [1], where buffer j, with level x j ( j = 1,K , m) , receives
the production from the department i, working at rate ui ( i = 1,K , n) units, and delivers the raw material to department i+1, working at rate ui+ 1 units; b j ,i + 1 ⋅ ui + 1 units are consumed from buffer j for each unit of production ui+ 1. This work is based on the case study of the flowsheet of the Centro Fabril de Viana da Portucel, E. P., represented in fig. 5. ui
xj
ui + 1
Fig. 1. Flowsheet example with 2 departments and 1 buffer. Pulp mills are rather complex systems where shutdowns and disturbances propagate and influence very easily all the mill. This will lead to mass and energy losses due to chemical incorrect dosing and consequently to production losses and quality breakdowns. A production control system must then follow the mill’s actual state so that the production targets are achieved. During the last decade the optimization area has undergone a considerable growth in such a way that many of engineering problems can now be solved with the aid of non-deterministic methods. Genetic algorithms are included in the probabilistic methods’ family which, generally, are considered more robust than the random ones, since they incorporate their own search techniques. In this work a GA approach is used based on multicriteria constraint-handling techniques. Several methods exist for handling constraints by genetic algorithms in optimization problems. The technique used here [5] is based on preserving feasibility of solutions using specialized operators which are closed on the feasible part of the search space. These operators, namely crossover and mutation, transform feasible solutions into other feasible solutions. The basic idea behind this method lies in (i) the elimination of the equalities present in the constraint set and in the (ii) use of specific operators which guarantee that individuals are kept in the feasible space. In the next sections a set of considerations will be presented as for the production scheduling as for
the genetic algorithms’field which will lead to a simulation of the overall system. MATHEMATICAL FORMULATION The stock equation (1) represents the overall model for the production co-ordination where B is the mass balance matrix, control u and state x are the departments production rates and the intermediate level buffers, respectively. T is the discretization interval and N is the number of discretizations in the planning horizon. x( k + 1) = x( k ) + B ⋅ T ⋅ u( k ) , k = 0,K , N − 1
(1)
Both control u and state x are physically constrained by equations (2) and (3). 0 ≤ u min ( k ) ≤ u( k ) ≤ u max ( k ) ≤ R
(2)
0 ≤ x min ( k ) ≤ x( k ) ≤ x max ( k ) ≤ K (3) The energy production and consumption in the mill can be represented by an energy balance matrix, issued from a careful study of the energetic balances in each department. The total consumption of electrical energy is expressed by the equation (4) EE total = BEE ⋅ T ⋅ u
(4)
where BEE is the energy balance matrix. There are some issues that should be attained in the production scheduling, as stated in [2][3]: • the final productions must be accomplished in the planning time horizon, since delays in delivery times lead to economic losses; ‚ the storage capacities should be used in order to avoid over and underflows and also to ƒ avoid production rate changes, since these are responsible for additional costs due to efficiency breakdowns in almost all departments; „ all the maintenance shut-downs should be carefully planned which will benefit the entire mill; … the end of a schedule plan should be seen as the beginning of the next one and therefore the final storage levels should be pre-determined; † some attention should be paid to the energy consumption since the pulp and paper industry is highly energy demanding. The mathematical formulation must take into account the aspects mentioned above. From these, it is rather essential to distinguish between an objective and a constraint. From the above statements, it is seen that in this problem two criterias are needed, given by equation (5): Obj1 = min
N− 1
∑ { BEE ⋅ T ⋅ u( k ) }
k= 0
and Obj2 = min
N− 1 n
∑ ∑
k = 1 i= 1
ch ( k , i )
(5)
where ch ( k , i ) , as stated in [4], is the production rate change function (department i and instant k) defined in equation (6). 1 ⇐ ch ( k , i ) = 0 ⇐
ui ( k ) ≠ ui ( k − 1)
ui ( k ) = ui ( k − 1)
(6)
The formulation will be complete with a constraint set definition: Ÿ the accomplishment of final production, during the planning time horizon, must corroborate equation (7) x mpap ( N − 1) − x mpap ( 0) = K fpap (7) where x mpap stands for the paper machine buffer level and K fpap represents the finished paper intended production; Ÿ the planned maintenance shutdowns and the production restrictions expressed by equation (2); Ÿ the minimum and maximum safety limits of all storage buffers as stated in equation (3); Ÿ the buffers’final state which should be pre-determined, that is: x( N ) = x final
