Conceptual optimisation of utility networks for operational variations—II. Network development and optimisation

PII:

Chemical Engineering Science, Vol. 53, No. 8, pp. 1609—1630, 1998 ( 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009–2509(97)00432–6 0009—2509/98 $19.00#0.00

Conceptual optimisation of utility networks for operational variations—II. Network development and optimisation S. P. Mavromatis and A. C. Kokossis* Department of Process Integration, UMIST, P.O. Box 88, Manchester M60 1QD, U.K. (Received 22 January 1997; in revised form 7 November 1997; accepted 25 November 1997) Abstract—The paper proposes a new approach for the optimisation of utility networks. The approach makes use of the targeting and conceptual developments presented in Part I, and the available knowledge is exploited to accomplish a decomposition scheme that accommodates for both simple and complex units. The decomposition establishes a common modelling basis, and provides for variable efficiencies in the operation of the steam turbines. Its principles are exploited to set up a network superstructure that is comprised by a managable number of generic elements. The units are developed to take into account operational variations against which the optimisation objectives determine the component sets that need be used. As a result of the insights thrown into the formulation and the targeting models used, the optimisation effort is simply facilitated by a Mixed Integer Linear Programming formulation whose application is illustrated with several examples and an industrial case study. ( 1998 Published by Elsevier Science Ltd. All rights reserved. Keywords: Process design; utility systems; steam turbine network synthesis; operational variations.

INTRODUCTION

The optimal selection of the steam levels has been discussed in Part I. Once the steam levels have been selected, the design can proceed with the development of the network layout. This task comprises a large combinatorial problem, as a result of the large number of design alternatives available. Candidate systems involve layouts of simple and/or complex turbines, arranged in series or parallel. Each alternative configuration results in a different overall efficiency at a different capital cost. The optimal turbine network has to be determined with respect to the turbine type (simple or complex), number of turbines, turbine size and allocation of units. In this paper a methodology is discussed for addressing the design of the steam turbine network. The methodology consists of three stages that include: (i) the development of design components to be used for the synthesis structure. Unlike general purpose applications, the development is contextual to the problem in that it systematically processes the available

*Corresponding author. Tel.: 0161 200 4384; fax: 0161 236 7439; e-mail: [email protected].

information to set up an effective system representation; (ii) the optimisation of the above structure with reference to its individual components so that to minimise efficiency losses due to the variation in the operation. The optimum model is a result of the THM application and is formulated as an MILP model; and (iii) the analysis and synthesis of complex turbines, as the results from the optimisation stage can be further processed to systematically reveal compact utility networks that facilitate the objectives and analyse the results for competitive alternatives. At the first stage, a decomposition strategy allows the reduction of the numerous alternative configurations into a single superset of design components. In addition, a preliminary analysis allows the discretisation of the design components of every expansion zone into units of specific sizes. This superset of component turbines forms the basis from which the optimum configuration is derived. The component superset is subject to optimisation in order to determine the optimum set of component turbines. This set features the optimum trade-off between the revenue raised by the production of power and the capital cost incurred with the installation of the units. In order to establish this trade-off, a simple mathematical optimisation model is developed. In case the process requirements are expressed as heat loads, the solution

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of the problem involves an iterative optimisation procedure for the accurate conversion of the heat loads into mass loads. The optimum component set consists of a number of simple component turbines of optimal sizes for every expansion zone. In the last stage of the proposed methodology, the component turbines are synthesised into practical, complex turbines. The synthesis procedure relies on the optimum operation schedule of the components and may result in several equivalent configurations, from which the designer may choose the final structure of the turbine network. Alternatively, the synthesis procedure can be automated, in which case consideration can be made of the reduction in the capital cost when merging two units. The approach is illustrated with a number of examples used to present the application of the design methodology to several problem cases, including the allocation of steam turbines as process drivers. The need to consider the effect of the process variations on the optimum design of the turbine network is highlighted through an industrial case study. DESIGN COMPONENTS OF THE SYNTHESIS STRUCTURE

The cogeneration potential available between the steam levels of an industrial site can be exploited through several combinations of turbine networks and layouts. Figure 1 shows the case of a site repre-

sented schematically by the site utility grand composite curve, for which several alternative turbine networks are considered. Different types of turbines may cross each expansion zone and realise the power cogeneration potential. A specific area of cogeneration potential, as the one appearing shaded in Fig. 1, can be taken up by complex or simple turbines, arranged in series or in parallel. Each turbine taking up the specific area may cover additional areas from other expansion zones. The position and extend of these areas give rise to numerous combinations. Even for moderate problems, the number of possible combinations is practically impossible to handle. ¹urbine decomposition As discussed in Part I, a complex turbine is equivalent to a cascade of simple turbines. As a consequence, complex turbines are decomposed into component cylinders, each taking up potential from a single expansion zone, as shown in Fig. 2. The same principle applies to multi-stage simple turbines, that is turbines without passout capacity, that expand steam across several zones. Thus, multi-stage, simple turbines are also replaced by a cascade of component turbines, as seen in Fig. 3. In the case of complex turbines, the component cylinders are normally of different capacities, while simple multi-stage units consist of cylinders of the same capacity.

Fig. 1. A specific area of cogeneration potential may be exploited by several alternative turbines.

Fig. 2. Complex turbines are considered as a cascade of simple turbines.

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Fig. 3. Multi-stage, simple turbines are considered as a cascade of simple, one-stage turbines.

Fig. 4. All possible configurations are represented by a single superset of component turbines.

On the grounds of the equivalence expressed by Figs 2 and 3, all possible combinations of turbine layouts are reduced to a single superset of component cylinders as illustrated by Fig. 4. It is only this superset of design component that needs to be considered, in order to derive the optimum network. The number and size of the turbines in each expansion zone of the superset will depend on the specific problem and in particular the operation scenarios considered, as explained in the following section. Discrete elements The power output of the turbine is given by the expression: E"E.!9

A

B

6 M 1 ! . 5 M.!9 5

The above expression, relates the power output proportionally to the mass load, and with an inverse proportionality to the size of the unit. As both the load and the size of the equipment need to be optimised, straightforward modelling would result in an Mixed-Integer Non-Linear Programming (MINLP) formulation. The discussion that follows next, however, reveals that a linear (MILP) problem could instead be solved as the economic analysis of the turbine operation suggests discrete levels of importance for the sizes of the candidate turbines.

Fig. 5. The effect of part-load operation prevails over the increase of efficiency with turbine size.

Consider a component turbine that is required to operate under a single scenario. As the isentropic efficiency increases with size but decreases at part load, the highest efficiency is obtained when a turbine is sized to operate at full load under the specific demands, as shown in Fig. 5. A larger unit would result in lower efficiency, since the effect of part-load operation is generally stronger than that of a larger turbine size. Smaller units operating on a complementary basis will also involve lower efficiencies.

