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Reliable and flexible steam and power system design
Applied Thermal Engineering 79 (2015) 184e191
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Reliable and flexible steam and power system design Li Sun a, *, Chang Liu b a
Centre for Process Integration, School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M13 9PL, UK State Key Laboratory of Fine Chemicals, Research and Development Center of Membrane Science and Technology, Dalian University of Technology, Dalian, Liaoning 116024, PR China b
h i g h l i g h t s Steam power system design including reliability analysis under uncertain demands. Equipment failure and mode transfer affect both system reliability and operating cost. Effect of uncertainty on optimization model by compensation items and penalty costs. System configuration and operation scheduling specified to uncertainties in the design. Five equipment modes in system configuration design and operating scheduling.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 July 2014 Accepted 29 November 2014 Available online 12 December 2014
Steam and power systems should be designed with high reliability and flexibility to satisfy process energy and power demands and reduce penalty costs due to equipment failures and steam and power demand variations. Uncertainties of equipment failure and flexible process steam and power demands have different impacts on system reliability, steam and power generation, individual equipment operation performance, and process production loss due to utility deficits. Measures adopted to respond to uncertainties implementation include compensation options of equipment operating load sharing, equipment startup, and equipment (in failure) repair, and penalties both of electricity import from the grid and production loss. This paper has proposed a procedure of the system design based on simultaneously modelling and optimizing of the structure and operation with system reliability analysis, and a mixed-integer linear programming (MILP) model is formulated associated with compensation costs and penalty costs to obtain both system configuration with spare equipment (in hot or cold standby) and spare capacities, and operating scheduling specification to account for equipment failures and process steam and power demand fluctuations. In this optimization, the effect of equipment failures on system operation performance and costs is analyzed by system reliability, and uncertain steam and power demands are formulated by a multi-period stochastic programming. A case study shows the application and effectiveness of the proposed methodology. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Steam and power systems Failure Reliability Scheduling Uncertainty
1. Introduction Deterministic methods have been developed and widely used in steam and power system design. However, process steam and electricity demand fluctuations and unavoidable equipment shutdowns might influence system operation, and lead to steam and power generation deficits and production loss based on the conventional deterministic design. Therefore, steam and power
* Corresponding author. Tel.: þ44(0) 161 306 8750; fax: þ44(0) 161 236 7439. E-mail address: [email protected] (L. Sun). http://dx.doi.org/10.1016/j.applthermaleng.2014.11.076 1359-4311/© 2014 Elsevier Ltd. All rights reserved.
systems should be designed flexibly with reliably analysis under uncertainty. Conventionally, the steam power system is designed for economic and environment optimization without reliability, availability and maintainability (RAM) consideration firstly, and then issues relative to RAM and process control are assessed and implemented based on the existing system configuration. Significant effort has been made to address utility system uncertain optimization without RAM consideration. Papoulias and Grossmann [1] proposed a MILP approach for the synthesis of flexible utility systems under uncertain demands. Clay et al. [2] presented production planning under cost uncertainties in chemical
L. Sun, C. Liu / Applied Thermal Engineering 79 (2015) 184e191
production. Iyer and Grossmann [3,4] explored multi-period operating planning for varying utility demands with two stage approach. Bruno et al. [5] formulated a mixed-integer nonlinear programming model (MINLP) to optimize utility systems considering different operating scenarios to avoid economic penalty under uncertain energy price. O. Adarijo Akindele et al. [6] utilized two-level stochastic programming approach for utility system retrofitting under uncertain steam and power demands Zhang et al. [7] developed a probability analysis to optimize uncertain power systems. Carpaneto et al. [8,9] formulated a comprehensive approach based on multiple time frames for cogeneration system planning under uncertain electrical/thermal load and electricity price. Chen et al. [10] designed network structures and operating conditions as variables to optimize steam systems. P. VelascoGarcia et al. [11] presented an optimization model accounting for the shut-downs and startups of utility operating units for decision making on running utility systems. Mitra et al. [12] proposed optimal scheduling accounting for different equipment operating modes under time-sensitive electricity prices. Lee [13] used a twostage stochastic program to address coupling energy supply planning and supply chain design. Kitamura et al. [14] used a triangle function to approximate the probability distribution of uncertainties in demands for cogeneration system. However, these uncertain optimization methods did not take into account equipment failures and system reliability analysis. Researches on RAM have been developed in the P2P network [15], satellite swarms [16], database system [17], energy supply , and heat exchange networks [18,19], etc. The researches mainly focused on equipment failure rate prediction, redundant system design, and process control. Different failure rate distribution models have been developed to predict the equipment failure rate, such as an exponential distribution, a normal distribution, and Weibull distribution [20], etc. More precise prediction model might cause calculation difficulties. Redundancy system with spare equipment is one of the efficient methods for increasing system reliability. Unfortunately, redundant devices in hot standby and cold standby result in higher capital cost and more complex operation. Strategies on the redundant system design [21] and reliability estimation [22] mainly are applied in some safety-critical systems, such as fly-by-wire and hydraulic systems in aircraft [23], space systems, and missile systems. The concept of redundancy and its application would be extended to the key units in other fields. In the steam and power system, key equipment like a boiler is the source of the steam and power generation. Once a boiler shuts down, the deficit of steam generation would cause production loss. Spare boilers in hot or cold standby would be switched to the operating condition to compensate the steam deficit. However, the increase of capital and operating costs would lead to bad system economy even though the system reliability is higher. The development of the researches on process control has great influence on process production, process operation, and system energy and power consumption. The researches and application of process control, such as process control on steam turbine blades [24,25] and distillation column [26,27]; normally are implemented for the existing system configuration. Equipment failures and system reliability issues have significant effects on the utility system design and operation [28], and would be considered at the system design stage. Olsommer et al. [29,30] studied waste incineration cogeneration facility economic performance including reliability factors with the passive and/or active redundancy. Haghifam et al. [31] proposed reliability and availability model employing Markov model in the combined heat and power systems. However, the system design based on these proposed methodologies was not an optimization but only generated
185