(8)
where x final represents the intended buffers’final state; Ÿ the contracted electrical power, which is time variant, should not be exceeded, that is: EE EDP ( k ) ≤ Pc ( k ) (9) where Pc ( k ) is the contracted power limit at instant k and EE EDP can be computed by equation (10) EE EDP = EE total − EE turbogenerator (10)
where EE turbogenerator is the electrical energy production of the turbogenerator. The mathematical formulation of the problem is not practical to traditional optimization techniques. Other approach must be sought for. THE GENETIC ALGORITHM The GAs have been used particularly in single objective problems but, nevertheless, most of the practical applications exhibit more than one objective to attend to. In this work the Pareto ranking method is used in order to properly select the next generation. This technique, which makes use of the definition of Pareto optimality, was first introduced by [6] and later redefined as a slightly different scheme in [7]. As proposed by Fonseca, an individual’s rank corresponds to the number of individuals in the current population by which it is dominated and, therefore, the dominated individuals are given a worse chance for reproduction. This process ends with the fitness assignment by interpolating from the best individual to the worst according to an exponential function, but possibly of other types. Here it was used the function expressed in equation (11) c − 1 P− i fi = P c ; i ∈ { 1,K , P} (11) c − 1 where P is the rank of the best individual and 0 < c < 1 is a constant. The crossover and mutation operators employed in this algorithm were chosen among those found in the literature which, by simulation, proved to be the set with the best convergence time and with the best diversification in the trade-off surface. The uniform crossover, based on [8] and [9], where, in instant k, 2 a( i )
vectors with dimension m, x ka and xkb , exchange each other genes i, that is x k i = 1,2, K , m , with probability p. Fig. 2 represents this crossover. a( 1)
p
a( 2)
xk
p
xk
xk xk
M a ( m)
xk
M
for
b( 1) b( 2 )
M p
b( i )
and x k
M b( m)
xk
Fig. 2. Uniform crossover with probability p The mutation phase is formed by a set of 4 strategies: uniform, boundary, non-uniform [5] and exchange mutations. Let C = ( c1 , K , ci , K , cl ) be a chromosome of length l and ci ∈ [a i , bi ] be the gene to whom the mutation operator will be applied resulting in gene ci′; then in the uniform mutation ci′ is a random value, according to a uniform probability distribution, from [a i , bi ] . In the boundary mutation ci′ is either ai or bi , with equal probability. In the non-uniform mutation, if g max is the maximum number of generations, ci′ is given by equation (12) c + ∆ ( k,b − c ) ⇐ α = 0 i i i ci′= ci − ∆ ( k , ci − a i ) ⇐ α = 1
(12)
b where α ∈ { 0, 1} is a random binary digit, ∆ ( k , y ) = y ⋅ β ⋅ 1 − g k , β is random number from interval max [0,1] and b is a parameter determining the dependence degree in the number of generations. Finally the
exchange mutation where 2 consecutive genes ci and ci+ 1 exchange each other. This last type can be seen as a particular case of uniform mutation where interval [a i , bi ] is simply ci+ 1 and [a i + 1 , bi + 1 ] is ci . The stochastic universal sampling is used in this work since it is considered the standard algorithm for sampling which exhibits null distortion and minimum spread. For the reinsertion the elected mechanism was the generational reproduction [10] where all the population is replaced in each generation. The scheme of sharing was introduced in [11], known as fitness sharing, and its main purpose is the population distribution in a set of niches of the search space. With this procedure the existence of similar individuals are avoided which denunciates the redundancy, enemy of diversity. Equation (13) represents the shared fitness function where nni is the niche number of individual i and given in equation (14). f
f ishare = nni i nni =
∑
(13)
(
Sh d ( i , j )
j∈ P
)
(14)
Function d ( i , j ) enables the computation of the distance between individuals i e j and Sh ( d ) is the sharing function as expressed by equation (15). σ share represents the niche radius which, as stated in [7], can be determined by equation (16) where n is the number of objectives, m and M are the minimum and the maximum of all objectives from the non-dominated set and α share is a positive real. α share Sh ( d ) = 1 − d σ share ( ) Sh d = 0 n
N ⋅ σ share n − 1 −
∏ ( Mi − i= 1
mi + σ share ) − σ share
n
⇐
d ≤ σ share
⇐
d > σ share
∏ ( Mi − i= 1
(15)
mi ) = 0
for σ share > 0
(16)