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Fig. 6. Typical plot for the capital cost of steam turbines vs the maximum power capacity [based on the correlation by Mroz (1979)].

Fig. 7. For the case of two operation scenarios, the design options involve three discrete sizes of turbines.

Hence, in terms of efficiency, the optimum turbine size for a single scenario is that which exactly matches the demands of the particular scenario. The capital cost of a turbine follows an exponential relation of the type presented in Fig. 6. The curved shape of the plot indicates that a single turbine would cost less than two or more turbines that are to accommodate the same total load. A larger unit, on the other hand, would incur a higher capital cost, in addition to the losses associated with part-load operation. Therefore, in cases of single scenarios, the optimum selection is one turbine sized to the demands of that scenario. Consequently a discretisation scheme is proposed for multiple scenarios so that turbines are sized to match the loads of every scenario as well as all their possible combinations. Consider the case of two

scenarios with the steam flows across a particular expansion zone illustrated in Fig. 7. The first combination involves the selection of turbine 1 sized to the largest demand, that of scenario B, while operating at part load under scenario A. This option features the lowest capital cost, but incurs a loss in the production of power due to the part-load efficiency at scenario A. Alternatively, turbine 2, can be installed to operate at full load under scenario A. This option achieves the highest overall efficiency, but in the same time requires the highest capital investment. Turbine 2 can instead be selected, along with a turbine sized to take up the remaining load for scenario B. As both turbines 2 and 3 are smaller than turbine 1, the combined efficiency during scenario B will be smaller than that of the first two options. This option, though,

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Fig. 8. The incremental capital and power cost will most likely favour the turbines of the discrete sizes.

achieves the highest efficiency for scenario A and requires a capital investment less than the second option. A final combination is to select turbine 2 only, in case scenario B does not take place for a period long enough to justify the installation of a turbine. Consider now a turbine smaller than turbine T2. Such a turbine will operate under both scenarios at full-load. According to eq. (21) of the THM, as its size increases, the power output will increase linearly as shown in Fig. 8(a), until the demands of scenario A are matched. On the other hand, the incremental capital cost decreases with size, so that on the grounds of the shape of the total cost curve, such a turbine should either be sized to the demands of scenario A, or not be selected at all. In other words, it is not beneficial to select a turbine smaller than turbine T2. As the size of the candidate turbine is increased beyond that of T2 and towards the size of T1, such a turbine would operate at full load only during scenario B, while it will have to operate at part-load during scenario A. The combined effect of these two operation modes on the basis of the THM is reflected by the slightly curved shape of the power savings curve of Fig. 8(a). Depending on the relative time fractions of the two scenarios, the curve may end at a point lower or above that for the demands of scenario A, as seen in Figs 8(a) and (b), respectively. In both cases though, despite the slight change of curvature, the trend of the total cost is to favour a turbine sized to the demands of either scenario A or scenario B. Apparently, specific combinations of the time fractions for the scenarios involved will alter the trade-off and the curvatures of Fig. 8 that may give rise to shapes shown in Fig. 8(c).

OPTIMISATION PROCEDURE

The optimisation problem involves the following definitions for sets and parameters: Sets ¹"Mi or j D component turbinesN, Z"Mz or s D expansion zonesN, O"Mk or l D operating scenariosN. Parameters M.!9 : maximum steam flow through turbine i of z,i zone z, Mt : total mass flow of steam across zone z under z,k scenario k, N : number of discrete turbines per zone, N : number of operation scenarios. k ¹urbine sets The design components of the superset are defined in terms of their discrete sizes. These sizes reflect on the discretisation scheme of the previous section and involve all scenarios and all combinations of incremental differences. The number, N, of the turbines needed to model the problem is N (N #1) N" k k . 2

(1)

The mass flows are determined iteratively using the strategy introduced in Part I, which generally requires just one iteration. The sizes of the discrete turbines are defined for each zone z, so that the first N units are k sized to the mass flows Mt of the scenarios z,k k"1,2, N across the zone z: k M.!9"Mt (2) z,i z,k

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Fig. 10. The component turbines are optimised independently for each expansion zone. Fig. 9. Discrete turbine sizes for the case of three operation scenarios.

where i"i(k)"k. The remaining units are sized to the combinations of the differences in the loads of the scenarios, M.!9"DMt !Mt D 1'k"1, 2,2, N !1 z,i z,l z,k k

The integer variables include variables that denote the:

(3)

(i) selection of each turbine in a zone, (ii) operation of a turbine under each scenario,

(4)

Equations (i) The power output of turbine i in zone z under scenario k is provided by the THM. In this expression the maximum power and maximum flow are parameters determined during the discretisation step:

where i is the following function of k and l: k(k!1) i"k(N !1)! #l. k 2

(ii) power output of each turbine (for each scenario), (iii) capital and operating cost.

For a case of three scenarios, six discrete turbine sizes are defined, according to eq. (1). These are presented in Fig. 9, with their numbering following the above sequence. For the case of 5 scenarios (it is quite unlikely that more than 5 scenarios are to be considered for practical purposes), 15 possible turbines need to be considered. In order to formulate the problem constraints, the following subsets of candidate turbines are defined: ¹A "M(z, k): turbines in zone z sized to match or z,k fill-up scenario kN ¹B "M(z, k): turbines in zone z that cross the zone of z,k additional load for scenario kN Model formulation The outcome of the decomposition step and the discretisation process, is a set of component turbines of specified sizes for each expansion zone. The turbines of each zone are decoupled since the selection of one does not interfere with the selection of the units in the other zones. Therefore, the optimisation of the component set can be conducted independently for every expansion zone, as illustrated in Fig. 10. The model is formulated with respect to continuous and integer variables. The continuous variables relate to the: (i) steam flowrates through turbines and throttle valves (for each scenario),

A

B

6M 1 z,i,k! w , E "E.!9 z,i,k z,i 5 M.!9 5 z,i,k z,i z 3 Z, i 3 ¹, k 3 O.