by parameter sensitivity study. Yin [32] firstly incorporated failure rate distribution and preventive maintenance into chemical process design, but the effect of the system operation modes and system transition conditions during equipment operation mode transfer on the system reliability were ignored in the design. Aguilar et al. [33] developed an availability index to embed reliability analysis in utility system optimization. Del Nogal et al. [34,35], Smith et al. [36] proposed the concept of down time as penalty costs in the utility system design. Lin et al. [37,38] developed utility system optimization methodologies taking account of equipment different standby phases for the operating optimization and system retrofit. Luo et al. [39] presented real and virtual periods representing normal operation and emergency conditions in operational planning optimization in petrochemical complex excluding system component standby phase analysis. Based on the reliable design, operation condition specifications to deal with equipment failures should be developed. The steam and power system should be designed to increase its reliability and to reduce operating and capital cost. The important task for steam and power system design is to analyse the effect of stochastic equipment failures and uncertain steam and power demands on system reliability and steam and power generation. Compensation measures are explored to reduce steam and power deficits and productivity reduction due to the implementation of uncertain equipment failures and flexible steam and power demands. The system might be designed to determine redundant system configuration and operation scheduling specification to account for uncertainty implementation with system reliability analysis. 2. System design methodology This paper proposes a stochastic model for steam and power system design composed of both system configuration and operating scheduling. Equipment selection (number, size and type), individual equipment characters (efficiency varies with operating load, equipment failure, repair and maintenance, etc), and its operation mode (in operating, standby, or failure) determine system configuration and operating performance in terms of power and steam generation. The operating scheduling including compensation adjustments and penalties are specified at the design stage to deal with equipment failures and uncertain steam and power demands. The system superstructure shown in Fig. 1, is composed of boilers (B), gas turbines (GT), heat recover steam generators (HRSG), and steam turbines (single-stage back pressure turbine (BPT), multi-stage turbines (MT), and condensing turbines (CT)). A redundant system with spare equipment and spare capacities can achieve reliable and flexible operation. 2.1. Uncertainties in the system design Uncertain factors are classified into two categories in this design: operation system external uncertainties and internal uncertainties. Process steam and power demands and prices might vary predictably with seasons, and they are system external uncertainties. Equipment loading and standby equipment mode transfer are adjustment options to meet these predictable uncertainties. In the optimization, this multi-period problem is formulated with probability st in the tth period in response to seasonal changing of steam and power demands, t ¼ 1, 2, …, Nperiod. Operation system internal uncertainties, such as changes of operating parameters such as fluid flow, operating temperatures, pressures, and equipment failures, would cause process operating
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Fig. 1. Steam and power system superstructure.
fluctuations and equipment efficiency variation. Uncertain operating parameters would cause variations of process utility demands following continuous or discrete probability functions based on the summarization from experience or historical data, and can be discretized into the process demand j, with the probability sj, j ¼ 1, 2, …, Ndemand. Equipment load sharing is an effective adjustment for this uncertainty. Equipment failures are random and inevitable. As soon as equipment shuts down, loads of other operating equipment would be adjusted to compensate steam and power deficits. Standby equipment start up and failure equipment repair are employed as compensation options as well. If steam and power generation cannot meet process utility demands after these compensations, electricity import from the grid, steam import if possible, as well as production scale reduction (production loss), are uncertainty penalties. The effect of equipment failures on system operation performance and costs is analyzed by system reliability, which is contributed by the system configuration, system redundancy, individual equipment failure and repair characteristics, operation mode and mode transfer. There are five operation modes: operating (O) in full load or part load; cold standby (C); hot standby (H); startup (S); and failure (F). The efficiency of the equipment in O normally increases with higher operating load. Equipment in H consumes fuel during the reheating period to maintain its standby mode. It can be switched to the O mode for steam generation immediately. Equipment in C takes several hours to be switched into the O mode without steam and power generation. Possible equipment mode transfer might be operating equipment failure (O to F), equipment in startup failure (S to F), hot standby equipment failure (H to F), a shutdown equipment repair (F to C), and equipment start up (S to O or H to O). Maintenance time and cost is required during equipment repair (F to C). Both equipment mode and mode transfer determine the system operation state i and system reliability probability Pi, i ¼ 1, 2, …, Nstate. The total state number Nstate is 5n, where n is the number of equipment embedded in the system. At each system state i, the operating cost is determined by equipment operational mode, operating load, efficiency, and equipment operation mode transfer cost. Pi can be calculated employing Monte Carlo model and Markov model. Even though the Markov model is based on the
assumption of the fixed equipment failure and repair rates, and would cause errors in the equipment failure rate prediction, its effect on the system design is limited. Thus, the Markov model is selected to calculate Pi due to its relative simple calculation to provide a good guidance for system design taking account to system reliability. Based on the Markov model, as expressed in Eq. (1), system reliability probability coming into the state i equals that of leaving system state i. Eq. (2) indicates the sum of system state probabilities is the unity [37,38]. N state X
Pi
k¼1;ksi N state X
Ratei/k þ
N state X
Ratek/i Pk ¼ 0
(1)
k¼1;ksi
Pi ¼ 1
(2)
i¼1
2.2. System optimization model The design is an optimization of the cost minimum. For the simultaneously modelling and optimizing of system structure and operation, the optimization objective is the minimum cost, as shown in Eq. (3), including the capital cost Ccap and the recurring cost Ccur. Ccur is the expectation value of the costs taking into account compensation adjustments and penalty costs under uncertainty.
min
Ccap ðxÞ þ Eq2U ½CCur ðx; y; qÞ
(3)
Uncertainty factors q cover system external uncertainty qout (seasonal changed steam and power demands), and internal uncertainty qin (equipment failures and steam and power demands fluctuation due to operation condition variation). Equipment decision variables x comprise equipment types, sizes, numbers and operational modes. Sizes of boilers, gas turbines and HRSGs are discrete following manufacture specifications, and steam turbines are customized design with sizes in continuous variables. The equipment mode is discrete, representing one of five operation modes: O, H, C, F, S. Operating decision variables y contain fuel consumption in boilers or gas turbines, boiler feed water (BFW) makeup, equipment mode transfer, and electricity import from the grid.
L. Sun, C. Liu / Applied Thermal Engineering 79 (2015) 184e191
Ccur in Eq. (4) constitutes the operating cost qa, equipment mode transfer cost qm, and system penalty cost qp. qa is contributed by fuel combustion and BFW makeup. qm is the combination of equipment startup cost, hot standby maintaining cost, and failure equipment repair cost. qp comprises electricity import cost and production loss. Equipment operating load sharing and equipment mode transfer as compensation options to account for equipment failures have an immediate impact on the steam and power generation, and system costs. In Eq. (4), the system state i, the multi-period t, and the discrete utility demand j are independent.