Once the sharing scheme is applied to the population, the crossover between individuals belonging to different niches may result in descendents in any niche. The mating restriction scheme [12] involves the parameter σ mate which is quite similar to σ share . The simplest mechanism using this approach is the mating radius which chooses for second progenitor the individual from the mating pool in a distance less than σ mate from the first progenitor. If none is in this situation then a random individual is chosen. SIMULATION RESULTS With the simplifications introduced before, scheduling of 3 from the 10 departments of the mill can be determined subsequently and, therefore, the resulting scheduling problem is formed by 7 departments. A discretization interval of 4 hours is used in a planning horizon of 48 hours which leads to 84 variables in the system. Each chromosome is then coded as real multiparameters constructed from the concatenated codes. The population is composed by 50 individuals and the initial ones are feasible and random generated. The initial and final buffers’ state are imposed to be 50% of their maximum capacity and the final state for the finished paper to be 90%. It is also imposed a shutdown in the paper mill during the third discretization interval and a reduction to 30% in the causticizing during the second discretization interval. Due to the limitations of the floating point representation a change in a production rate (equation (6)) is considered only if greater than 2% of the maximum. In fig. 3 it is presented the evolution of the best individual in the population along 100,000 generations. The fig. 4 depicts the population in 3 generations and the cumulative trade-off surface. Finally in fig. 5 it is represented the solution marked in fig. 4, in generation 100,000, being one of the possible solutions from the optimal Pareto set.
CONCLUSIONS This work is a contribution to the development of a computerised management system to the mass and energy production with application to a kraft pulp and paper mill. To solve this problem an evolutionary procedure was used, namely the genetic algorithms, since the dimension of this problem isn’t favourable to traditional methods. Nevertheless the main obstacles were the multiobjective characteristic and the presence of a high order constraint set and for that the Pareto ranking method and a technique which preserves the feasibility of the solutions were used. In agreement with other studies [13], these methods and the operators mentioned above (crossover, mutation, sharing and mating restriction) were those that revealed the best convergence time and the best diversification in the trade-off surface. Although this system has been applied to a linear problem in one objective (Obj1) it was easily extended to all kind of criteria since the second objective (Obj2) is strongly non-linear. In this way, if this system had been designed using a conventional optimization technique its functionality would had been compromised by the quite diverse nature of the objective functions. If a non-linear component would be present in the constraint set, the overall system could easily be adapted to this new situation using the proposal in [14]. In this way, the technique broached in this paper exhibits a flexibility otherwise achieved by traditional optimization methods. ACKNOWLEDGEMENTS The authors express their gratitude to Eng. José Luis Amaral (Portucel-Viana) for the used plant data and information and to Dr. C. M. Fonseca for the useful discussion. REFERENCES [1]
[2] [3] [4]
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Algoritmos Genéticos. Master Thesis (in Portuguese), Dep. de Engenharia Informática da Universidade de Coimbra, 1996. [14] Z. Michalewicz and G. Nazhiyath. Genocop III: A Co-evolutionary Algorithm for Numerical Optimization Problems with Nonlinear Constraints. In D. B. Fogel, editor, Proceedings of the Second IEEE International Conference on Evolutionary Computation, pp. 647-651, IEEE Press, 1995. 141.5
55.0
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Obj2
Obj1
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139.0 0
20,000
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Fig. 3. Evolution of Obj1 and Obj2 along the 100,000 generations. 60 58
38
generation 100
28 generation 10,000
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generation 100,000
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Obj2
Obj2
Obj2
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140.1
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140.7
18 138.9 139.1 139.3 139.5 139.7 139.9 140.1 140.3 Obj1
Fig. 4. The population and the cumulative trade-off surface in generations 100, 10,000 and 100,000.
Fig. 5. One solution from the trade-off surface in generation 100,000.
