Effect of reliability considerations on the optimal synthesis, design and operation of a cogeneration system

Energy 29 (2004) 309–329 www.elsevier.com/locate/energy

Effect of reliability considerations on the optimal synthesis, design and operation of a cogeneration system Christos A. Frangopoulos ∗, George G. Dimopoulos Dept. of Naval Architecture and Marine Engineering, National Technical University of Athens, Heroon Polytechniou 9, 157 73 Zografou, Greece

Abstract In most of the publications on optimization of energy systems, it is considered that the equipment is available for operation at any instant of time (i.e. it is not subject to failure) except, perhaps, of predetermined periods of maintenance. Thus, it is left to the designer to decide empirically how to provide the system with redundancy, which is necessary in case of equipment failure. However, in this way, the final configuration may not be optimal. In the present work, reliability and availability are introduced in the thermoeconomic model of the system, so that redundancy is embedded in the optimal solution; in addition, more realistic values are obtained for the cost and profit, if any. The state-space method (SSM) of reliability analysis is used. The optimization problem is formulated at two levels: (A) synthesis and design, (B) operation under time-varying conditions. For the solution of the problem at level A and also at level B with no failure, a genetic algorithm coupled with a deterministic one is used. In case of partial failure, the optimization problem is solved by the Intelligent Functional Approach (IFA). The use of IFA combined with SSM is proved to be very efficient for decision making regarding systems under partial failure. It turned out that reliability aspects have a direct and significant impact on the optimal result at each one of the three levels: synthesis, design and operation.  2002 Elsevier Ltd. All rights reserved.

1. Introduction A cogeneration system has to cover certain electrical and thermal loads reliably. Backup power is usually provided by grid connection (for electrical loads) and auxiliary boilers (for thermal loads). If higher reliability is required, the cogeneration system itself is designed and built with redundancy. Then, the question arises: how many units should the system consist of, and which

∗

Corresponding author. Tel.: +30-10-772 1108; fax: +30-10-772 1117. E-mail address: [email protected] (C.A. Frangopoulos).

0360-5442/$ - see front matter  2002 Elsevier Ltd. All rights reserved. doi:10.1016/S0360-5442(02)00031-2

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Nomenclature aij B˙ el C Cd,yr C˙ ᐉ Cmrt C˙ op FOS F Fyr fL ft h K m ˙g Ne Ncgmax Nopi QI q r SII SSGa SSGp SSM SV Ta Tg t ˙n W w x y yrt Z˙ cg Zrt z

transition rate matrix; profit from selling excess electricity; equity contribution (equal to the investment cost, if there is no subsidy); part of capital cost depreciated in year yr (for tax reduction); penalty rate for not covering energy needs due to failure of equipment; maintenance cost; operation cost rate for covering energy needs; fully operational state; objective function; annual profit of operation in year yr; load factor; operation profit rate before taxes during the time interval t; independent binary variable for synthesis optimization; number of components in a system; mass flow rate of exhaust gases from a gas turbine; period of economic analysis (years); the maximum number of cogeneration packages that the system may have (number of packages in the generic system); number of operating components at state i; probability of state I; independent binary variable for operation optimization; the r th unit in the functional diagram; state index (index of state i); state-space graph in case of active redundancy; state-space graph in case of passive redundancy; state-space method; present worth of the salvage value of the system at the end of Ne years; ambient temperature; exhaust gas temperature at the turbine exit; time; nominal electric power output of a cogeneration package; set of independent decision variables for synthesis; there is only one variable of this type for each unit; zero value of this variable means that the corresponding unit does not exist in the optimal system configuration; set of independent decision variables for operation; set of all the functions in the functional diagram; product of unit r in time interval t; amortized capital cost rate of the cogeneration system; amortized capital cost of unit r ‘spent’ during the time interval t; set of independent decision variables for design;