(5)

(ii) The mass balance across zone z for scenario k involves the steam through the turbines and the steam throttled through the let-down valves (in case the installation of a turbine is not cost effective). The balances yield: + M "Mx "Mt , z 3 Z, k 3 O. (6) z,i,k z,k z,k i|T (iii) The logical constraints relate the existence of the turbine to its operation and the limitations of the hardware: + w !y º)0 z 3 Z, i 3 ¹ (7) z,i,k z,i k|O M !w º)0 z 3 Z, i 3 ¹, k 3 O. (8) z,i,k z,i,k M )M.!9 z 3 Z, i 3 ¹, k 3 O. (9) z,i,k z,i (iv) The exclusivity constraints express the impossibility of co-existence of turbines of the same set: + y )1 z 3 Z, k 3 O z,i i|T Az,k + w )1 z 3 Z, k 3 O. z,i,k i|T Bz,k

(10) (11)

Conceptual optimisation of utility networks—II

(v) The economic functions refer to the savings achieved through the cogeneration of power and the capital cost incurred for the installation of the equipment: Cs "K + E F H, z 3 Z (12) z e z,i,k k i|T k|O Cc,505" + Cc y , z 3 Z. (13) z z,i z,i i|T (vi) The objective function used is the annualised cost of power generation: Cc,505 min C505" z !Cs , z 3 Z. z z p b

(14)

Remarks on the mathematical formulation The above formulation consists of linear constraints of continuous and integer variables, and comprises a Mixed-Integer Linear Programming model (MILP). The integer variables, y and w , are z,i z,i,k necessary to select only the optimum turbines in the capital cost function and switch the selected turbines off when they do not produce any net power output, respectively. The exclusivity constraints refer to the existence or operation of particular units. Constraints (10) express the logic that among the turbines that are sized to match or fill-up a scenario, only one can exist in the optimum component set. For example in the case of scenario B in Fig. 9, turbine 2 is sized to match the entire demand of this scenario, while the capacity of turbine 4 is such that it can fill-up the gap when turbine 1 is operating. If turbine 2 is selected, it will provide maximum efficiency for scenario B. In case, though, the capital investment is not justifiable, turbine 4 may be preferred. Since both turbines are sized to fit scenario B, at the most one among the two could be selected. Their counter-exclusive existence is expressed as: y #y )1. (15) 2 4 The same principles apply for turbines 3, 5 and 6 of Fig. 9. Only one the three, at the most, could be selected and the contingency constraint of eq. (10) is developed as: y #y #y )1. (16) 3 5 6 The constraint (11) expresses the logic that among the turbines that cross the zone of the additional load for a scenario, only one can operate at that scenario. In reference to the example in Fig. 9 for scenario A, the zone of additional load is identical to the entire load of the scenario. This zone is crossed by turbines 1—3. Though more than one of these turbines could be selected in the optimum component set, only one could operate under scenario A. If turbine 1 is selected, it will provide the maximum efficiency for that scenario, and therefore, should be the one to be operated. Otherwise, one of turbines 2 and 3 should operate. A combined operation would result in higher degrees of part-load for both units and, consequently,

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lower efficiency. This contingency constraint is expressed as: w #w #w )1. (17) 1,A 2,A 3,A The zone of additional load for scenario B is the difference between the load of scenario B and that of the next smallest scenario A. Across this zone, four turbines can be found, turbines 2—5. If turbine 2 is selected in the optimum component set, it will certainly be the one to operate under scenario B. In case turbine 4 is selected, there is no reason for turbine 5 to operate, since turbine 4 will operate at full load. Regarding turbine 3, if selected, it will cover the whole load of scenario B to minimise the part-load effect. Operation along turbines 4 or 5 would result in higher degrees of part-load. The pertinent contingency constraint is therefore: w #w #w #w )1. 2,B 3,B 4,B 5,B

(18)

Optimisation procedure The optimal selection of the turbine network is proposed as a procedural search that: (i) initialises parametric variables associated with steam flow, (ii) sets up maximum power output E.!9, (iii) applies mathematical programming in the form of an MILP model, (iv) restores parametric values and iterates. These steps are outlined as follows: (1) Initialisation: With the steam requirements, expressed as heat loads, the mass flows at every level are estimated assuming constant specific heat loads, equal to that of the VHP steam entering the system. As such, Q*/$ (19) M*/$" z,k , z 3 Z, k 3 O z,k q c Qe Me " z,k , z 3 Z, k 3 O. (20) z,k q c The total mass flows Mt crossing each expansion z,k zone z at scenario k are calculated as: Mt " + (Me !M*/$ ), z, s 3 Z, k 3 O. (21) z,k s,k s,k s*z (2) Discretisation: On the basis of the total flows Mt the sizes of the discrete turbines, expressed as z,k maximum steam capacities, M.!9, are specified for every expansion zone z according to the algorithm outlined in Part I. For every component turbine i in zone z of size M.!9 the corresponding maximum z,i power output is calculated according to eq. (21) of the Turbine Hardware Model (THM): 1 E.!9" (*H M.!9!A ), z 3 Z, i 3 ¹. z,i *4 z z,i z B z

(22)

In order to estimate the capital cost of Cc of every z,i component turbine, the correlation proposed by Mroz (1979) is applied:

A

B

E.!9 0.68 z,i Cc "Cc,3%& , z 3 Z, i 3 ¹ . (23) z,i E.!9,3%&

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(3) Optimisation: Having specified the superset of component turbines for every expansion zone, the model consisting of eqs (5)—(14) is optimised independently for every zone z. The elements M.!9, E.!9 and z,i z,i Cc are treated as parameters defined by Step 2. The z,i formulation comprises a MILP problem, and its solution determines the optimum set of turbines for every expansion zone. In addition, their steam flows M and power outputs E during each operation z,i,k z,i,k scenario are provided, along with the total capital, power and annual cost. (4) Restoration and Iteration: The results from step 3 are next used to recalculate the specific heat loads according to the expressions: + E i|T z,i,k q065 "q*/ #c *¹4!5! , z,k z,k pz z Mx #+ M z,k i|T z,i,k z 3 Z, k 3 O (24)

q065 (Mx #+ M )#Q*/$ z,k , i|T z,i,k q*/ " z,k z,k z`1,k Mx #M*/$#+ M z,k z,k i|T z,i,k z 3 Z, k 3 O. (25) The specific heats are used next to readjust the mass requirements at every level according to: Q*/$ M*/$" z,k , z 3 Z, k 3 O (26) z,k *H -',z Qe Me " z,k , z 3 Z, k 3 O (27) z,k q065 z,k and the total steam flows across the expansion zone as: Mt " + (Me !M*/$), z, s 3 Z, k 3 O. z,k s,k s,k s*z

(28)

On the basis of the new total flows across each expansion zone, the procedure is repeated from the discretisation step, as shown schematically in Fig. 11. The outcome of the optimisation step is a set of turbines for each expansion zone that features the optimum trade-off between capital investment and power savings.

