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Short-Term Multiperiod Optimal Planning of Utility Systems Using Heuristics and Dynamic Programming Jeong Hwan Kim and Chonghun Han* Department of Chemical Engineering, Pohang University of Science and Technology, San 31 Hyoja, Pohang, Kyungbuk, Korea 790-784
In this paper, a new approach for short-term multiperiod planning of nonlinear systems is proposed. To find a more exact solution for the multiperiod planning problem of utility systems within an allowable computation time, a three-step approach has been introduced. In the first step, alternatives for optimum configuration in each period are generated, and a nonlinear programming problem is solved for generated configurations. In the second step, the optimal configuration sequence that minimizes the sum of the operating cost and the switching cost is determined using dynamic programming. In the third step, a fine search for the optimum is performed using an iterative search to consider transition cost. With the decomposition of the original MINLP (mixed integer nonlinear programming) problem into NLP (nonlinear programming) subproblems and a dynamic programming problem, a more reliable and accurate solution that considers nonlinear characteristics is obtained and the computation time is greatly reduced. The case study has shown that the proposed approach shows good performance in finding the optimum solutions considering changeover costs between periods, and this approach can be applied to other various MINLP-type problems by adopting appropriate heuristics. 1. Introduction Optimal multiperiod planning for utility and power plants that consume huge amounts of fuel has been an active research issue in chemical industries because of the soaring and consistently changing oil prices. The operating decision choices for different periods can have a large economic impact on operation profit. Without proper operational planning, companies cannot avoid paying high transition costs, resulting in drastic production rate changes, and may fail to satisfy the process demands. Under the highly competitive enterprise environment, optimal multiperiod planning for cost minimization has been a challenging and indispensable research issue. Many research studies have been made to operate the utility plants at their maximum efficiency. Those works can be grouped into two approaches: thermodynamic1-3 and mathematical programming. The thermodynamic approach uses thermodynamic analyses such as steam cycles and steam-turbine cycles to decrease the loss of available energy as much as possible and has the advantage of a better understanding of the characteristics of the system and simple calculations. However, a thermodynamic approach does not provide a common framework for solving different classes of problems in a systematic manner. Its major limitation is in accounting for the interactions that exist when the system is composed of several major components.4 In this context, since the 1980s, the mathematical programming approach has been widely used for the design and planning of the utility plant. In the formulation of the multiperiod planning problem, binary vari* To whom correspondence should be addressed. Tel.: +82562-279-2279. Fax: +82-562-279-3499. E-mail: chan@ postech.ac.kr.
ables and continuous variables are required. Binary variables are required to represent the existence of equipment or start-up/shutdown of equipment. Continuous variables are required to represent the operational conditions such as flow rates, temperatures, pressures, or design variables such as equipment sizes. Therefore, multiperiod planning problems are usually formulated as MILP (mixed integer linear programming) or MINLP (mixed integer nonlinear programming). Most of the chemical processes including utility plants have nonlinear characteristics such as efficiency of the equipment and energy balances, and thus the problem can be formulated as a MINLP problem. Various methods have been suggested to solve the MINLP problem, for example, branch and bound, outer approximation, generalized benders decomposition, extended cutting plane, and disjunctive programming. Gupta et al.5 proposed the branch and bound method, which finds optimal solution by tree enumeration and NLP relaxation, but this approach is only attractive when the NLP subproblems are relatively easy to solve. Outer approximation (OA6) and the generalized benders decomposition approach (GBD7) solves the MILP master problem that gives a nondecreasing sequence of lower bounds and NLP subproblems that give upper bounds. The difference of these methods is in the definition of the MILP master problem. The GBD method, where only active constraints are considered, commonly requires larger iterations than the OA method. Usersupplied constraints are generally required in GBD, especially when the number of integer variables is large, such as in a scheduling problem.8 The extended cutting plane method (ECP9) finds optimal solution by successively adding the most violated constraint at the predicted point to the MILP problem. Because the discrete and continuous variables are converged simultaneously, the ECP method may require a large number of itera-
10.1021/ie000344y CCC: $20.00 © 2001 American Chemical Society Published on Web 03/16/2001