Ccur ðx; y; qÞ ¼ min
Table 1 2-period utility demands. Period t
Base period 1
Summer period 2
Probability st Electricity/MW VHP/t$h1 HP/t$h1 MP/t$h1 LP/t$h1
67% 13 30 50 50 40
33% 14 20 60 35 50
Table 2 Price data.
NX period N state NX demand n X
Electric import/$$kWh1 Electric export/$$kWh1 Natural gas/$$t1 BFW/$$t1
stj Pi
t¼1
i¼1
j¼1
h qatij ytij ; qout;t ; qin;t þ qmtij ytij ; qout;t ; qin;t io þ qptij ytij ; qin;t (4) Equality constraints and inequality constraints cannot be violated during the optimization. Equality constraints are composited by system steam balance, power balance, mass balance of BFW for steam generation, and energy balance of fuel combustion in boilers and gas turbines with HRSG for power and steam generation. For boilers, gas turbines, HRSGs, and steam turbines, equipment performance models provide relationships among steam and power generation, fuel consumption, operating load, and efficiency [40,41]. For individual equipment, there is the minimum operating load constraint as an inequality constraint to guarantee high operation efficiency. In this model, the operating load of every boiler is designed no less than 60% of the maximum load, while GT and ST are required to be larger than 40% of the maximum load. From the view of the system, steam and power generation are designed no less than process demands while no equipment failure happens.
Mtjdem
XX i
pro
Mtj ; Pðqin Þ ¼ 0
(5)
j
Once equipment shuts down, the operating load-sharing as the compensation adjustment has priority over start up equipment startup. Only when all adjustments cannot compensate utility deficit, the production scale has to be reduced to cause production loss as the penalty. This constraint presents compensation adjustments and penalty sequence is an inequality constraint. Electricity import/export to the grid are limited by site constraints. exp Etij imp Etij
exp Elim imp Elim
187
(6)
Uncertainties factors in the design are flexible steam and power demands and equipment failures. Steam and power demands changing with seasons are divided into two periods corresponding to basic demands (period 1, eight months per year, s1 ¼ 67%) and summer demands (period 1, four months, s2 ¼ 33%), Table 1 presents the two-period utility demands. Steam and power demands changing with process operating and market fluctuations follow normal distributionsNðmt ; d2t Þ. The expectation mt is the normal demand value in the t period. dt takes 0.05mt. Prices of fuel, BFW and electricity are listed in Table 2. Steamheader data are presented in Table 3. The system allows the maximum electricity import from the grid of 1 MW and electricity export to the grid of 10 MW. Options of boilers, gas turbines and HRSGs, and their reliability data from the literature [34,38,42,43] are listed in Table 4. Steam turbines are quite reliable assuming no failure during the operation. For the equipment mode transfer cost, when equipment is in hot standby, the fuel consumption cost during the reheating is assumed
Table 3 Steam headers.
VHP HP MP LP Condensation
Equipment Size
HRSG
2.3. The model solution GT
3. Case study A petrochemical steam and power system is designed to provide process utility demands.
Pressure/bar
Temperature/ C
101 20.6 4.1 2.7 0.1
539 333 186 150 45
Table 4 Equipment data.
Boiler
The MILP model is solved using CPLEX7.5 in GAMS23.6 to achieve optimal system configuration and specified operating scenarios to deal with uncertain utility demands and equipment failures.
0.08 0.05 250 0.43
ST
MTBF/y MTTR/h Startup MTBFs/y Efficiency/% duration/h
100 t$h1 1.5 200 t$h1 1 24.2 t$h1 0.667 38.8 t$h1 0.951 14,320 kW 0.25 23,270 kW 0.25 20,000 kW >4
48 48 60 30 36.96 36.96 36.96
8 10 1.5 1.5 1.5 1.5 2
2 2 1 1 1 1 >4
90.0e91.0 92.0e93.0 52.7e54.6 53.7e54.9 30.5e31.6 31.2e31.9 60.0e85.0
Table 5 Equipment selection. Equipment
B1/t$h1
B2/t$h1
B3/t$h1
BPT1/kW
BPT2/kW
BPT3/kW
Size
100
100
100
6783
10,045
11,087
188
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Fig. 2. System operation conditions in two periods (a) Base condition (in period 1) (b) In summer (in period 2).
Table 6 Equipment operating loads under uncertain demands in summer. Demand j
Probability
B1/t$h1
B2/t$h1
B3/t$h1
BPT1/t$h1
BPT2/t$h1
BPT3/t$h1
Electricity Import/kW
Normal demands Min demands Max demands Uncertain demands
0.591 3.125E-7 3.125E-7 5.625E-6
100 100 100 100
65 40.25 89.75 63.5
0 0 0 0
60 51 69 69
35 29.75 40.25 35
50 42.5 57.5 42.5
4318 780 7855 3757
Note: The normal demands: VHP ¼ 20 t$h1, HP ¼ 60 t$h1, MP ¼ 35 t$h1, LP ¼ 50 t$h1, Power ¼ 14 MW. The min demands: VHP ¼ 17 t$h1, HP ¼ 51 t$h1, MP ¼ 29.75 t$h1, LP ¼ 42.5 t$h1, Power ¼ 11.9 MW. The max demands: VHP ¼ 23 t$h1, HP ¼ 69 t$h1, MP ¼ 40.25 t$h1, LP ¼ 57.5 t$h1, Power ¼ 16.1 MW. Uncertain demands: VHP ¼ 17 t$h1, HP ¼ 69 t$h1, MP ¼ 35 t$h1, LP ¼ 42.5 t$h1, Power ¼ 16.1 MW.