Remarks and discussion As applied to a variety of problems the proposed procedure has been found to require minimum effort on iterations. In almost all of the examples tested, the approach converged within a single iteration; two iterations required in the worst cases. Based upon the propositions of this work, the specific heat capacity of the VHP steam, q , is used to convert both the inc duced and extracted loads into mass flows according to eqs (19) ad (20). Clearly, the flows of the induced steam raised against the process can be accurately converted from the first step, rather than step 5, as the latent heat of steam at each level is already known. However, the use of q results in better estimates of the c steam flows as the actual specific heat of the expanded steam increases zone-by-zone. As the assignment of the inlet value to the actual specific heat is already an underestimation, if the latent heat of the induced steam had been considered, the resulting weighted value would have been even further from the actual. In case the steam loads are given as mass flows, no iteration procedure is required. The discrete turbines are defined directly and the component superset is optimised as in step 3. We should summarise that a problem originally in the form of a Mixed-Integer Non-Linear Programming problem (MINLP), has been addressed as a Mixed-Integer Linear Programming problem (MILP). This has been possible through the use of a discretisation scheme and a solution procedure based upon a decomposition of the problem into component turbines and optimisation per operation zone. However, even with these tools the problem would still have remained a MINLP if not without the use of Turbine Hardware Model which inherently reflects linearly on the efficiency of the turbines and as such it does not require non-linear components. Example 1 In Example 4 of Part I the optimum steam levels were selected for a site operating under three scenarios. Having specified the placement of the steam levels, the optimum configuration of the steam turbine network can be determined by applying the methodology presented in this paper. (i) Initialisation and discretisation. First, the heat load targets for every steam level are obtained by superimposing the selected steam levels on the TSPs, from which the SUGCC shown in Fig. 12 are extracted. The initial estimates of the mass flows across each expansion zone are calculated assuming constant specific heats. On the basis of these flows, there are 18 turbine candidates (6 for each zone) whose discrete sizes are presented in Table 1. Note that the sizes of the first three units in every zone correspond to the mass flows of each scenario.

Fig. 11. Schematic representation of the iterative optimisation procedure.

(ii) Optimisation. The MILP model is applied to obtain the optimum set of component turbines. The model has been developed using the general algebraic

Conceptual optimisation of utility networks—II

modelling system (GAMS) and the optimisation has been conducted by employing the OSL solver. The mathematical model involved 68 variables for every expansion zone, out of which 24 were integer. The

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solution is obtained in less than 3 s on a HP-9000 workstation. The resulting optimum set of component turbines is shown in Fig. 13, in which the maximum inlet capacities and the corresponding maximum outputs are also provided. The numbers of the units correspond to those of the candidate turbines of Table 1 with the sizes readjusted after the first iteration. The annual power savings achieved by the set of Fig. 13 are k$1874 at a total capital cost divided by the payback period, of k$363. The economic parameters used include the electricity price, assumed to be $60/MWh, a five years pay-back period, and 8500 operating hours per year. As part of the solution, the optimum load distributions among the turbine components under each scenario are also obtained. These are given on Table 2. As it can be seen, turbines T and T in the upper two 1.2 2.1 zones operate under the first two scenarios of similar

Table 1. Initial sizes of units in the component supersets in t/h Candidate turbines (maximum size) Zone

1

2

3

4

5

6

1 2 3

12.9 26.2 49.4

13.8 21.8 41.5

28.5 6.1 26.9

0.9 4.4 7.9

15.6 20.1 22.4

14.7 15.7 14.6

Table 2. Optimum load distributions under each scenario in t/h

Fig. 12. The SUGCCs of the three scenarios for the optimum steam levels.

Turbine

Scenario A

Scenario B

1.2 1.3 2.1 2.3 3.1 3.2 3.3

12.3

12.7

25.6

20.7

Scenario C

26.4 5.2 48.8 40.3 26.1

Fig. 13. The optimum set of component turbines for Example 1. Figures represent the maximum inlet capacities in t/h and the corresponding power outputs in MWs.

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loads. During the third scenario, where the load varies significantly, units T and T are operated. In the 1.3 2.3 last expansion zone, every scenario involves a separate turbine which operates at full load and achieves maximum efficiency. SYNTHESIS OF COMPLEX TURBINES BY INSPECTION

The outcome of the optimisation stage is a set of component turbines for each expansion zone that feature the optimum degree of flexibility to exploit the cogeneration potential efficiently under the various operation scenarios. It is reminded that these component turbines are the result of a decomposition stage, based on the observation of Figs 2 and 3 that complex or multi-stage units are equivalent to a cascade of component turbines. Apparently, this observation applies both ways, so that the component turbines can be used to synthesise practical complex or multi-stage turbines. The synthesis of the complex turbines relies on the operation schedule of the component turbines as provided by the optimisation stage. For two cylinders to be merged into one complex unit, both have to be loaded during the same scenarios. The reason is that it is not possible for a cylinder to rotate dry, namely without any steam flowing through, as this would damage the unit. Depending on whether the steam flow through the upper cylinders of a complex turbine is larger or smaller than that in the lower sections, the turbine will be of the extraction, induced, or extraction/induced type. Turbines that involve induction of steam at intermediate levels, though, are generally more difficult to control and may not be desired in the turbine configuration. On the basis of such considerations, the synthesis of the complex units is conducted in a way that the engineer has the freedom to select the most appropriate scheme for his/her plant. As noted earlier the derived network of simple and complex turbines is not unique (Fig. 14). In fact, a number of alternative schemes is possible, as illustrated by Fig. 15. All these networks are equivalent and it is aspects of topology or controllability as the

ones outlined above that comprise the criteria for the selection of the final configuration. The synthesis of complex units form the component turbines is illustrated in the examples to follow. Example 2 Using the information of Table 2 on the operation of the individual units, complex turbines can be synthesised. First, it is observed that turbines 1.3, 2.3 and 3.3 all operate under scenario 3 and, therefore, can be merged into one shell. The same is true for units 1.2 and 2.1, which both operate during the first two scenarios. On the other hand, turbines 3.1 and 3.2 cannot be combined with any other units, as they have a different operation pattern. Had turbine 3.1 for example been merged with turbines 1.2 and 2.1, it would have no steam flowing through during scenario C, which, as mentioned earlier, is not allowed in practice. On the basis of the above observations the network of complex and simple turbines of Fig. 15(a) is derived. As the steam flow in unit 2.1 is larger than that of unit 1.2, the complex turbine that results from the synthesis of the two units is an induction turbine. Similarly, the turbine comprising of cylinders 1.3, 2.3 and 3.3 involves the induction of steam at the third steam level, while steam is extracted at the second level. Consequently, this is an induction/extraction turbine. In case no more than one extraction or induction point is desired for the turbines, one possible option is that shown in Fig. 15(b), where the three-stage turbine is broken down into a simple and an induction turbine. Alternatively, the turbine could be decomposed into an extraction unit and a simple in the lower expansion zone. It is often the case for turbine networks involving very high pressures at the top steam level, to have simple turbines that expand the steam down to the next level. If a network with simple ‘topping’ turbines were desired, the configuration of Fig. 15(c) is most appropriate. As all of the above alternatives involve induction turbines, in case this type of turbines is not desired in the network for controllability reasons, the scheme of Fig. 15(d) can be

Fig. 14. Alternative schemes involving complex turbines, resulting from the components of the optimum set. Several alternative networks involving complex turbines can be derived from the component set.