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tions. Another approach for the MINLP problem is disjunctive programming,10-12 which uses logic propositions consisting of the variables and the logic operations to describe the logical relationships among discrete variables such as connections and interactions between units. Boolean variables (True/False) are used instead of binary variables (0/1), and the logic-based approach has the advantage of facilitating the modeling and reducing the computation time. Because of its importance, many applications of the mathematical programming approach to the industrial utility plant have been reported. Hui and Natori13 proposed the MILP technique for the optimization of the Mitsubishi chemical utility plant to find out the best combination of equipment to be added to the existing system to cope with the changing circumstances, such as the government’s deregulation policy. Significant operating cost savings were reported by the optimized operation of the utility plant. Ito et al.14 considered the start-up and shutdown costs in the gas turbine cogeneration plant, which finds a more exact optimum point reflecting the real situation. Papoulias and Grossmann4 proposed a structural optimization approach using MILP where a superstructure is proposed and the optimal configuration is selected from the superstructure, and Kalitventzeff15 proposed the MINLP formulation for the management planning of utility networks for chemical plants. Lee et al.16 proposed hierarchical on-line data reconciliation and optimization approach for the utility plant, which is based on the hierarchical decomposition approach. The variables are classified into common variables, linearization variables, and internal variables according to their characteristics, and upper level NLPs and lower level LPs are solved until the optimum is found. The computation time has been reduced by the decomposition of the whole problem and linearization of nonlinear equations. Shang et al.17 proposed a systematic method for solving optimal planning and scheduling, considering the maintenance for a utility system. One of the hot issues in multiperiod planning is optimization under uncertainty that comes from the modeling uncertainty and the demand uncertainty. Uncertainty issues become more important for the longterm planning problem. Papalexandri et al.18,19 suggested a multiperiod optimization method where the variable energy demands and uncertainties are considered. Different case studies were made to explore the effect of modeling uncertainty on the optimization result. Sahinidis and Liu20 proposed the constraint generation and projection approach using a cutting plane algorithm to solve the long-range planning problem under uncertainty. The multiperiod planning problem for industrial application or under uncertainty generally becomes a large size problem, and computational efficiency becomes the critical issue. Global optimization methods offer rigorous solutions, but suffer from combinatorial explosion of computational requirements, while heuristics offer fast solutions, but no guarantee of optimality.21 Tradeoff exists between computation time and optimality. Our conjecture is that most multiperiod planning problems cannot be solved within polynomial time; therefore, the incorporation of heuristics into the global optimization approach is required. Iyer and Grossmann22,23 proposed the two-stage algorithm for utility planning using a bilevel decomposi-
tion method and the modified shortest path algorithm with partial enumeration to solve the MILP problem considering changeover cost between periods. Sahinidis et al.24 presented a MILP multiperiod long-range planning model based on cutting planes, branch and bound method, and other heuristics. Optimal selection and expansion of processes given time-varying forecasts for the demands and prices of chemicals are solved. Comparisons among the computation results of proposed heuristics were made, and the results show a combination of the heuristics and optimization approach can produce good results. However, previous approaches based on mixed integer programming are difficult to formulate and use linearized models instead of nonlinear models for the MILP formulation case. The utility system shows nonlinear characteristics such as energy balance and boiler and turbine efficiencies. Therefore, important nonlinear characteristics of the process should not be linearized. On the other hand, the MINLP approach may require too much computation time if the problem structure is not analyzed and fully utilized. Therefore, a new efficient algorithm is needed that can solve the optimal planning problem within an allowable computation time and consider the nonlinear characteristics of the system to determine the optimum configuration and optimum value simultaneously. The optimization approach using appropriate heuristics can reduce the computation time by preliminary screening of the search space of candidate alternatives. Kim et al.25 proposed a two-level decomposition strategy that consists of the nonlinear programming level and discrete decision-making level where the optimal configuration is determined using the heuristics combined dynamic programming. In this paper, we improved the previous approach by introducing successive refinement of the optimum solution using a threestep approach to consider the transition cost that cannot be ignored for the exact optimal solution for the utility plant. This paper is presented as follows. In section 2, a utility plant under study is described and the optimal multiperiod planning problem is formulated as an optimization problem. In section 3, the proposed algorithm is presented. In section 4, the proposed approach is applied to the case study of multiperiod planning for a utility system. 2. Problem Formulation 2.1. Utility System. The schematic diagram for a utility plant is shown in Figure 1. The utility plant produces steams and electricity to meet process demands for various qualities of steam and electricity. The plant is composed of the steam generation part and the steam distribution part. In the steam generation part, a boiler produces superheated high-pressure steam (XPS) by burning several types of fuels such as B-C oil and PFO (process fuel oil). In the steam distribution parts, generated steam is distributed to the HPS (highpressure steam), MPS (medium-pressure steam), and LPS (low-pressure steam) header to meet the downstream steam requirements. Turbines generate electricity to satisfy the electricity demand. The excess amount of electricity is sold to the external electric company, while the insufficient electricity is supplied from an external electric company. A boiler feedwater steam pump is also operated using
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Figure 1. Schematic diagram of a utility plant.