5% of that in full load operation. Equipment startup burns fuel 5% of that in full load operation. When equipment is in failure, repair cost is 4% of the purchasing cost. 3.1. Reliable design under uncertainty The proposed methodology is applied to design the system. Equipment selection is listed in Table 5. 3.1.1. Compensation adjustments under uncertain steam and power demands The design provides equipment operating loads adjustment to compensate steam and power demand variations. Fig. 2 shows two special operation conditions to meet the basic demands (in period 1) and summer demands (in period 2). In the first period, the boiler B1 is in full load operation, B2 operates at 70% load (70 t$h1), and B3 is a spare unit in hot standby. In summer (in period 2), the load of B2 reduction of 60 t$h1 can save the fuel consumption. As presented in Fig. 2, operating loads of other equipment like steam turbines are also adjusted in different periods. Table 6 lists different operating loads as an example to present operating scheduling under uncertain demands without equipment shutdown. In this case, extra electricity export is allowed to the grid to make profit. 3.1.2. Compensations and penalties under equipment failures If equipment shut down, the process production scale might reduce. In this case, the utility deficit and production loss are presented in Table 7. Fig. 3 compares system conditions and the penalties when one boiler failure and two boilers shut down happen. Table 8 lists boiler failure(s) and compensation adjustments. If B2 shuts down, B3 in hot standby is transferred into operating mode immediately. For the case of two boilers shut down, the
utility generation is not enough for process heating and power demands even though B3 in full load operating. The penalties contain two items: 1 MW electricity import and production scale reduction to cause a production loss of 0.023 M$$y1. The probability of two boilers failing is smaller than 0.001%. In reality, equipment in hot standby sometimes is a liability due to its maintenance. A suboptimal solution in this case is a similar design except the hot standby replaced by cold standby equipment. In this case, the production loss increases to 0.554 M$$y1.
3.2. Comparison with conventional design methodology The proposed methodology has been compared against conventional deterministic method without reliability analysis. Fig. 4 shows the system configuration and normal operation condition based on a conventional deterministic design for the base demands (in period 1). Its equipment selection and operating loads to satisfy different demands are listed in Table 9. Equipment failures are not included in the conventional design. A boiler failure implies utility system shut down in this design without steam and power generation. The process loss is 8.567 M$$y1 while the system is shut down. Empirically, two small boilers (size 100 t$h1) substituting for one large boiler (size 200 t$h1) increases system reliability with higher capital cost. One
Table 7 Production loss result of steam and power shortage. Utility deficit
Production loss
VHP HP MP LP Electricity
828.2/$$t1 702.2/$$t1 618.9/$$t1 598.1/$$t1 2.9/$$kWh1
L. Sun, C. Liu / Applied Thermal Engineering 79 (2015) 184e191
189
Fig. 3. Boiler failure(s) and operation conditions (a) One boiler failure (in period 1) (b) Two boiler failures (in period 1).
Table 8 Equipment failure and compensation adjustments (normal demands, in summer). Condition i
Probability Pi
B1/t$h1
B2/t$h1
B3/t$h1
BPT1/t$h1
BPT2/t$h1
BPT3/t$h1
Electricity import/kW
(O, O, H) (O, F, O) (F, F, O)
0.987 3.603E-3 1.318E-5
100 100 0
65 0 0
0 65 100
60 60 0
35 35 35
50 50 49
4318 4318 1000
Note: (O, O, H): Two boilers in operating and one in hot stand-by. Other equipment in operating modes. (O, F, O): Two boilers in operating and one failure. Other equipment in operating modes. (O, F, F): One boiler in operating and two boilers failure. Other equipment in operating modes.
small boiler failure and the others operating in full load still lead to the production scale reduction with process loss of 3.611 M$$y1. Another empirical design by adding a same sized standby boiler can reduce the production loss to be 1.374 M$$y1 due to equipment failure. System capital cost and recurring cost based on different design methods are compared in Table 10. The proposed method can achieve the design of the minimum cost. 4. Conclusions The proposed method deals with power and steam system design, which is composed of redundant equipment in five possible operation modes.
As illustrated in the case study, the optimization can achieve a redundant design with operation scheduling for normal operational conditions and emergency conditions (equipment shut down). Equipment failures and system reliability analysis in the system design can realize accurate evaluation of system costs, individual equipment performance in term of operation mode, operating load, steam and power generation, and its operation efficiency. The operating scheduling of both equipment load sharing and equipment operation mode transfer as compensation adjustments bring significant cost savings on the production loss in the
Table 9 Equipment selection and operating loads based on deterministic design (in summer). Equipment
B1
BPT1
BPT2
BPT3
Size Max demands Normal demands Min demands
200 t$h1 94.9% 82.5% 70.1%
6783 kW 100% 83.8% 67.5%
10,045 kW 63.8% 52.8% 41.8%
11087 kW 100% 84.5% 68.9%
Table 10 Cost comparisons based on different methodologies. Strategy
Proposed method
Deterministic method
Optimal Sub-optimal One boiler
Fig. 4. System configuration and operating by conventional deterministic method (in period 1).