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Fig. 15. Several alternative networks involving complex turbines can be derived from the component set.

obtained. The only possible complex turbine here is that consisting of units 1.3 and 2.3. From the discussion made so far, it becomes clear that, although the optimisation of the component turbines have provided economic targets for the operation, there are decisions left for the engineer as regards: (i) the interpretation of the solution toward the synthesis of complex turbines, (ii) the consideration of problem constraint and preferences, and (iii) the incorporation of qualitative aspects such as the operability and flexibility of the turbine operation. These challenges call upon the use of optimisation techniques to scope and screen among these challenges and are discussed in the following sections.

SYNTHESIS OF COMPLEX TURBINES BY OPTIMISATION

Although the one to one translation from component turbines to a single complex turbine still holds, the

translation of component turbines to networks of complex turbines is not unique as multiple scenarios can be developed in the general case. The situation raises additional challenges for the optimisation whose functions are now recalled to screen for the most convenient and flexible schemes available. In merging the component cylinders in complex turbines, apart from the practicality aspect, there is also an economic benefit. Normally, the cost of a number of separate units is less than that of a complex turbine comprising of these units. Therefore, there is an incentive to explore component sets that allow for more units to be merged into complex turbines. By attempting to match the operation pattern of the turbines, such sets will be inferior in terms of efficiency, but the benefit in the capital cost could outweigh the loss in efficiency. As the synthesis of two turbines depends on their operation pattern, it is necessary to include appropriate

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constraints within the optimisation model that will promote the merger of the component units. Inevitably, this also leads to the simultaneous optimisation of all the expansion zones. The application of optimisation for the synthesis of complex turbines requires additional continuous and integer variables in the formulation. The continuous variables consist of bonus in merging units from different zones and the integer variables to denote logic that reflects on the assignment of turbines for different scenarios. The integer variables are mutually exclusive, namely: xp #xn "1, z 3 Z, i, j 3 ¹. (29) z,i,j z,i,j A set of constraints is introduced to address the synthesis of two units i and j operating in zones z and z#1. The synthesis is only possible when the units follow the same operating pattern. As such the following set of constraints is introduced: + w 2k! + w 2k)xp º!xn , z,i,k z`1,j,k z,i,j z,i,j k|O k|O z 3 Z, i 3 ¹ (30) + w 2k! + w 2k*xp !xn º, z,i,k z`1,j,k z,i,j z,i,j k|O k|O z 3 Z, i 3 ¹ (31) Bx "a(Cc #Cc )(1!xp !xn ), z,i,j z,i z`1,j z,i,j z,i,j z 3 Z, i, j 3 ¹.

(32)

In the above constraints the binaries w that z,i,k denote the operation mode of a unit under scenario k are multiplied by the element of the binary series 2k, so that the summation of w over all the scenarios z,i,k provides a unique number for each specific operation pattern. In other words, this number will reflect the exact operation pattern for every unit. As a result, units with the same value of the above summation are allowed to merge. It is assumed that whenever two units are merged into one shell, a bonus equal to a% of their capital cost is achieved, which is subtracted from the total capital cost of the component set. In particular, if both units i and j follow the same pattern, both xp and xn are zero and eqs (30) and z,i,j z,i,j (31) hold as trivial cases, while the potential bonus for merging is non-zero, according to eq. (32). Otherwise, if + w 2k' + w 2k z,i,k z`1,j,k k|O k|O then from eq. (29): xp "1 and xn "0, and from z,ij z,i,j eq. (32): Bx "0. z,i,j Else if + w 2k( + w 2k z,i,k z`1,j,k k|O k|O then from eq. (29): xp "0 and xn "1 and from z,i,j z,i,j eq. (32): Bx "0. z,i,j Apparently, two units that are not selected in the optimum component set will seem to follow the same operating pattern. In order to avoid the inclusion of

a bonus for units that are not selected, the following equations are used: B !Bx )0, z 3 Z, i, j 3 ¹ (33) z,i,j z,i,j B !y º)0, z 3 Z, i, j 3 ¹ (34) z,i,j z,i B505" + + B , z3Z (35) z z,i,j j|T i|T where B505 is the total bonus in capital for every z expansion zone. According to the above formulation, a turbine that is merged with a unit in the above level and the level below will present a double bonus. This is considered to be realistic, as both the front and the back end of a unit that comprises the middle section of a complex turbine are spared. However, in order to avoid a unit i taken as merged with more than one units in the subsequent zone, the following equation is included: + B )max (Bx ), z 3 Z, i 3 ¹. (36) z,i,j z,i,j j j|T The above constraint is not rigorous, in that it might allow a fraction of a bonus to be included in the total bonus for a second unit in zone z#1, in addition to unit j that is merged with unit i in zone z. However, a rigorous constraint would require the introduction of a large number of additional integer variables which would unduly increase the size of the problem even further. As mentioned earlier, the component set has to be optimised simultaneously for all expansion zones, and the objective function will be:

A

B

Cc,505!B505 z z !Cs . min C505" + (37) z z p b z|Z The simultaneous optimisation of the turbine set, along with the introduction of the binary variables xp and xn causes a significant increase in the size z,i,j z,i,j of the problem. As it will be shown in the example to follow, this has a strong bearing in the computational time. Therefore, for large problems, and particularly in cases where the capital cost is not of prime importance, it may still be preferable to conduct the optimisation independently for every zone and subsequently explore near-optimum solutions that allow for more turbines to be merged. Example 3 In Example 1 the derived component set featured the optimum trade-off between energy savings and capital investment for which opportunities to synthesise complex units were explored. In this example, the constraints introduced above are included in the optimisation model to encourage the relaxation of the efficiency performance in order to obtain schemes with more complex turbines. As discussed earlier, the consideration of merging constraints is expected to decelerate the solution algorithm. Indeed, a dramatic increase in the solution times is noticed compared to those of Example 1. Apart from the simultaneous optimisation of all zones and the introduction of

Conceptual optimisation of utility networks—II

additional binary variables, for this particular example one of the underlying reasons is that many of the discrete candidate turbines are of similar sizes. As seen from Table 1, four of the turbines in the first zone have a size between 12.9 and 15.6 t/h. The situation

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results in a large number of similar solutions and a flat optimum that complicates the application of a branch and bound algorithm. As some turbine sizes of Table 1 fall close to each other, the optimisation efficiencies can be improved by dropping from the superset, (i)

Fig. 16. The component set with the optimum degree of possible turbine merging.

Fig. 17. Alternative schemes with complex turbines, resulting from the component set of Fig. 16.