steam. Letdown valves are used to maintain the pressure and temperature of the process steams. Steams that are not used to supply the process demands are condensed by cooling water, recycled to the deaerator, and reused as boiler feedwater. In the deaerator section, the resolved air in the water is removed. According to the season and the time of day, the demands for each steam, electricity, and unit cost of fuel and electricity change. To determine the optimum operation for the amounts of fuel to the boilers, the distribution of the steam, the turbine operation while satisfying the changing process demands, multiperiod optimal planning is required in terms of steam recycles, boiler load allocation, and optimized steam distribution. Each boiler has different efficiencies, and the optimization of boiler allocation is required. 2.2. Problem Definition. The short-term multiperiod planning problem for a utility system is defined as follows. Given a fixed flowsheet configuration of a utility system and process demands for steam and electricity, the choice of operation units and optimal operation policy for each period is determined while satisfying the process demands in an optimal manner. The process operation variables include the steam pressures and temperatures for HPS, MPS, and LPS. A multiperiod scenario is considered, where the process demand changes as a piecewise constant function defined over each time period. The boiler feedwater pressure and temperature are specified. Start-up/shutdown costs are given, and the equation that quantifies the transition cost is defined. The length of time period may be different for each period, and weighting is given to be proportional to the length of the period. The optimum number of boilers can be changed according to the steam demand for the
purpose of efficient and economic operation. The following points are considered in this problem. 1. Changeover cost is defined and considered. Changeover cost occurs when the equipment experiences a changeover such as start-up/shutdown (switching cost) or drastic changes of operation value (transition cost). When the equipment experiences start-up or shutdown, it involves related costs. For example, start-up of a boiler requires preheating, cleaning, inspection, hydrostatic testing, calibration of equipment, steam line cleaning, and valve testing, and the normal steady state operation cannot be reached sharply because of the physical limitation, and this also incurs costs. Another type of changeover cost is a transition cost. When the equipment experiences repeated drastic operation changes, the equipment, which is usually expensive, may go out of order because of the accumulated thermal fatigue. Therefore, this kind of operation is not preferable in terms of maintenance. 2. The transition cost is smaller than either the operating cost or the switching cost by an order of magnitude. 3. We assumed the electricity can be purchased from the external electric company and also can be sold. 4. The boiler efficiency curve is regressed as a quadratic form according to the steam flow rate. 2.3. Multiperiod Planning Problem Formulation. Most previous works on the multiperiod optimal planning of a utility plant have been performed using MILP because MINLP often requires too much computation time. However, the MILP approach may miss the important characteristics of the system because of aggressive linearization and may lead to local optimum. In this paper, this MINLP problem has been solved using a three-step approach, and the planning problem
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is formulated as follows. See the Notation section for the descriptions of the variables. Objective Function
larger than the steam demand from the process in tth period (QDr,k,t).
∑k QGr,k,t g QDr,k,t
Min[Total Cost] ) Min[ XD
∑t OCt + ∑t SCt + ∑t TCt]
6. Boiler efficiency equation for equipment k at time
[∑ ∑(
h
)
N
∑t ∑k (Xk,t - Xk,t-1)2yk,t
(1)
The objective is to minimize the total cost over the planning horizon. Total cost consists of operating cost, switching cost, and transition cost. Operating cost is composed of fuel cost, electricity cost, and water cost. Fuel cost is determined by the amount of fuel consumed multiplied by unit fuel cost. The amount of fuel consumption is determined by boiler efficiency and heating value of the fuel. Switching cost occurs when the equipment experiences start-up or shutdown, and it is expressed as the multiplication of switching cost and binary variable, which indicates the status of the equipment. The transition cost is expressed as the square of the difference between the flow rates of successive periods. The decision variables are boiler flow rates and turbine flow rates in each period and integer variables that specify the configurations of equipment in each period. Switching cost is assumed constant and transition cost is defined as having the quadratic relationship with the steam production change. Constraints 1. Material balance for each unit k in each period
∑Fin,k,t - ∑Fout,k,t ) 0
∀ k, t
(2)
2. Energy balance for equipment k in each period
∑Hin,k,tFin,k,t - ∑Hout,k,tFout,k,t ) 0
∀ k, t
(3)
Note that the H (enthalpy of steam) is the nonlinear function of temperature and pressure. 3. Operation region constraints Each piece of equipment has feasible operation regions bounded by the minimum and the maximum capacity.
