Jiří Jaromír Klemeš, Petar Sabev Varbanov and Peng Yen Liew (Editors)
Proceedings of the 24th European Symposium on Computer Aided Process Engineering – ESCAPE 24
June 15-18, 2014, Budapest, Hungary.© 2014 Elsevier B.V. All rights reserved.
Modeling Multi-period Operations using the
P-graph Methodology
István Heckla, László Halásza, Adrián Szlamaa, Heriberto Cabezasa,b, and Ferenc
Friedlera,*
a
Department of Computer Science and Systems Technology, University of Pannonia, 10
Egyetem utca, Veszprém, 8200, Hungary
b
Office of Research and Development, U.S. Environmental Protection Agency, 26 West
Martin Luther King Drive,Cincinnati, OH 45268, USA
[email protected]
Abstract
A new modeling technique is presented here for handling multi-period
operations in process-network synthesis (PNS) problems by the P-graph
(process graph) framework. Until now, the P-graph framework could only
handle single-period operating units. It means that the operating conditions and
the load of each unit remain unchanged throughout its operation. This
assumption is usually true for the chemical industry but may be false in
agriculture or in other areas where seasonal effects are important. Hence, the
current work proposes the notion of multi-period operation wherein the load of
an operating unit may vary from period to period. Subsequently, a modeling
technique is proposed to represent operating units in the multi-period operation.
The idea is to represent separately the physical body of an operating unit and the
operations in each period. Surprisingly, to achieve these tasks, there is no need
to dramatically change or augment the basic structure of the P-graph
methodology, e.g. with a new type of multi-period unit, but the already
available constituents are adequate.
Keywords: process-network synthesis, P-graph, multi-period, optimization
1. Introduction
A process system aims at producing certain products from a set of raw materials
through a sequence of processing steps. The major developments in this area
have been comprehensively reviewed by Sargent (2004). Usually, a multitude of
alternative feasible network structures is capable of producing the desired
products, because of the combinatorial nature of the problem. Normally, process
synthesis seeks the optimal network structure in terms of some objective
function. The determination of the optimal network structure is frequently
referred to as process-network synthesis (PNS) or flowsheet design.
The significance of PNS is highlighted by numerous publications in the
technical literature. The structural properties, especially the redundancy, of the
super-structures of the processes of interest have been explored (Farkas et al.,
2005). A novel representation, the state-task network, has been introduced
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I. Heckl et al.
originally for scheduling problems (Kondili et al., 1993). This representation
includes explicitly both the states (feedstocks, intermediates, and final products)
and the tasks (operations) as network nodes. The state-task network and stateequipment network has been applied to aid process synthesis (Yeomans and
Grossmann, 1999).
The P-graph framework aims to solve the PNS problem rigorously (Bertok et
al., 2013). The P-graphs are bipartite graphs, comprising one class of nodes for
materials, and the other, for operating units, as well as arcs linking them. The
framework constitutes three cornerstones: the representation of processnetworks with P-graphs; the five axioms stating the underlying properties of the
combinatorially feasible process-networks; and effective algorithms. These are:
the algorithm for the maximum-structure generation, MSG (Friedler et al.,
1993); the algorithm for solution-structure generation, SSG; and the algorithm
for the optimal-structure determination, ABB (Friedler et al., 1995).
A MILP mathematical model is generated automatically from the structural
model. It contains a continuous and a binary variable for each operating unit.
The former signifies the capacity of the unit, the latter marks the existence of
the same unit. The mathematical model is capable of determining the n-best
optimal solution.
The P-graph framework was developed originally for chemical processes.
Nevertheless, due to the versatility and flexibility of the P-graph framework, it
has been deployed for a wide range of problems involving synthesis, e.g.,
separation-network synthesis (Heckl et al., 2010), steam supply system (Halasz
et al., 2002), emission reduction (Klemeš and Pierucci, 2008), and the usage of
renewable energy sources (Lam et al., 2010) among others.
The current work aims at PNS using the P-graph framework involving
multi-period operations. It is presumed that steady-state operating conditions are
strictly maintained within each period, which nevertheless vary from period to
period to accommodate the change in demand and availability of the raw
materials. The current work also proposes a modeling technique for
representing operating units involved in multi-period operation.
2. Comparison of the single and multi-period operations
Multi-period operation is important on many areas, including economic
portfolio selection (Wu et al., 2014), supply network synthesis (Čuček et al.,
2013), and district energy systems (Fazlollahi et al., 2013). A motivational
example is presented herein to illustrate the notion of multi-period operation in
PNS problems. The example involves a single task, the peeling of 30 tons of
apples. It is assumed that the weight of peels is negligible.
In single period operation, the feed is unchanging, 2.5 t in each month. In
multi-period operation, there are three periods, October - February, March July, and August - September, with 5, 10, and 15 t/period demands,
Modeling Multi-period Operations using the P-graph Methodology
3
respectively. Both the single and the multi-period operations produce 30 tons of
peeled apples in a year. The cost data of this specific example are compactly
expressed algebraically in the following equations;
cc = 140 + 20m
[€]
(1)
acc = 140 / 10 + (20 / 10)m
[€/y]
(2)
oci = (6 + 3ai)pli, for all i
[€/y]
(3)
ai ≤ m, for all i
[t/y]
(4)
tc = acc + Σoci
[€/y]
(5)
where cc, acc, oci, pli, and tc stand for the capital cost, the annualized capital
cost, the operating cost in period i, the period length of period i, and the total
cost, respectively. Moreover, ai and m signify the actual deployed, or simply
actual capacity, in period i, and the maximal capacity, respectively.
Table 1. Pertinent data for the single and multi-period operation.