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Greek letters ⌫0kr ⌬ki δki η λ µ µ0krt ˙t Φ φ

cost due to the kth input to unit r from the environment; condition of component k at state i in case of passive redundancy, Eq. (11); condition of component k at state i in case of active redundancy, Eq. (4); efficiency; failure rate; repair rate; marginal cost of the kth input to unit r from the environment; operation cost rate of the system after taxes in time interval t; tax bracket;

Subscripts cg el fo HP i k lb LP ncg t ub

cogeneration; electric; fully operational state; high pressure steam; operational state of the system; the kth component of the system; lower bound; lower pressure steam; no cogeneration; time interval t; upper bound;

Superscripts ∗ #

optimum solution; near optimum solution.

should the capacity of each one be? This question is followed by another one: at what load should each unit be operated at any instant of time, taking into consideration that the grid can absorb any excess electricity or supply with extra electricity, if needed? A rational answer to the preceding questions can be determined by optimization procedures. In most of the related publications, it is considered that the equipment is available for operation at any instant of time (i.e. it is not subject to failure) except, perhaps, of pre-determined periods of maintenance. Thus, the designer has to decide empirically on how to provide the system with

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redundancy in order for the loads to be covered in case of partial equipment failure. But an empirical decision may result in a non-optimal configuration of the system. An attempt to introduce reliability and availability into an optimization procedure for energy systems has been published in Refs. [1–3]. However, no optimum results had been generated including reliability and availability. Instead, a sensitivity analysis using the developed procedure had been performed and reported in the same publications. In the present work, reliability and availability are introduced in the thermoeconomic model of the system, so that redundancy is embedded in the optimal solution; in addition, more realistic values are obtained for the cost and profit, if any. The methodology is presented in Sections 2 and 3 in general terms, not restricted to cogeneration systems. An application to a particular example is presented in Section 4, followed by certain conclusions. 2. Mathematical formulation of the general optimization problem The questions posed in the Introduction show that the optimization of an energy system can be considered at three levels: A. Synthesis optimization. The term ‘synthesis’ implies the components appearing in a system and their interconnections. B. Design optimization. The word ‘design’ is used here to imply the technical characteristics (specifications) of the components and the properties of the substances entering and exiting each component at the nominal load of the system. The nominal load is usually called the ‘design point’ of the system. One may argue that design includes synthesis too. However, in order to distinguish the various levels of optimization and due to the lack of a better term, the word ‘design’ will be used with the particular meaning given here. C. Operation optimization. For a given system (i.e. the synthesis and design are known) under specified conditions, the optimal operating point is requested, as it is defined by the operating properties of components and substances in the system (speed of revolution, power output, mass flow rates, pressures, temperatures, composition of fluids, etc.). Of course if complete optimization is the goal, each level cannot be considered in complete isolation from the others. Consequently, the complete optimization problem can be stated by the following question:What are the synthesis of the system, the design characteristics of the components and the operating strategy that lead to an overall optimum? There are several methods for formulating and solving the optimization problem. In the present work, two formulations have been developed: one for direct application of numerical optimization techniques, and another one based on the Intelligent Functional Approach (IFA). In both formulations, the objective function of the optimization problem is written in the general form1 min F ⫽ F(x,y,z,w,t) (1) There are equality constraints, which are revealed by the simulation of the system, and the 1

Symbols are explained in the nomenclature.