Operation cost/M$$y1 25.956 Mode transfer 0.341 cost/M$$y1 2.089 Electricity import/M$$y1 0.023 Production loss/M$$y1 Capital cost/M$$y1 7.489 1 Total cost/M$$y 31.720
Two small Adding boilers stand-by
25.945 0.193
25.254 0.231
25.866 0.176
25.395 0.263
2.085
2.070 2.064
2.085
0.554 7.489 32.096
8.567 5.723 37.705
1.374 7.962 32.909
3.611 6.154 33.743
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case of equipment failures and process demand fluctuations, and achieve the minimum system cost. The proposed methodology and computational tools comprise the development of systematic methods and tools for practical application of an integrated approach to economic and operational optimization. Acknowledgements The support of EC Project EFENIS (contract ENER/FP7/296003/ EFENIS) is sincerely acknowledged. Nomenclature Ccap Ccur exp Elim exp Etij
the equipment capital cost, M$$y1 recurring cost, M$$y1 electricity export limit, MW electricity export to the grid, MW
imp Elim
electricity import limit, MW
Etij
electricity import from the grid, MW
i j n Ndemand Nperiod Nstate Mdem Mpro MTBF MTBFs MTTR Pi qa qm qp Ratei/k Ratek/i t x y
the index of system state the index of discrete utility demand the number of equipment total demand conditions total period in seasons the total system state in the utility systems steam and power demands steam and power production mean time between failure, year mean time between failure during startup, year mean time to repair, h the probability of system state i for reliability analysis the operating cost, M$$y1 the mode transfer cost, M$$y1 the penalty cost, M$$y1 transfer rate from state i to k transfer rate from state k to i the index of period equipment variables operation variables the probability of discrete utility demand j the probability at the t period the standard deviation of utility demand in the t period system internal uncertainty system external uncertainty the expect utility demand in the t period the scope of uncertain variables
imp
sj st dt qin qout mt U
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Reliable and flexible steam and power system design Li Sun a, *, Chang Liu b a
Centre for Process Integration, School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M13 9PL, UK State Key Laboratory of Fine Chemicals, Research and Development Center of Membrane Science and Technology, Dalian University of Technology, Dalian, Liaoning 116024, PR China b
h i g h l i g h t s Steam power system design including reliability analysis under uncertain demands. Equipment failure and mode transfer affect both system reliability and operating cost. Effect of uncertainty on optimization model by compensation items and penalty costs. System configuration and operation scheduling specified to uncertainties in the design. Five equipment modes in system configuration design and operating scheduling.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 July 2014 Accepted 29 November 2014 Available online 12 December 2014
Steam and power systems should be designed with high reliability and flexibility to satisfy process energy and power demands and reduce penalty costs due to equipment failures and steam and power demand variations. Uncertainties of equipment failure and flexible process steam and power demands have different impacts on system reliability, steam and power generation, individual equipment operation performance, and process production loss due to utility deficits. Measures adopted to respond to uncertainties implementation include compensation options of equipment operating load sharing, equipment startup, and equipment (in failure) repair, and penalties both of electricity import from the grid and production loss. This paper has proposed a procedure of the system design based on simultaneously modelling and optimizing of the structure and operation with system reliability analysis, and a mixed-integer linear programming (MILP) model is formulated associated with compensation costs and penalty costs to obtain both system configuration with spare equipment (in hot or cold standby) and spare capacities, and operating scheduling specification to account for equipment failures and process steam and power demand fluctuations. In this optimization, the effect of equipment failures on system operation performance and costs is analyzed by system reliability, and uncertain steam and power demands are formulated by a multi-period stochastic programming. A case study shows the application and effectiveness of the proposed methodology. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Steam and power systems Failure Reliability Scheduling Uncertainty
1. Introduction Deterministic methods have been developed and widely used in steam and power system design. However, process steam and electricity demand fluctuations and unavoidable equipment shutdowns might influence system operation, and lead to steam and power generation deficits and production loss based on the conventional deterministic design. Therefore, steam and power
* Corresponding author. Tel.: þ44(0) 161 306 8750; fax: þ44(0) 161 236 7439. E-mail address: [email protected] (L. Sun). http://dx.doi.org/10.1016/j.applthermaleng.2014.11.076 1359-4311/© 2014 Elsevier Ltd. All rights reserved.
systems should be designed flexibly with reliably analysis under uncertainty. Conventionally, the steam power system is designed for economic and environment optimization without reliability, availability and maintainability (RAM) consideration firstly, and then issues relative to RAM and process control are assessed and implemented based on the existing system configuration. Significant effort has been made to address utility system uncertain optimization without RAM consideration. Papoulias and Grossmann [1] proposed a MILP approach for the synthesis of flexible utility systems under uncertain demands. Clay et al. [2] presented production planning under cost uncertainties in chemical
L. Sun, C. Liu / Applied Thermal Engineering 79 (2015) 184e191
production. Iyer and Grossmann [3,4] explored multi-period operating planning for varying utility demands with two stage approach. Bruno et al. [5] formulated a mixed-integer nonlinear programming model (MINLP) to optimize utility systems considering different operating scenarios to avoid economic penalty under uncertain energy price. O. Adarijo Akindele et al. [6] utilized two-level stochastic programming approach for utility system retrofitting under uncertain steam and power demands Zhang et al. [7] developed a probability analysis to optimize uncertain power systems. Carpaneto et al. [8,9] formulated a comprehensive approach based on multiple time frames for cogeneration system planning under uncertain electrical/thermal load and electricity price. Chen et al. [10] designed network structures and operating conditions as variables to optimize steam systems. P. VelascoGarcia et al. [11] presented an optimization model accounting for the shut-downs and startups of utility operating units for decision making on running utility systems. Mitra et al. [12] proposed optimal scheduling accounting for different equipment operating modes under time-sensitive electricity prices. Lee [13] used a twostage stochastic program to address coupling energy supply planning and supply chain design. Kitamura et al. [14] used a triangle function to approximate the probability distribution of uncertainties in demands for cogeneration system. However, these uncertain optimization methods did not take into account equipment failures and system reliability analysis. Researches on RAM have been developed in the P2P network [15], satellite swarms [16], database system [17], energy supply , and heat exchange networks [18,19], etc. The researches mainly focused on equipment failure rate prediction, redundant system design, and process control. Different failure rate distribution models have been developed to predict the equipment failure rate, such as an exponential distribution, a normal distribution, and Weibull distribution [20], etc. More precise prediction model might cause calculation difficulties. Redundancy system with spare equipment is one of the efficient methods for increasing system reliability. Unfortunately, redundant devices in hot standby and cold standby result in higher capital cost and more complex operation. Strategies on the redundant system design [21] and reliability estimation [22] mainly are applied in some safety-critical systems, such as fly-by-wire and hydraulic systems in aircraft [23], space systems, and missile systems. The concept of redundancy and its application would be extended to the key units in other fields. In the steam and power system, key equipment like a boiler is the source of the steam and power generation. Once a boiler shuts down, the deficit of steam generation would cause production loss. Spare boilers in hot or cold standby would be switched to the operating condition to compensate the steam deficit. However, the increase of capital and operating costs would lead to bad system economy even though the system reliability is higher. The development of the researches on process control has great influence on process production, process operation, and system energy and power consumption. The researches and application of process control, such as process control on steam turbine blades [24,25] and distillation column [26,27]; normally are implemented for the existing system configuration. Equipment failures and system reliability issues have significant effects on the utility system design and operation [28], and would be considered at the system design stage. Olsommer et al. [29,30] studied waste incineration cogeneration facility economic performance including reliability factors with the passive and/or active redundancy. Haghifam et al. [31] proposed reliability and availability model employing Markov model in the combined heat and power systems. However, the system design based on these proposed methodologies was not an optimization but only generated