ΩL,k,t e Uk,t(F,T,P) e ΩU,k,t ∀ k, t
(4)
4. Satisfaction of electricity demand in period t The sum of power that is generated in the utility plant using kth equipment in period t (PGk,t) and that is purchased from the external electric company in the tth period (PPt) should be higher than the downstream power demand in the tth period (PDt) through all planning horizons.
∑k PGk,t + PPt g PDt
(6)
t
h N H out,m,tFout,m,t - Hin,m,tFin,m,t CFtyk,t + ) Min XD t k ηk,tCPk,t EtCEt + WtCWt + CSUkzsuk,t + CSDkzsdk,t +
a
∀ r, k, t
∀ k, t
(5)
5. Satisfaction of steam demand in period t The amount of steam for type r, which is generated using kth equipment in period t (QGr,k,t), should be
ηk,t ) aFk,t2 + bFk,t + c ∀ k, t
(7)
7. Start-up conditions for equipment k in period t When the equipment experiences a start-up, the integer variable zsuk,t has a value of 1. Of course, it occurs when the equipment is turned off in period t - 1 and turned on in period t.
zsuk,t g yk,t - yk,t-1 ∀ k, t
(8)
where yk,0 ) 0. 8. Shutdown variable for equipment k in period t
zsdk,t g yk,t - yk,t+1 ∀ k, t
(9)
where yk,p+1 ) 0. 3. Multiperiod Heuristics Combined Dynamic Programming (MHCDP) 3.1. Overall Structure of the Proposed Algorithm. In the multiperiod planning problem where transition cost is considered, it is impossible to solve the problem using the decomposition approach because of the linking constraints between periods. If the whole original planning problem can be decomposed into several subproblems, the computation time can be greatly reduced. In this paper, we note that the original problem can be decomposed into several subproblems by introducing two assumptions: first, the switching cost is constant. This assumption is reasonable because start-up/shutdown of the equipment incurs some fixed amount of money regardless of the process conditions. It is rather related to the equipment type and capacity. Second, the transition cost is relatively so small that it is not likely to change or determine the optimal configuration of equipment in the multiperiod planning problem. The ratio of transition cost to the other costs is small, and it is not bigger than the difference of the sum of operating cost and switching cost between each configuration. Therefore, transition cost does not affect the determination of binary variables that represent the configuration of equipment, but affects only continuous variables. This assumption is verified through the case study. With the introduction of the above two assumptions, the original planning problem can be solved by decomposing the original MINLP problem into NLP subproblems, master DP problem, and refinement problem. Therefore, the proposed approach is composed of three steps where these problems are solved to find the optimal solutions. The overall structure of the multiperiod planning procedure is shown in Figure 2. Demands for steam and electricity and oil and electricity unit cost are predicted for the planning horizon based on the plant history data. Utility plant historical data show that the demand for steam and electricity shows a seasonal demand pattern. The electricity demand peaks during the summer while the steam demand is usually higher during the winter.
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Figure 2. Structure of proposed multiperiod planning strategy.
Demand profiles also show characteristic trends for the weekday/weekend and night/daytime also, which is considered in the short-term multiperiod planning. In this research, we used the seasonal-enhanced exponential smoothing model to predict the demand profiles based on historic operation data. External information such as scheduled maintenance of equipment or expected demand changes can be incorporated into the prediction model by increasing the smoothing constant, which makes the model more responsive to the change. Although many approaches have been proposed for exact prediction, perfect prediction is impossible and an unexpected event can occur. Therefore, the uncertainties in demand are an unavoidable and inherent problem for long-range planning, and often make the solutions for optimal planning problem infeasible or nonoptimal. For the short-term planning case, however, uncertainties can be minimized and the planning result can be more realistic and applicable. For the predicted demand, the proposed multiperiod planning procedure is applied through three steps.
When the decomposition strategy is applied, the computation time is greatly reduced, and nonlinear characteristics are well reflected by solving a NLP subproblem. If the predicted demand shows significant deviation from the real demand, and as a result, the future real demand, which was predicted when the planning was first made, shows a deviation from the predicted one, then new planning is made according to the newly predicted demand. The detailed description for each step is given in the following section. 3.2. Step I: Alternative Generation. In this step, alternatives are generated and infeasible alternatives are removed based on the heuristics. The detailed procedure for step I is as follows. (i) Generate a superstructure that includes all the possible alternative structures. (ii) Eliminate the infeasible and nonoptimum alternatives based on the heuristics. The applied heuristics are as follows: (a) boiler capacity and total demand for steam; (b) the efficiency relationship among boilers. Detailed descriptions are given in the case study section.