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functional diagram (in the IFA formulation), and inequality constraints, which are imposed by the operability of the system, safety considerations, environmental regulations, etc. For the synthesis optimization, a super-configuration is considered with all the units that might exist in the system. The solution of the problem will indicate which units really exist in the optimal configuration. The IFA has been described in detail in preceding publications [4,5]. The main features of the method are presented here in brief, for the reader’s convenience. The word intelligent implies (a) the information (intelligence) which is obtained by the analysis and during the optimization procedure, and (b) the intelligent (i.e. guided or directed by intellect: rational) use of the information for the solution of the optimization problem. Here, function is the purpose or product of a unit or of the whole system, and functional analysis is the formal, documented determination of the function of the system as a whole and of each unit individually [6]. Based on the analysis, the functional diagram of the system is drawn, which is a picture of the system composed of the units and a function distribution network. The network establishes the interrelations between units, as well between the system and the environment. For the complete mathematical formulation and the solution procedure, the method of Lagrange multipliers is used. One of the essential aspects of the approach is the fact that certain Lagrange multipliers are defined in such a way that they are economic indicators (marginal costs of products) and their value obtained during the optimization procedure is used for decisions regarding the optimal operation of the system. In this way, the computing time for the numerical solution of the optimization problem, which otherwise might be prohibitively long, is significantly reduced, thus facilitating the solution. 3. Reliability analysis in energy systems Industrial plants and energy systems are made up of a large number of components, with multiple interactions and functional dependencies. Failure of a component may result in failure of a sub-system or of the whole system with various detrimental consequences: loss of power may result in loss of production, in damage of production equipment (e.g., in case of solidification of liquid metals), and it may cause accidents. Therefore, reliability has to be considered in the design and implementation of energy systems. Many reliability analysis methods have been developed throughout the years [7,8], that can be grouped into qualitative and quantitative methods. In the present work the State–Space Method has been selected for the following reasons: it is appropriate for quantitative assessment of availability, reliability and maintainability of systems; it can be used with large, complex systems; it is not only useful, but often irreplaceable, for assessing repairable systems [8]. For the reader’s convenience, it is described in brief in the following. 3.1. The State–space method The aim of the SSM is to determine the probabilities of each failure condition of a system. There are two variations of the method, one for active redundancy and one for passive redundancy, i.e. existence of standby units [1,7–9].

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3.1.1. The SSM with active redundancy The method consists of three steps: (Step 1) Identification of all functional and failure modes of the system by making an inventory of all possible states. For a system of K components the number of possible states is I ⫽ 2K

(2)

A table of numbers characterizing each state (logic table) is formulated, which is referred to as the state⫺space graph: SSGa ⫽ [d1i…dKi,Nopi,SIi]

(3)

In Eq. (3) each δki represents the condition of component k at state i and has two possible values:

再

dki⫽

1 running 0 failed

(4)

The number of operating components at state i is Nopi ⫽

冘

δki

(5)

k

A state index is defined by the equation SIi ⫽

冘

δki·2k⫺1 ⫹ 1

(6)

k

An example of a State–Space graph of a system of three components is depicted in Fig. 1. (Step 2) Establishment of all rules for transitions between states and formulation of the transition rate matrix (TRM). Forward transitions (from states with higher Nop to states with lower Nop,) represent failures while backward transitions represent repair actions. The rules for each component and the corresponding component transition rates βk are given in Table 1. The overall transition rate from state i to state j is given by the transition rate matrix aij K

aij ⫽ k

⌸ βk ⫽1

(7)

where βk ⫽ f(δki,δkj)

(8)

as shown in Table 1. (Step 3) time t.

Evaluation of the probabilities for the system being in each state i at any instant of

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Fig. 1. Example state–space graph of the SSM with active redundancy.

Table 1 Component transition rates rules for SSM with active redundancy Transition

0→0 1→0 1→1 0→1

Component transition rate βk Forward

Backward

1 λk 1 0

1 0 1 µk

It is proven [7,8] that for the vast majority of systems the probabilities are obtained by solving the homogenous linear system of equations:

冘 I

j⫽1

Qj(t)·aji ⫽ 0 ∀i ⫽ 1,…,I

(9)

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3.1.2. The SSM with passive redundancy In case of systems with passive redundancy units, the possible operating (or failure) states for a unit in general are those of Table 2. The method consists of the following steps: Step 1a: Application of the first step of the SSM with active redundancy (the existence of passive redundancy in some of the units is not taken into consideration in this step). Step 1b: Transition from the state-space graph of SSM with active redundancy to the one of the SSM with passive redundancy. There are no analytical rules for substitution of binary (active redundancy only) states with the additional states of Table 2. The transition rules depend on the following: 앫 앫 앫 앫

System structure. Functional dependencies between units. Maintenance strategy. Strategy for operation under partial failure.