185
by parameter sensitivity study. Yin [32] firstly incorporated failure rate distribution and preventive maintenance into chemical process design, but the effect of the system operation modes and system transition conditions during equipment operation mode transfer on the system reliability were ignored in the design. Aguilar et al. [33] developed an availability index to embed reliability analysis in utility system optimization. Del Nogal et al. [34,35], Smith et al. [36] proposed the concept of down time as penalty costs in the utility system design. Lin et al. [37,38] developed utility system optimization methodologies taking account of equipment different standby phases for the operating optimization and system retrofit. Luo et al. [39] presented real and virtual periods representing normal operation and emergency conditions in operational planning optimization in petrochemical complex excluding system component standby phase analysis. Based on the reliable design, operation condition specifications to deal with equipment failures should be developed. The steam and power system should be designed to increase its reliability and to reduce operating and capital cost. The important task for steam and power system design is to analyse the effect of stochastic equipment failures and uncertain steam and power demands on system reliability and steam and power generation. Compensation measures are explored to reduce steam and power deficits and productivity reduction due to the implementation of uncertain equipment failures and flexible steam and power demands. The system might be designed to determine redundant system configuration and operation scheduling specification to account for uncertainty implementation with system reliability analysis. 2. System design methodology This paper proposes a stochastic model for steam and power system design composed of both system configuration and operating scheduling. Equipment selection (number, size and type), individual equipment characters (efficiency varies with operating load, equipment failure, repair and maintenance, etc), and its operation mode (in operating, standby, or failure) determine system configuration and operating performance in terms of power and steam generation. The operating scheduling including compensation adjustments and penalties are specified at the design stage to deal with equipment failures and uncertain steam and power demands. The system superstructure shown in Fig. 1, is composed of boilers (B), gas turbines (GT), heat recover steam generators (HRSG), and steam turbines (single-stage back pressure turbine (BPT), multi-stage turbines (MT), and condensing turbines (CT)). A redundant system with spare equipment and spare capacities can achieve reliable and flexible operation. 2.1. Uncertainties in the system design Uncertain factors are classified into two categories in this design: operation system external uncertainties and internal uncertainties. Process steam and power demands and prices might vary predictably with seasons, and they are system external uncertainties. Equipment loading and standby equipment mode transfer are adjustment options to meet these predictable uncertainties. In the optimization, this multi-period problem is formulated with probability st in the tth period in response to seasonal changing of steam and power demands, t ¼ 1, 2, …, Nperiod. Operation system internal uncertainties, such as changes of operating parameters such as fluid flow, operating temperatures, pressures, and equipment failures, would cause process operating
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Fig. 1. Steam and power system superstructure.
fluctuations and equipment efficiency variation. Uncertain operating parameters would cause variations of process utility demands following continuous or discrete probability functions based on the summarization from experience or historical data, and can be discretized into the process demand j, with the probability sj, j ¼ 1, 2, …, Ndemand. Equipment load sharing is an effective adjustment for this uncertainty. Equipment failures are random and inevitable. As soon as equipment shuts down, loads of other operating equipment would be adjusted to compensate steam and power deficits. Standby equipment start up and failure equipment repair are employed as compensation options as well. If steam and power generation cannot meet process utility demands after these compensations, electricity import from the grid, steam import if possible, as well as production scale reduction (production loss), are uncertainty penalties. The effect of equipment failures on system operation performance and costs is analyzed by system reliability, which is contributed by the system configuration, system redundancy, individual equipment failure and repair characteristics, operation mode and mode transfer. There are five operation modes: operating (O) in full load or part load; cold standby (C); hot standby (H); startup (S); and failure (F). The efficiency of the equipment in O normally increases with higher operating load. Equipment in H consumes fuel during the reheating period to maintain its standby mode. It can be switched to the O mode for steam generation immediately. Equipment in C takes several hours to be switched into the O mode without steam and power generation. Possible equipment mode transfer might be operating equipment failure (O to F), equipment in startup failure (S to F), hot standby equipment failure (H to F), a shutdown equipment repair (F to C), and equipment start up (S to O or H to O). Maintenance time and cost is required during equipment repair (F to C). Both equipment mode and mode transfer determine the system operation state i and system reliability probability Pi, i ¼ 1, 2, …, Nstate. The total state number Nstate is 5n, where n is the number of equipment embedded in the system. At each system state i, the operating cost is determined by equipment operational mode, operating load, efficiency, and equipment operation mode transfer cost. Pi can be calculated employing Monte Carlo model and Markov model. Even though the Markov model is based on the
assumption of the fixed equipment failure and repair rates, and would cause errors in the equipment failure rate prediction, its effect on the system design is limited. Thus, the Markov model is selected to calculate Pi due to its relative simple calculation to provide a good guidance for system design taking account to system reliability. Based on the Markov model, as expressed in Eq. (1), system reliability probability coming into the state i equals that of leaving system state i. Eq. (2) indicates the sum of system state probabilities is the unity [37,38]. N state X
Pi
k¼1;ksi N state X
Ratei/k þ
N state X
Ratek/i Pk ¼ 0
(1)
k¼1;ksi
Pi ¼ 1
(2)
i¼1
2.2. System optimization model The design is an optimization of the cost minimum. For the simultaneously modelling and optimizing of system structure and operation, the optimization objective is the minimum cost, as shown in Eq. (3), including the capital cost Ccap and the recurring cost Ccur. Ccur is the expectation value of the costs taking into account compensation adjustments and penalty costs under uncertainty.
min
Ccap ðxÞ þ Eq2U ½CCur ðx; y; qÞ
(3)
Uncertainty factors q cover system external uncertainty qout (seasonal changed steam and power demands), and internal uncertainty qin (equipment failures and steam and power demands fluctuation due to operation condition variation). Equipment decision variables x comprise equipment types, sizes, numbers and operational modes. Sizes of boilers, gas turbines and HRSGs are discrete following manufacture specifications, and steam turbines are customized design with sizes in continuous variables. The equipment mode is discrete, representing one of five operation modes: O, H, C, F, S. Operating decision variables y contain fuel consumption in boilers or gas turbines, boiler feed water (BFW) makeup, equipment mode transfer, and electricity import from the grid.
L. Sun, C. Liu / Applied Thermal Engineering 79 (2015) 184e191
Ccur in Eq. (4) constitutes the operating cost qa, equipment mode transfer cost qm, and system penalty cost qp. qa is contributed by fuel combustion and BFW makeup. qm is the combination of equipment startup cost, hot standby maintaining cost, and failure equipment repair cost. qp comprises electricity import cost and production loss. Equipment operating load sharing and equipment mode transfer as compensation options to account for equipment failures have an immediate impact on the steam and power generation, and system costs. In Eq. (4), the system state i, the multi-period t, and the discrete utility demand j are independent.