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(iii) For the remaining alternatives, solve NLP subproblems that minimize the operating cost for each period. The NLP subproblems are as follows. S1. NLP Subproblems Objective function
) Min[Operating Cost] for fixed yk,t ∀ k, t XD ) Min XD
[∑ ∑(
)
h
N
Hout,m,tFout,m,t - Hin,m,tFin,m,t
t
k
ηk,tCPk,t
CFtyk,t +
EtCEt + WtCW (10)
subject to Constraints (2)-(7) 3.3. Step II: Multiperiod Planning Using DP. In step II, the dynamic programming is used to determine the optimal configuration for the multiperiod optimal planning problem. The deterministic DP problem is formulated under the assumptions introduced in section 2.2. The alternatives in each period constitute the state space (S). Associated with each state S is the decision set D(S) of decisions that can be taken at S. The transitions from period t - 1 to t always occur from one state to another state. In this paper, the definition of states is so formulated that the operating cost in each period and the next state that the system moves to after a decision depend only on the current state and the decision and not on the path of past states through which the system arrived in the current state. (This property is known as Markovian Property.26,27) Because we assumed the constant switching cost, the whole multiperiod planning problem can be decomposed, and the state space S can be partitioned as S1 ∪ S2 ∪ S3, ..., Sh. In this step, optimal configuration for the whole planning horizon is determined using DP by considering the operating cost and switching cost. The detailed procedure for step II is as follows. (i) Start from the last period. Read the binary variables for the last period, yk,h, for all k ) 1...N. TC(s) ) 0. TC(si,t) ) minimum total cost that is incurred by pursuing an optimum operation policy beginning with si mode as the initial state at time t. TC(s) ) 0 if s is a terminal state; boundary condition. (ii) Starting from the TC(s) in the terminal state, compute the value of TC using a recursive method by moving backward one state at a time until the original master DP problem is solved. S2. Master DP Problem Objective Function
tions for the remaining periods are independent of the configurations adopted in previous periods. By using dynamic programming, the optimal configuration is determined with a reduced number of calculations. As the proposed approach uses the DP, the number of calculations increases linearly. In this step, the operating cost and switching cost are considered in the determination of the optimum configuration, but the transition cost is not considered. The optimum configurations for each period are determined in this step. 3.4. Step III: Refinement of Optimal Plan Using an Iterative Search. After determining the optimum configurations in step II, the refinement of the optimum values in each period considering transition cost is made using an iterative search. We assumed that the refinement of the optimum value does not change the optimum configuration that has been determined in step II, as the magnitude of the refinement effect is not large enough to change the optimum configuration determined in step II. Compared with the operating cost and switching cost, the transition cost is small, but the absolute value is not so small that it can be ignored. In the case study, the importance of step III is shown. To find out the exact optimum value considering transition cost, successive refinement of optimum value for operation mode determined in step II is performed. The iteration is repeated until the sum of differences between new optimums and previous ones in each period are smaller than the tolerance limit. Figure 3 shows the schematic diagram of step III, and the detailed procedure for step III is as follows. (i) Read the optimum values calculated in step II. (ii) Solve the refinement problem where transition cost is considered. (iii) Check if the sum of differences between optimum values of successive periods is within the tolerance limit. (iv) Generate the final optimum solution. S3. Refinement Problem Objective Function
Min[Total Cost] for fixed yk,t, zsuk,t, zsdk,t for ∀ k,t ) Min [ XD
∑t OCt + ∑t SCt + ∑t Tt]
[∑ ∑(
h N
R
until
Doptimal ) D1 + D2 + ‚‚‚ + Dh
(12)
From the principle of optimality, the optimal configura-
]
||Xi,t - Xi-1,t||e ∑ t)1
(13)
subject to
using recursive method
(iii) Optimal configuration for the whole planning horizon is
∑t ∑k (Xk,t - Xk,t-1)2yk,t h
) Min[Operating Cost + Switching Cost] for fixed yk,t for ∀k,t TC(si,t-1) ) min{SCi,t + TC(Si,t); i ) 1..k, t ) 1...h} (11)
)
h N H out,m,tFout,m,t - Hin,m,tFin,m,t CFtyk,t + ) Min XD t k ηk,tCPk,t EtCEt + WtCWt + CSUkzsuk,t + CSDkzsdk,t +
Constraints (2)-(7) 4. Case Study In the case study, MHCDP has been applied to the multiperiod planning problem for a utility system, and the efficiency of the proposed approach is compared with other previous approaches. In the case study, multiperiod planning for a 1 week horizon, which is composed of seven periods, has been performed. The proposed
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Figure 3. Schematic diagram of step III. Table 1. Power and Steam Demand Prediction period demand
1
2
3
4
5
6
7
HPS [ton/h] MPS [ton/h] LPS [ton/h] electricity [MW]
180 75 65 85
150 60 40 75
180 120 50 90
145 85 55 75
185 110 65 95
140 85 60 80
180 110 45 95
Table 2. Operating Conditions for Steam XPS HPS MPS LPS
Figure 4. Efficiency curve of each boiler (+ boiler 1; - boiler 2; ‚‚‚ boiler 3; - - - boiler 4; * boiler 5).