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S. P. Mavromatis and A. C. Kokossis

Table 3. Optimum load distributions for the component set of Fig. 16 Turbine

Scenario A

Scenario B

1.2 1.3 2.1 2.3 3.1 3.3

12.4

12.5

25.7

20.5

48.9

40.2

Scenario C

26.4 5.2 26.1

turbines 1, 2, 6 for zone 1, (ii) turbines 5 and 6 for zone 2. The optimal solution is obtained in 15 minutes on a HP-9000 workstation. The resulting component set, assuming a percentage of merging savings a"20%, is given in Fig. 16 and the respective operating schedule is provided on Table 3. Compared to the set of Example 1, this set contains two cylinders in the last expansion zone instead of three. In addition, the operation of these two cylinders is in alignment with that of the respective cylinders in the upper zones, so that all units can be merged into two three-stage turbines as shown in Fig. 17(a). Note that the slight change in the steam flows and the sizes of the cylinders is due to the different expansion efficiencies involved. As a result of the fewer units in the last zone, the efficiency of the network is slightly lower, as indicated by the reduced power output. However, this effect is overweighed by the savings in the capital investment which are two-fold. First, due to the omission of one component turbine and, second, due to the opportunity to merge all cylinders into two complex units. In case this opportunity is realised fully, the savings in capital amount to k$ 70 and the resulting annualised cost is !k$ 1,815. In view of considerations such as those discussed for Example 1, alternative schemes can be obtained that realise the potential for merging the cylinders only partly. Such schemes are shown in Figs 17(b)—(d).

take up some of the cogeneration potential available, and therefore, the non-allocated turbines would have to be sized to the remaining potential. This means that the allocated turbines at each expansion zone need to be known. In many cases, the position of an allocated unit, that is the steam levels to which it is connected, is specified as a result of particular design constraints and safety considerations (Peterson and Mann, 1985). In others, the position of the allocated units comprises part of the optimisation problem. Each of these two cases is handled in a different manner. Allocated turbines of known position For allocated units whose position is specified in advance, the mass flows that correspond to their power loads are subtracted from the steam available at the particular expansion zone. For the remaining steam flows, the discretisation is conducted in the same manner as described earlier. In the general case, the power requirements, of a process unit are different for every operation scenario k. Using eq. (21) of the THM, the size of the allocated turbine is calculated for the maximum power load, E.!9,a, over all the scenarios: z,i B E.!9,a#A z , z 3 Z, i 3 ¹. M.!9,a" z z,i z,i *H *4, z

(38)

Equation (22) of Part I is then applied to calculate the mass flows, Ma , required from the allocated z,i,k turbine i in zone z at each operation scenario k.

A

B

1 5 Ea z,i,k # wa , Ma " M.!9,a z,i,k 6 z,i E.!9,a 5 z,i,k z,i z 3 Z, i 3 ¹, k 3 O

(39)

ALLOCATED TURBINES

where integer wa denotes the operation mode of z,i,k allocated turbine i in zone z under scenario k. These flows are then subtracted from the total available and the remaining loads are used to define the discrete sizes of the non-allocated turbines. Figure 18 shows the discrete turbine sizes for the three scenarios of Fig. 9, where one allocated turbine has been

In the analysis so far, it has been assumed that the turbines to be selected can be freely sized to match the potential unveiled between the steam levels of each operation scenario. In other words, the purpose of these units is to exploit the cogeneration potential and generate power that is directed to the central grid. In many cases, however, there is a need to allocate some steam turbines as drivers of particular pieces of equipment, such as pumps and compressors. These units require a specific amount of power under every operation scenario, which the allocated turbine should provide. Consequently, allocated turbines cannot be freely sized as the turbines considered so far, but need to match the power demands of the process unit. In cases the allocation of such units needs to be considered, the discretisation algorithm has to be adjusted. The reason is that an allocated turbine would

Fig. 18. Discrete turbine sizes for the case of three operation scenarios and an allocated turbine.

Conceptual optimisation of utility networks—II

placed within the particular expansion zone requiring different loads for every scenario. In case more than one allocated turbines exist within the zone, the respective loads of all turbines are subtracted from the total loads at each scenario. The discrete turbine sizes are defined in the same manner as above. Since the allocated units have to be selected, no integer variables are used and, essentially, they are not subject to optimisation. Allocated turbines at optimum position If it is desired to optimise the position of an allocated turbine within the steam system, in other words select the optimum steam levels between which the turbine should operate, the number of candidate turbines in the component set increases significantly. The reason is that an allocated unit could be placed at any expansion zone. Therefore, when defining the discrete sizes of the non-allocated turbines, cases of existing and non-existing allocated units should be considered. In the general case where several units are to be allocated, different discrete sizes will arise for every possible combination of the allocated units in each zone. Regarding the allocated turbines, a vector ya of z,i integer variables has to be introduced for every unit to express the possibility of unit i falling within zone z, followed by the mutual exclusivity constraint: + ya "1, i 3 ¹. (40) z,i z|Z Note that in this case, the turbines of all expansion zones have to be optimised simultaneously due to the coupling effect of the allocated units and the above equation. For the case of example 1, the allocation of two turbines as drivers to process equipment is considered. In view of practical considerations it is determined that one turbine has to be placed in the second expansion zone and the other in the last. The power requirements in MWs for the process units under the three scenarios are: Scenario A Ea z,2 Ea z,3

Scenario B Scenario C

0.5

0.4

0

1.2

1.0

0.7

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The optimum set of turbines that best exploit the remaining cogeneration potential is shown in Fig. 19. Table 4 provides the corresponding operation schedule under the various scenarios. The turbines are numbered and sized according to the reasoning illustrated by Fig. 18. Turbines 2.7 and 3.7 are the units allocated to the process equipment. The allocation of the two units of specific power requirements has resulted in a reduction in the overall efficiency of the turbine network. The network produces 6% less power than that of Fig. 13, which is due to the suboptimal partitioning of the cogeneration potential imposed by the allocation of the two turbines. This means that more units share the load under a particular scenario, entailing lower efficiencies. For example, the network of Fig. 13 involves only turbine 2.1 operating under scenario A, while in the network of Fig. 19 this load is shared by unit 2.1 and the allocated turbine 2.7. It is also interesting to note that although no units are allocated in the upper zone, the mass flows are slightly less than in the original case. The reason is that the allocation of turbines in the lower zones causes a reduction in the efficiency of expansion, which results in higher specific heats at the exhaust of the turbines. Consequently, the mass loads required at every zone are less than in the case without allocated units. Regarding the synthesis of complex units, no combinations are possible for the network operating according to the schedule of Table 4. If, however, the load of turbine 2.1 under scenario C was lead through a let-down valve, a three-stage turbine could be composed from units 1.2, 2.1 and 3.1. Such options can be explored by introducing the merging constraints discussed above. Remarks and discussion The discretisation procedure described in this paper presents a relative advantage for cases of three or four scenarios, where no allocated units are involved. In such cases the number of discrete units is relatively small and the sizes easy to derive. In the consideration of allocated turbines one of the complexities involved is associated with the discrete turbines included in the component superset. The number of these turbines may become large for cases of many operation scenarios and allocated units. In

Fig. 19. The optimum component set including the allocated units.