Ccur ðx; y; qÞ ¼ min
Table 1 2-period utility demands. Period t
Base period 1
Summer period 2
Probability st Electricity/MW VHP/t$h1 HP/t$h1 MP/t$h1 LP/t$h1
67% 13 30 50 50 40
33% 14 20 60 35 50
Table 2 Price data.
NX period N state NX demand n X
Electric import/$$kWh1 Electric export/$$kWh1 Natural gas/$$t1 BFW/$$t1
stj Pi
t¼1
i¼1
j¼1
h qatij ytij ; qout;t ; qin;t þ qmtij ytij ; qout;t ; qin;t io þ qptij ytij ; qin;t (4) Equality constraints and inequality constraints cannot be violated during the optimization. Equality constraints are composited by system steam balance, power balance, mass balance of BFW for steam generation, and energy balance of fuel combustion in boilers and gas turbines with HRSG for power and steam generation. For boilers, gas turbines, HRSGs, and steam turbines, equipment performance models provide relationships among steam and power generation, fuel consumption, operating load, and efficiency [40,41]. For individual equipment, there is the minimum operating load constraint as an inequality constraint to guarantee high operation efficiency. In this model, the operating load of every boiler is designed no less than 60% of the maximum load, while GT and ST are required to be larger than 40% of the maximum load. From the view of the system, steam and power generation are designed no less than process demands while no equipment failure happens.
Mtjdem
XX i
pro
Mtj ; Pðqin Þ ¼ 0
(5)
j
Once equipment shuts down, the operating load-sharing as the compensation adjustment has priority over start up equipment startup. Only when all adjustments cannot compensate utility deficit, the production scale has to be reduced to cause production loss as the penalty. This constraint presents compensation adjustments and penalty sequence is an inequality constraint. Electricity import/export to the grid are limited by site constraints. exp Etij imp Etij
exp Elim imp Elim
187
(6)
Uncertainties factors in the design are flexible steam and power demands and equipment failures. Steam and power demands changing with seasons are divided into two periods corresponding to basic demands (period 1, eight months per year, s1 ¼ 67%) and summer demands (period 1, four months, s2 ¼ 33%), Table 1 presents the two-period utility demands. Steam and power demands changing with process operating and market fluctuations follow normal distributionsNðmt ; d2t Þ. The expectation mt is the normal demand value in the t period. dt takes 0.05mt. Prices of fuel, BFW and electricity are listed in Table 2. Steamheader data are presented in Table 3. The system allows the maximum electricity import from the grid of 1 MW and electricity export to the grid of 10 MW. Options of boilers, gas turbines and HRSGs, and their reliability data from the literature [34,38,42,43] are listed in Table 4. Steam turbines are quite reliable assuming no failure during the operation. For the equipment mode transfer cost, when equipment is in hot standby, the fuel consumption cost during the reheating is assumed
Table 3 Steam headers.
VHP HP MP LP Condensation
Equipment Size
HRSG
2.3. The model solution GT
3. Case study A petrochemical steam and power system is designed to provide process utility demands.
Pressure/bar
Temperature/ C
101 20.6 4.1 2.7 0.1
539 333 186 150 45
Table 4 Equipment data.
Boiler
The MILP model is solved using CPLEX7.5 in GAMS23.6 to achieve optimal system configuration and specified operating scenarios to deal with uncertain utility demands and equipment failures.
0.08 0.05 250 0.43
ST
MTBF/y MTTR/h Startup MTBFs/y Efficiency/% duration/h
100 t$h1 1.5 200 t$h1 1 24.2 t$h1 0.667 38.8 t$h1 0.951 14,320 kW 0.25 23,270 kW 0.25 20,000 kW >4
48 48 60 30 36.96 36.96 36.96
8 10 1.5 1.5 1.5 1.5 2
2 2 1 1 1 1 >4
90.0e91.0 92.0e93.0 52.7e54.6 53.7e54.9 30.5e31.6 31.2e31.9 60.0e85.0
Table 5 Equipment selection. Equipment
B1/t$h1
B2/t$h1
B3/t$h1
BPT1/kW
BPT2/kW
BPT3/kW
Size
100
100
100
6783
10,045
11,087
188
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Fig. 2. System operation conditions in two periods (a) Base condition (in period 1) (b) In summer (in period 2).
Table 6 Equipment operating loads under uncertain demands in summer. Demand j
Probability
B1/t$h1
B2/t$h1
B3/t$h1
BPT1/t$h1
BPT2/t$h1
BPT3/t$h1
Electricity Import/kW
Normal demands Min demands Max demands Uncertain demands
0.591 3.125E-7 3.125E-7 5.625E-6
100 100 100 100
65 40.25 89.75 63.5
0 0 0 0
60 51 69 69
35 29.75 40.25 35
50 42.5 57.5 42.5
4318 780 7855 3757
Note: The normal demands: VHP ¼ 20 t$h1, HP ¼ 60 t$h1, MP ¼ 35 t$h1, LP ¼ 50 t$h1, Power ¼ 14 MW. The min demands: VHP ¼ 17 t$h1, HP ¼ 51 t$h1, MP ¼ 29.75 t$h1, LP ¼ 42.5 t$h1, Power ¼ 11.9 MW. The max demands: VHP ¼ 23 t$h1, HP ¼ 69 t$h1, MP ¼ 40.25 t$h1, LP ¼ 57.5 t$h1, Power ¼ 16.1 MW. Uncertain demands: VHP ¼ 17 t$h1, HP ¼ 69 t$h1, MP ¼ 35 t$h1, LP ¼ 42.5 t$h1, Power ¼ 16.1 MW.