approach has been applied to the simulated model of an industrial utility plant of Hyundai Petrochemical in Korea. The utility system for the case study is composed of five boilers and two turbines. The efficiency curve is generated based on the process operation data, and Figure 4 shows the efficiency curve of each boiler. As shown in the figure, the efficiency curve shows the nonlinear characteristics, and this fact should be properly reflected in the problem formulation. However, previous work22,23,14 has not considered the nonlinearities of the boiler efficiency and used constant efficiency for the operating region of each boiler. In addition, they used the fixed unit cost for producing steam. However, the cost to produce steam varies because of the nonlinear efficiency of the boiler. In reality, the fuel cost can be fixed for a short period, but the steam production cost changes as the boiler efficiency shows nonlinear characteristics in the operating region. Introducing nonlinear formulation to the planning problem makes the problem difficult to solve, but leads to a more accurate optimum solution. Especially for the utility plant, where the magnitude of the operating cost is large, this becomes more important. Table 1 shows the steam and electricity demand for the planning horizon, and Table 2 shows operating conditions for each steam. Table 3 shows the cost data for unit fuel, electricity, water, switching cost, and transition cost coefficient. In this case study, we assumed the unit costs for fuel, electricity, and water do
temperature [°C]
pressure [bar]
510 395 285 210
106.5 45 11.5 3.3
not change over the planning horizon because we are dealing with the short-term planning problem. Table 4 shows the reduction of the search space by introducing heuristics. The following heuristics have been applied. Heuristics 1. Boiler Capacity and Total Demand for Steam. To meet the steam demands under the mechanical constraints of the turbine, there is the minimum amount of steam to produce in the boiler part. However, each boiler has its own operation range, and there is a limit for the maximum steam production rate. Figure 5 shows the operation region and the expected steam consumption for a turbine that has HP extraction and condensing streams only. As shown in the figure, the inlet stream, HP extraction stream, and condensing stream have their limiting flow rates, and these constitute the mechanical operation region. Therefore, for the given demands of HP, MP, and LP steam and electricity, the minimum and maximum amounts of total steam produced in the boiler section are determined, and this information is used to eliminate infeasible configurations at the alternative generation step. Heuristics 2. The Efficiency Relationship among Boilers. All the boilers and turbines have different efficiencies, even though they had the same efficiencies when initially installed. When one of the boilers shows better efficiency than the other boilers over the entire operating range, this information is used to remove the definitely nonoptimum alternatives. For the alternatives generated in step I, dynamic programming is performed to find optimum solutions considering the switching cost by checking the status change of the boilers (on f off, off f on) in each period, and the sum of the operating cost and switching cost is compared, moving backward one period in a time until
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1935 Table 3. Cost Data
operating cost
Total Cost ) Operating Cost + Switching Cost + Changeover Cost ) Fuel Cost + Electricity Cost + Water Cost + Switching Cost + Changeover Cost unit fuel cost [won/ton B-C oil] unit electricity cost [won/MW] unit water cost [won/ton]
200 000 45 000 2 000
switching cost
start-up cost [won] shutdown cost [won]
107 106
transition cost coefficient
R
0.005
Table 4. Reduction of the Search Space by Using Heuristics period minimum amount of steam produced in the boiler section [ton/h] minimum no. of boilers to satisfy the steam demand no. of alternatives w/out using heuristics no. of alternatives using heuristics
1
2
3
4
5
6
7
513 4 25 3
427.5 4 25 3
542.4 5 25 1
413.3 4 25 3
527.3 5 25 1
399 4 25 3
513 4 25 3
Table 6. Total Cost Comparison: Transition Cost Is Considered (Unit: 106 won) operating cost
switching cost
transition cost
total cost
transition cost is 1087.3 1 1.4 1089.7 not considered transition cost 1087.5 1 0.55 1089.1 is considered cost saving effect by considering transition cost: 0.6 Table 7. Optimal Configuration of Boilers period
Figure 5. Operable region of turbine. Table 5. Reduction of the Number of Required Calculations by Using DP required calculations enumerative search dynamic programming
243 17