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S. P. Mavromatis and A. C. Kokossis

these cases, a range of turbine sizes at constant capacity increments can be used to alternatively develop supersets of turbine components or, as a more practical approach, the sizes of these units could be those of the standard turbines available in the market, within the range of capacities involved in the specific problem. An underlying assumption beneath the above analysis on the consideration of allocated turbines is that these units operate within a single expansion zone at which the entire power demand is produced. If the turbine were to operate across several zones, the resulting model would be an MINLP, unless the proportion of work within each zone is specified. In the latter case, the segmented loads can be considered as individual loads for which the same analysis as for the units of known position applies. Another assumption in the model is that the cogeneration potential within each expansion zone is sufficient to cover the demands of the allocated units. If this were not the case, additional steam would have to be raised or, depending on the economic figures, an electric motor could be installed instead. This aspect has not been considered in this work, as it falls within the more general problem of the utility system design. Here, emphasis is given to turbine networks that exploit the cogeneration poten-

tial between the steam levels. Nevertheless, the present analysis can be used with the appropriate adjustments as part of a methodology to address the general problem.

AN INDUSTRIAL CASE STUDY

The proposed methodology for the optimum design of steam turbine networks has been applied to an industrial complex, with an existing network of steam turbines as part of its cogeneration plant. The purpose of the case study is to evaluate the effect of the anticipated variations on the optimum turbine configuration and compare the results of the applied methodology to the existing installation. The turbine network of the cogeneration plant consists of five passout steam turbines operating between four mains as shown in Fig. 20. The processes require steam at the three lower levels, which is produced by expanding VHP steam through the turbines, in order to cogenerate power. The equivalent set of component turbines for the existing network and their respective capacities are given in Fig. 21. The set of data provided by the operator involve hourly measurements of the steam demands at the three steam levels for the whole of January and July.

Table 4. The operation schedule for the network with allocated units

VHP-HP HP-MP MP-LP Year fraction

Scenario A winter weekend

Scenario B winter weekday

Scenario C winter peak

Scenario D summer weekend

Scenario E summer weekday

270 230 200 0.14

330 275 250 0.30

400 350 310 0.02

150 120 110 0.22

220 170 150 0.32

Fig. 20. The turbine network of the industrial complex.

Conceptual optimisation of utility networks—II

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These are depicted in Fig. 22 as the accumulated steam flows across the three expansion zones. For the purpose of the case study, these measurements are classified under five discrete operating scenarios of

specific average demands. It is assumed that the measurements are representative of the entire yearly cycle and take place for the same proportion of the year as for the two months. The resulting operating scenarios and the mass loads across every expansion zone are provided on Table 5. On the basis of the information of Table 5, the proposed design methodology is applied to obtain the optimum set of component turbines that would best exploit the cogeneration potential available in the industrial site. Note, that as the steam loads are given in mass flows, the optimisation procedure is straightforward without the need for iterations.

Fig. 21. The equivalent component set for the existing turbine network. Figures denote the maximum inlet capacities in t/h.

Case A: Optimum configuration The five different operation scenarios give rise to 15 candidate turbines per expansion zone, many of which

Fig. 22. Actual measurements of steam demands at the various levels for the industrial complex.

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S. P. Mavromatis and A. C. Kokossis Table 5. Operation scenarios for the industrial case study (loads in t/h) Scenario A winter weekend 1.1 1.2 1.3 2.1 2.2 2.3 3.1 3.2 3.3

Scenario B winter weekday

Scenario C winter peak

Scenario D summer weekend

150

Scenario E summer weekday 70

220 270

330

330 75

230

275

150

150

275 110 150 50

120 170 110 150

Fig. 23. The optimum component set for the industrial case.

Table 6. Operation schedule for the optimum component set

1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4

Scenario A winter weekend

Scenario B winter weekday

150

65 155

Scenario C winter peak

155 115

60 10 110

Scenario D summer weekend

Scenario E summer weekday

20

20 70 155 155 60 20 50 110 110 90 50 60 110

155 155 55

11 110 90

110 110 90

110 110 90 50

110

110

60 110

are of similar sizes. Therefore, an elimination step is adopted, as in previous illustrative examples, to remove units of similar sizes. By doing so, approximately half of the candidate units are eliminated and optimal solutions for every expansion zone are obtained within 10 s. The optimum set of component turbines is shown in Fig. 23, and the respective operation schedule is given in Table 6. The economic

parameters used involve the price for electricity, $60/MWh, 8000 operating hours per year and a payback period of 5 yr. Capital costs are estimated according to eq. (23). In view of the operating schedule, the alternative configurations of Fig. 24 can be derived. They involve the maximum number of complex turbines for cases where induction turbines are allowed or cases they are not.

Conceptual optimisation of utility networks—II

1627

Fig. 24. Alternative configurations with complex turbines.

Fig. 25. A design that does not consider the operational variations results in significant power losses.

Case B: Design neglecting variations The average power output predicted for the optimum network is 32.1 MW. Had a network been selected consisting of one turbine cylinder per expansion zone and sized to the maximum steam demands, as shown in Fig. 25, the corresponding power output would be 28.3 MW at an investment cost of k$ 940. Although such a network would employ larger and more efficient units, the extensive operation at partload would result in a loss of power in the order of 12%. The difference in the annual cost for the two cases would be 11%. Case C: Maximum capacity constraints The optimum component set features three cylinders per expansion zone, compared to five in the existing installation. The reason is that the existing network involves smaller units, the capacity of which does not exceed 155 t/h for the first two zones and 110 t/h for the last. In order to produce comparable results, it is assumed that there is a restriction in the maximum capacity of the units to be selected, corre-

sponding to the above figures, and a new design is produced. The consideration of a maximum allowable capacity requires the adjustment of the procedure to determine the sizes of the discrete turbines. In particular, units that, according to the original discretisation procedure, exceed the maximum capacity, are replaced by an appropriate number of units of maximum capacity, plus a turbine sized to take up the remaining load. For example, a turbine, originally of 150 t/h is replaced by a unit of the maximum size, 110 t/h, plus a unit of 40 t/h. The pertinent formulation for the adjusted sizes is: M.!9{"M.!9!INT(M.!9/S.!9)S.!9.

(41)

The optimum turbine network is depicted in Fig. 26. As expected, the network features more units per expansion zone than that of Fig. 23. The inclusion of smaller units leads to a reduction in the power output by 3%, which is not fully outweighed by the decreased capital cost. Note, that the original network has a greater overall capacity, which is justified by the improved efficiency of the larger units. In

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S. P. Mavromatis and A. C. Kokossis

Fig. 26. The optimum component set considering maximum allowable capacities for the component turbines.