5% of that in full load operation. Equipment startup burns fuel 5% of that in full load operation. When equipment is in failure, repair cost is 4% of the purchasing cost. 3.1. Reliable design under uncertainty The proposed methodology is applied to design the system. Equipment selection is listed in Table 5. 3.1.1. Compensation adjustments under uncertain steam and power demands The design provides equipment operating loads adjustment to compensate steam and power demand variations. Fig. 2 shows two special operation conditions to meet the basic demands (in period 1) and summer demands (in period 2). In the first period, the boiler B1 is in full load operation, B2 operates at 70% load (70 t$h1), and B3 is a spare unit in hot standby. In summer (in period 2), the load of B2 reduction of 60 t$h1 can save the fuel consumption. As presented in Fig. 2, operating loads of other equipment like steam turbines are also adjusted in different periods. Table 6 lists different operating loads as an example to present operating scheduling under uncertain demands without equipment shutdown. In this case, extra electricity export is allowed to the grid to make profit. 3.1.2. Compensations and penalties under equipment failures If equipment shut down, the process production scale might reduce. In this case, the utility deficit and production loss are presented in Table 7. Fig. 3 compares system conditions and the penalties when one boiler failure and two boilers shut down happen. Table 8 lists boiler failure(s) and compensation adjustments. If B2 shuts down, B3 in hot standby is transferred into operating mode immediately. For the case of two boilers shut down, the
utility generation is not enough for process heating and power demands even though B3 in full load operating. The penalties contain two items: 1 MW electricity import and production scale reduction to cause a production loss of 0.023 M$$y1. The probability of two boilers failing is smaller than 0.001%. In reality, equipment in hot standby sometimes is a liability due to its maintenance. A suboptimal solution in this case is a similar design except the hot standby replaced by cold standby equipment. In this case, the production loss increases to 0.554 M$$y1.
3.2. Comparison with conventional design methodology The proposed methodology has been compared against conventional deterministic method without reliability analysis. Fig. 4 shows the system configuration and normal operation condition based on a conventional deterministic design for the base demands (in period 1). Its equipment selection and operating loads to satisfy different demands are listed in Table 9. Equipment failures are not included in the conventional design. A boiler failure implies utility system shut down in this design without steam and power generation. The process loss is 8.567 M$$y1 while the system is shut down. Empirically, two small boilers (size 100 t$h1) substituting for one large boiler (size 200 t$h1) increases system reliability with higher capital cost. One
Table 7 Production loss result of steam and power shortage. Utility deficit
Production loss
VHP HP MP LP Electricity
828.2/$$t1 702.2/$$t1 618.9/$$t1 598.1/$$t1 2.9/$$kWh1
L. Sun, C. Liu / Applied Thermal Engineering 79 (2015) 184e191
189
Fig. 3. Boiler failure(s) and operation conditions (a) One boiler failure (in period 1) (b) Two boiler failures (in period 1).
Table 8 Equipment failure and compensation adjustments (normal demands, in summer). Condition i
Probability Pi
B1/t$h1
B2/t$h1
B3/t$h1
BPT1/t$h1
BPT2/t$h1
BPT3/t$h1
Electricity import/kW
(O, O, H) (O, F, O) (F, F, O)
0.987 3.603E-3 1.318E-5
100 100 0
65 0 0
0 65 100
60 60 0
35 35 35
50 50 49
4318 4318 1000
Note: (O, O, H): Two boilers in operating and one in hot stand-by. Other equipment in operating modes. (O, F, O): Two boilers in operating and one failure. Other equipment in operating modes. (O, F, F): One boiler in operating and two boilers failure. Other equipment in operating modes.
small boiler failure and the others operating in full load still lead to the production scale reduction with process loss of 3.611 M$$y1. Another empirical design by adding a same sized standby boiler can reduce the production loss to be 1.374 M$$y1 due to equipment failure. System capital cost and recurring cost based on different design methods are compared in Table 10. The proposed method can achieve the design of the minimum cost. 4. Conclusions The proposed method deals with power and steam system design, which is composed of redundant equipment in five possible operation modes.
As illustrated in the case study, the optimization can achieve a redundant design with operation scheduling for normal operational conditions and emergency conditions (equipment shut down). Equipment failures and system reliability analysis in the system design can realize accurate evaluation of system costs, individual equipment performance in term of operation mode, operating load, steam and power generation, and its operation efficiency. The operating scheduling of both equipment load sharing and equipment operation mode transfer as compensation adjustments bring significant cost savings on the production loss in the
Table 9 Equipment selection and operating loads based on deterministic design (in summer). Equipment
B1
BPT1
BPT2
BPT3
Size Max demands Normal demands Min demands
200 t$h1 94.9% 82.5% 70.1%
6783 kW 100% 83.8% 67.5%
10,045 kW 63.8% 52.8% 41.8%
11087 kW 100% 84.5% 68.9%
Table 10 Cost comparisons based on different methodologies. Strategy
Proposed method
Deterministic method
Optimal Sub-optimal One boiler
Fig. 4. System configuration and operating by conventional deterministic method (in period 1).
Operation cost/M$$y1 25.956 Mode transfer 0.341 cost/M$$y1 2.089 Electricity import/M$$y1 0.023 Production loss/M$$y1 Capital cost/M$$y1 7.489 1 Total cost/M$$y 31.720
Two small Adding boilers stand-by
25.945 0.193
25.254 0.231
25.866 0.176
25.395 0.263
2.085
2.070 2.064
2.085
0.554 7.489 32.096
8.567 5.723 37.705
1.374 7.962 32.909
3.611 6.154 33.743
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case of equipment failures and process demand fluctuations, and achieve the minimum system cost. The proposed methodology and computational tools comprise the development of systematic methods and tools for practical application of an integrated approach to economic and operational optimization. Acknowledgements The support of EC Project EFENIS (contract ENER/FP7/296003/ EFENIS) is sincerely acknowledged. Nomenclature Ccap Ccur exp Elim exp Etij
the equipment capital cost, M$$y1 recurring cost, M$$y1 electricity export limit, MW electricity export to the grid, MW
imp Elim
electricity import limit, MW
Etij
electricity import from the grid, MW
i j n Ndemand Nperiod Nstate Mdem Mpro MTBF MTBFs MTTR Pi qa qm qp Ratei/k Ratek/i t x y
the index of system state the index of discrete utility demand the number of equipment total demand conditions total period in seasons the total system state in the utility systems steam and power demands steam and power production mean time between failure, year mean time between failure during startup, year mean time to repair, h the probability of system state i for reliability analysis the operating cost, M$$y1 the mode transfer cost, M$$y1 the penalty cost, M$$y1 transfer rate from state i to k transfer rate from state k to i the index of period equipment variables operation variables the probability of discrete utility demand j the probability at the t period the standard deviation of utility demand in the t period system internal uncertainty system external uncertainty the expect utility demand in the t period the scope of uncertain variables
imp
sj st dt qin qout mt U
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