the whole period is solved. In the case study, the number of calculations in step II is reduced by introducing dynamic programming (17 ) 3 + 1 + 3 + 1 + 3 + 3 + 3), compared with that from the enumerative search method (243 ) 3 × 1 × 3 × 1 × 3 × 3 × 3). Table 5 shows the reduction in the number of calculations using dynamic programming. Combining heuristics and dynamic programming greatly reduces the search space and results in the reduction of computation time. In step III, fine-tuning of the optimum, which minimizes the total cost which is the sum of the operating cost, switching cost, and transition cost, has been made by an iterative search. The optimum value found in step II has been changed by considering the transition cost in the objective function in step III by a small amount, and the total cost is also reduced. As shown in Table 6, the operating cost for the case where the transition cost is considered is higher, but the total cost is reduced, owing to the transition cost. The reduced total cost shows that a fine search for the optimum is done successfully. To check the efficiency of the MHCDP approach, the result of the MHCDP is compared with that of the MILP approach. Two approaches give different optimum val-
1
2
3
4
5
6
7
MILP
No. 1 No. 2 No. 3 No. 4 No. 5
on off on on on
on off on on on
on on on on on
on off on on on
on on on on on
on off on on on
on off on on on
MHCDP
No. 1 No. 2 No. 3 No. 4 No. 5
on on on on on
on on on on on
on on on on on
on on on on on
on on on on on
on off on on on
on off on on on
ues including different optimal configurations in each step and optimum boiler loads. Table 7 shows the optimal configurations of the boilers. The results show that the optimum configurations from the MHCDP experience less changeover (one shutdown) compared with MILP approach (two start-ups and two shutdowns) for the No. 2 boiler. Figure 6 shows the optimal boiler load profiles from the MILP approach, and Figure 7 shows the results from the MHCDP. Compared with the MILP results, the boiler load profile from the MHCDP shows smoother operation change, which leads to the smaller switching cost and transition cost. Table 8 shows the total cost comparison results. The MHCDP results show smaller total costs than the MILP with constant boiler efficiency approach. The relative magnitudes of operating cost, switching cost, and transition cost confirm that our assumption is valid. The transition cost is smaller than the other costs and does not affect the optimal configuration. Table 9 shows the MHCDP results for the major continuous variables. A total cost comparison is made between (1) normal operation (the boiler loads are equally allocated to all existing boilers without consideration of the boiler efficiency), (2) separate optimization (optimum configuration is determined in each period separately without considering the transition cost), (3) optimization with normal configuration (optimization using all boilers
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Figure 6. Optimal boiler load profiles from MILP.
Figure 7. Optimal boiler load profiles from MHCDP.
considering the switching cost), and (4) multiperiod optimal planning. Table 10 shows the total cost comparison. The results show that separate optimization results show the minimum operating cost. However, it experiences many switchings (three start-ups and three
shutdowns), thus involving a relatively high switching cost that, as a result, does not produce the optimum in terms of total cost. Note that the total cost of the separate optimization can be larger than the no optimization case because of the large changeover cost.
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001 1937 Table 8. Total Cost Comparison of MILP and MHCDP (Unit: 106 won) MILP MHCDP
sum of OC
sum of SC
sum of TC
total cost
1264.1 1087.5
22 1
2 0.55
1288.1 1089.1
Table 9. MHCDP Results of Major Equipment generated flow rate temp press vapor electricity [kg mol H2O] [°C] [bar] fraction [MW] boiler1•inlet boiler3•inlet boiler4•inlet boiler5•inlet boiler1•outlet boiler3•outlet boiler4•outlet boiler5•outlet turbine1•inlet turbine1•ext1 turbine1•ext2 turbine1 condensing turbine2•inlet turbine2•ext1 turbine2•ext2 turbine2 condensing HP•header MP•header LP•header deaerator boiler feed
3330.6 7008.9 4847.3 7017.0 3330.6 7008.9 4847.3 7017.0 9991 5995 1998 1998 12202 6101 3050 3051 12106 5270 3499 15876 13877
120 120 120 120 510 510 510 510 510 395 129 84 510 395 240 160 395 292 208 99 30
125 125 125 125 106.5 106.5 106.5 106.5 106.5 45 2.57 0.57 106.5 45 11.5 4.3 106.5 11.5 5.3 1 1
0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
25.12
28.52
Table 10. Comparison of Total Cost Savings Effect (Unit: 106 won)
no optimization separate optimization optimization with normal configuration multiperiod optimal planning