Fig. 27. A network with complex turbines derived from the component set of Fig. 26.

Table 7. The operating schedule for the network of Fig. 26 Turbine

Scenario A

Scenario B

1.2 1.3 2.1 2.7 3.1 3.3 3.7

12.1

12.3

14.3 11.0 27.3

11.0 9.2 21.6

21.2

18.3

Scenario C

26.2 5.0

12.0 13.9

the case of the network of Fig. 26, though, the overall capacity exactly matches the maximum steam demands. The optimum operating schedule for the network of Fig. 26 is given in Table 7. Since the only turbines that follow the same operating pattern are the pairs 1.3, 2.4 and 1.4, 2.5, it is these units only that can be merged into complex turbines as shown in Fig. 27. Networks involving more complex turbines can be obtained at the expense of the generation efficiency, as discussed in the previous example. Although no operation data of the existing network for the five scenarios are available for comparison, it can be said that the existing installation has a good degree of flexibility to address the highly variable process demands. Modelling the existing network according to the proposed analysis predicts an average power output of 31.2 MW and power savings of k$15,004 at a capital cost of k$2003. In other words the existing installation achieves virtually the same power savings as the opti-

mised network of Fig. 26, however at a significantly higher capital cost. The reason is that the existing installation involves two more turbine cylinders and a much larger overall capacity. In addition, the cylinders are not optimally sized, as opposed to the network of Fig. 26, which achieves the same savings at a lower capital investment. An important conclusion that can be drawn from the current case study is that process variations largely influence the efficiency of power cogeneration. Arbitrarily designed schemes as the one-cylinder-perzone can lead to substantial power losses. Therefore, it is essential that the anticipated variations are considered in the design of the steam turbine network. As the optimum network is the result of a trade-off between efficiency and capital, further compromises in efficiency may have to be accepted if particular features in the lay-out of the final configuration are desired.

CONCLUSIONS

A systematic methodology has been proposed for the conceptual design of industrial steam turbines networks. The design task is addressed in view of the anticipated variations in the process demands and the effect of the turbine size and the varying loads on the efficiency of the selected units. These aspects normally give rise to highly complex and large problems, featuring strong non-linear and combinatorial elements, that are generally impossible to handle. Instead, a three-stage procedure has been developed that utilises engineering knowledge and analytical insight to reduce the size and complexity of the problem into

Conceptual optimisation of utility networks—II

easily manageable dimensions. First, a decomposition strategy is employed that enables the independent optimisation of the turbines for every expansion zone. Only a restricted number of discrete units need to be considered, due to the trends revealed by the problem variables. The optimisation step involves an MILP model that relies on the use of the turbine hardware model to describe the individual turbines. The model is applied in an iterative fashion to allow for the accurate estimation of the mass loads required for the calculation of the power outputs. A single iteration has been found to be adequate for convergence. For the general problem, proven optimal solutions are obtained almost rapidly. Longer run times result from the consideration of optimally allocated turbines, which requires the simultaneous optimisation of all expansion zones. It should be emphasised that, had a conventional model for the turbines been applied, the use of MINLP formulations would be inevitable. As a result of a synthesis step, a number of alternative turbine networks can be derived from the optimum set of component turbines. All schemes feature the same degree of flexibility to exploit the cogeneration potential efficiently. The network most suitable for the particular case is then determined on the basis of practical considerations and controllability aspects. Alternative solutions can also be explored that involve slightly lower power savings but potential for reduced capital through the merging of more units into complex turbines. Acknowledgements The authors would like to acknowledge the support provided by the Department of Process Integration at UMIST. Gratitude is expressed to Mr A. Strouvalis for helping in revising the manuscript. NOTATION

Az , Bz Bxz,i,j Bz,i,j B505 z cp,z Cs z Cc,505 z C505 z Ccz,i Cc,3%& Ez,i,k Eaz,i,k E.!9 z,i

regression parameters of THM for inlet conditions of zone z potential capital bonus for merging unit i in zone z with unit j of next zone bonus in capital cost from merging unit i in zone z with unit j of next zone total capital bonus from merging the units of zone z with units in the next zone average specific heat capacity across zone z power savings across zone z total capital cost of turbines in zone z total annualised cost for zone z capital cost of turbine i in zone z reference capital cost of reference turbine power output of turbine i of zone z under scenario k power output of allocated turbine i of zone z under scenario k maximum power output of turbine i of zone z

E.!9,a z,i

1629

maximum power output of allocated turbine i of zone z E.!9,3%& maximum power output of reference turbine Fk time fraction of scenario k H operating hours per year isentropic enthalpy change across *H*4, z zone z *H-', z average latent heat of saturated water across zone z INT(M.!9/ the integer number of maximum capacity S.!9) units that fit under the capacity of the original unit K unit cost of electricity e M.!9{ the adjusted size of the discrete unit M.!9 the original size of the discrete unit M.!9 maximum steam flow through turbine z,i i of zone z Mtz,k total mass flow of steam across zone z under scenario k Mz,i,k steam flow through turbine i of zone z under scenario k Maz,i,k steam flow through allocated turbine i of zone z under scenario k Mxz,k steam throttled through a let-down valve across zone z under scenario k M.!9,a maximum steam flow through allocated z,i turbine i of zone z Mez,k mass flow of steam extracted from zone z under scenario k M*/$ mass flow of steam induced into zone z,k z under scenario k N number of discrete turbines per zone Nk number of operation scenarios O Mk or l D operating scenariosN pb pay-back period q*/ specific heat at the inlet of zone z under z,k scenario k q065 specific heat at the outlet of zone z under z,k scenario k Qez,k heat extracted from zone z under scenario k Q*/$ heat induced into zone z under scenario k z,k Qt total heat exiting zone z under scenario k z,k S.!9 the maximum allowable capacity ¹ Mi or j D component turbinesN ¹Az,k M(z, k): turbines in zone z sized to match or fill-up scenario kN ¹Bz,k M(z, k): turbines in zone z that cross the zone of additional load for scenario kN *¹4!5 saturation temperature difference across z zone z º a large positive number wz,i,k integers to denote the operation of turbine i of zone z under scenario k xp integer to denote that turbine i of zone z,i,j z operates under more scenarios than turbine j of zone z#1 xnz,i,j integer to denote that turbine i of zone z operates under fewer scenarios than turbine j of zone z#1

1630

yz,i Z

S. P. Mavromatis and A. C. Kokossis

integers to denote the selection of turbine i of zone z Mz or s D expansion zonesN

Greek letters a percentage of capital cost saved by merging two units

REFERENCES

Mroz, T. and Bailey, M. (1979) Rankine-cycle component characteristics in Handbook of Data on Selected Engine Components for Solar ¹hermal Applications. DOE/NASA/1060—78/1. June, pp. 13—83.