Period length, pli [y]
Monthly feed, mfj [t/month]
Periodic feed, pfi [t/period]
Actual capacity, ai [t/y]
Maximal capacity, m [t/y]
Ann. capital cost, acc [€/y]
Operating cost, oci [€/period]
Total cost, tc [€/y]
Single
period
Period 1
1
2.5
30
30
30
74
96
170
Multi-period
Period 1
5/12
1
5
12
17.5
Period 2
5/12
2
10
24
90
194
32.5
290
Period 3
2/12
7.5
15
90
46
In the single-period operation the maximal structure contains only the feed, the
peeler unit, and the peeled apple. The only question is the value of the capacity,
x, of the peeling machine. During the single period the capacity of the apple
peeler, is always fully utilized and matches the yearly demand 30 t/y. For the
payback period of 10 years, the annualized capital cost, acc, the operating cost,
oc1, and the total cost, tc, are calculated from Eqs. (2), (3), and (5), respectively.
The results are summarized in Table Error! Reference source not found..
The multi-period operation involves 3 periods of 5, 5, and 2 months. Unlike the
single period operation, the maximal capacity of the apple peeler, m, is not
necessarily fully deployed in all periods. The actual capacity in each period, ai,
can be calculated by dividing the periodic feed, pfi, by the length of the
corresponding period, pli.
ai = pfi / pli
[t/y]
(6)
Obviously, m = a3 = 90 t/y to accommodate the capacity needed in the peak
period operation. Naturally, the capital cost, cc, is concomitant only with E, and
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I. Heckl et al.
the operating cost, oci, needs to be computed for an individual period on the
basis of the actual capacity, ai.
The multi-period operation is more expensive than the single period one, 170
€/y vs. 290 €/y, because the capital cost of the latter is significantly greater. On
the other hand single period operation requires larger storage capacity which
would increase its cost.
A1
A2
A3
MU
m = 90
a1 = 12
a2 = 24
a3 = 90
E, x4 = 90
5/12
5/12
M1
1
1
1
M2
1
O1
x1 = 5
PA1
5 t/period
2/12
M3
1
1
1
O2
x2 = 10
PA2
10 t/period
1
1
O3
x3 = 15
PA3
15 t/period
Figure 1. Representation of a multi-period unit, MU.
3. The P-graph model of the multi-period operation
The multi-period operating unit can be modeled with traditional operating units.
First, three distinct operating units, O1, O2, O3, are needed to represent the
operations in each period. The capacities of these units, x1, x2, and x3,
correspond to the actual capacities of the multi-period unit, a1, a2, and a3.
Second, an operating unit, E, is needed to represent the equipment, i.e., the
physical body of the multi-period unit. Its capacity corresponds to the maximal
capacity of the multi-period unit, m. New materials, M1, M2, and M3 and the
proposed structure ensures that the capacity of the equipment is the same as the
largest of the capacities of the operations, see Figure 1.
Practically, the operations and the physical realization of a multi-period unit
have been separated. The operating cost of the multi-period unit is associated
with the operations; the capital cost is associated with the equipment.
Algorithm ABB can calculate the capacities and the costs for the suggested
structure. The results are in accord with those obtained with the manual
calculation as discernible in Table 1, thus indicating that the proposed method is
indeed valid.
Modeling Multi-period Operations using the P-graph Methodology
5
4. Case study: St. Margarethen
To explore these ideas with a practical case, we consider St. Margarethen, a
small town in Burgenland, Austria. The town considers environmental
protection seriously, thus, the use of alternative energy sources are carefully
weighted (Gwehenberger and Narodoslawsky, 2008). Corn is a typical
agricultural product of the area; consequently, the drying of corn and the use of
corn cubs are important tasks. The aim is to design a process-network capable
fulfilling these tasks, as well as produce heat for greenhouses. The available
resources are biogas, forest wood, fast growing wood, heat, and corn.
Processing units are present in this area such as biogas units, dryers, and so on.
The dryer can be used for both corn and wood drying.
Corn cannot be stored for long because it spoils quickly. As a result, the
demands changes within the year, and a design method is needed which can
take this into account. The PNS framework with the proposed representation for
multi-period operation is such a method.
Three time periods are defined, one for winter, one for summer, and one for the
corn harvest (termed as corn period). The lengths of the first two periods are
3600 hours, and the length of the last period is 1440 h. The P-graph model of
the problem is too large to visualize here given the space limitations. The model
contains 18 operating units and 21 materials. The biogas production and the
drying work in multi-period operation. The details of the model and all results
are available from the authors on request. The solution of the problem with
algorithm ABB results in objective value of 366,195 €/y. Until now the P-graph
framework could not be used in situations where a single unit, e.g., the dryer,
could be used for two purposes, e.g., drying either corn or wood but now it can.
5. Conclusions
The concept of multi-period operation of an operating unit in the P-graph
framework has been introduced. This is important because often in practice raw
materials cannot be stored indefinitely or the majority of the product is required
in a certain time period. A modeling technique has been proposed to represent
multi-period units. The main idea is to differentiate the operations of a
multi-period unit into a sequence of different time periods using the same
equipment. The elegance in this modeling technique is that: (1) the multi-period
units are represented by traditional operating units, and (2) the period length can
be adjusted as needed. The main result of this work is, therefore, the major
qualitative progress in the development of P-Graph theory rather than a
quantitative improvement in any particular model.
Acknowledgments
We acknowledge the financial support of the Hungarian State and the European
Union under the TAMOP-4.2.2.A-11/1/ KONV-2012-0072.
This research was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001
„National Excellence Program – Elaborating and operating an inland student
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I. Heckl et al.
and researcher personal support system” The project was subsidized by the
European Union and co-financed by the European Social Fund.
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