subproblems are solved for each configuration. When heuristics are used in the alternative generation step, infeasible configurations are excluded and this leads to the reduction of the computation time. In the second step, the global optimum that minimizes the sum of the operating cost and switching cost is determined using dynamic programming. In the third step, a fine search for the optimum is made using an iterative search including the transition cost. The proposed approach considers the changeover cost that is an important factor in the total cost. Consideration of the changeover costs makes the multiperiod planning problem linked period by period and makes the problem size bigger, thus making it difficult to solve. However, careful introduction of assumptions and dynamic programming help solve the problem using the three-step approach. The proposed multiperiod planning approach shows good characteristics by combining heuristics and dynamic programming; The maximum use of the information on the process and DP leads to significant reduction of the search space and results in the reduction of computation time. The case study verifies that the proposed approach shows good performance in finding the optimal operation plan efficiently, considering changeover costs between each period. The proposed approach can be applied to other general MINLP problems by introducing appropriate heuristics. Acknowledgment
OC
SC
TC
total cost
% total cost
1106.6 1085.5 1091.3
0 33 0
1.67 2.05 1.91
1108.3 1120.5 1093.2
1 1.011 0.986
1090.5
1
0.55
1089.1
0.982
Multiperiod optimal planning results show the lowest total cost. Although it has a larger operating cost than separate optimization, the total cost is low because of the low switching and transition costs. Optimization with normal configuration has a larger total cost than the planning results because it uses all the equipment, even though it is not necessary to use all the equipment when the demands for steam are small. One of the interesting points is that the normal configuration case gives a relatively low total cost. This implies that the normal configuration at the industrial site has an advantage in terms of changeover cost. When the switching cost is high, the normal configuration can be a good configuration, but requires the consideration of the process demand, which is not considered in the normal configuration approach. The proposed approach can easily handle the process equipment availability such as maintenance. When the scheduled maintenance constraint is required for the planning, this condition can be easily included in the DP formulation by fixing the corresponding binary variable and the DP is solved. 5. Conclusions To solve the nonlinear multiperiod planning problem for a utility plant more accurately within a limited computation time, a three-step approach has been introduced. In the first step, alternatives for optimum configuration in each period are generated, and NLP
This work was supported by the Brain Korea 21 project, and the authors appreciate the great help from Il-Kwon Kang (Hyundai Information Technology Co. Ltd.) and Sang Hyun You (Hyundai Petrochemical Co. Ltd.) Notation Sets t ) period (t ) 1, ..., h) k ) equipment (k ) 1, ..., N) XD ) decision variables: FB,k,t, FTb,k,t (t ) 1, ..., h, k ) 1, ..., N) Parameters CFt ) unit fuel cost in the tth period CEt ) unit electricity cost in the tth period CWt ) unit water cost in the tth period CPk,t ) heating value of fuel in the kth boiler in period t PDt ) power demand in the tth period QDr,k,t ) steam demand for type r using -kth equipment in period t ΩL,k,t ) lower operation bound for equipment k in period t ΩU,k,t ) upper operation bound for equipment k in period t CSUk ) start-up cost for equipment k CSDk ) shutdown cost for equipment k ak,t, bk,t, ck,t ) kth boiler efficiency coefficients in period t R ) transition cost coefficient h ) planning horizon N ) number of pieces of equipment ) tolerance Continuous Variables OCt ) operating cost in the tth period TCt ) transition cost in the tth period SCt ) start-up and shutdown cost in period t Ft ) fuel usage in the tth period Wt ) water usage in the tth period Et ) electricity usage in the tth period PPt ) power purchase in the tth period
1938
Ind. Eng. Chem. Res., Vol. 40, No. 8, 2001
QGr,k,t ) steam generation for type r using kth equipment in period t PGk,t ) power generation in the kth turbine in the tth period Fin,k,t ) inlet stream flow rate for unit k at time t Fout,k,t ) outlet stream flow rate for unit k at time t Hin,k,t ) inlet stream enthalpy for unit k at time t Hout,k,t ) outlet steam enthalpy for unit k at time t Uk,t (F,T,P) ) operation range of equipment k for flow rate, temperature, and pressure FB,k,t ) flow rate in the kth boiler at time t FTb,k,t ) flow rate in the kth turbine at time t Fdea,t ) flow rate in deaerator splitter part at time t Xit ) decision variable value in the ith iteration step in the tth period Xk,t ) continuous value of equipment k in the tth period ηk,t ) efficiency of boiler k at time t Integer Variables yk,t ) 1 if equipment k is in operation in period t; )0 otherwise zsuk,t ) 1 if equipment k experiences start-ups in period t; )0 otherwise zsdk,t ) 1 if equipment k experiences shutdown in period t; )0 otherwise
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Received for review March 20, 2000 Revised manuscript received January 8, 2001 Accepted January 24, 2001 IE000344Y