The strategy for operation under failure is of great importance, since it determines whether a standby unit will be put into service and, if yes, it specifies its operating mode. Each ⌬ki represents the condition of component k at state i, it is given by the equation ⌬ki ⫽ f(δki ,rules)

(10)

and takes values between 0 and 5 (see Table 2). The state–space graph of the SSM with passive redundancy is SSGp ⫽ [⌬1i,…⌬Ki,Nopi,SIi]

(11)

An example of a State–Space graph of a system of two active components and one in standby is depicted in Fig. 2. According to the strategy for operation under partial failure followed here, a standby unit is put into service, if one of the active units fails. 3.1.2.1. Step 2 The second step of the SSM with passive redundancy is identical to the one of the SSM with active redundancy. The only difference is the extension of the component transition rules in order to handle standby components. The rules are now given in Table 3.

Table 2 Possible component states for systems with passive redundancy State

Description

Redundancy

0 1 2 3 4 5

Failed Operating Failed during stand by Standby operational Failed upon demand Started upon demand

Active Active Passive Passive Passive Passive

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Fig. 2.

3.1.2.2. Step 3

317

Example state–space graph of the SSM with passive redundancy.

Identical to Step 3 of the SSM with active redundancy.

3.2. The SSM with IFA The main drawback and, at the same time, limitation of the previous methods in their application on energy systems is the fact that decisions regarding operation under partial failures are taken a priori. In the present work the IFA is used in order to decide optimally the course of actions (emergency procedures, strategy for operating under partial failure) in every possible failure state of the system at any instant of time. Thus, for every state of the system the IFA reveals the optimal decision as to whether a) standby units will start to operate and at which production level, b) the production level of active (and not failed) units will be adjusted or certain units will be shut down, c) no action is taken. The SSM with IFA method consists of the following steps: (Step 1) Step 1a: The same as Step 1a of the SSM with passive redundancy.

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Table 3 Component transition rates rules for SSM with passive redundancy Transition

0→0 1→0 1→1 0→1 2→2 2→3 2→4 2→5 3→2 3→3 3→4 3→5 4→2 4→3 4→4 4→5 5→2 5→3 5→4 5→5

Component transition rate βk Forward

Backward

1 λk 1 0 1 0 1 0 0 1 γk 1⫺γk 0 0 1 0 0 0 λk 1

1 0 1 µk 1 µk 1 0 0 1 0 1 1 0 1 µk 0 1 0 1

Step 1b: Application of the IFA for every possible state of the system in order to determine the optimal mode of operation. The decisions dictated by IFA along with considerations regarding maintenance actions (taking also into consideration the system configuration and functional dependencies) generate the State– Space graph of the SSM with IFA. It has the form of Eq. (11), where, however, each ⌬ki is determined by the IFA. The resulting State–Space graph is strongly connected to the technical and economic parameters of the system. (Steps 2 and 3) Identical to Steps 2 and 3 of the SSM with passive redundancy. The IFA method has been preferred to a numerical optimization one, due to its short computing time, general applicability and robustness. Furthermore, by applying IFA a better understanding of the internal economy of the system is obtained. 3.3. Output of the SSM The main output of the SSM is the set of probabilities Qi of the system being in state i at any instance of time. In the application of the method to energy (or process) systems, these probabilities may be used for three purposes:

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앫 to identify quantitatively the most probable failure states-modes, especially in conjunction with other qualitative or quantitative reliability analysis methods (such as Fault Tree Analysis); 앫 to calculate availability and reliability indices of the system; 앫 to produce more realistic values for the system’s cost, profit and output by the use of expected values. The expected value of cost, profit or any type of energy output of the system is the weighted sum of every value for every possible operating state using as weights the probabilities Qi: X¯ ⫽

冘

Qi·Xi

(12)

i

Examples of X are the cost, profit, energy, mass flow rate, etc. Expected values are more realistic, even though they are probabilistic figures, because all possible failure states will eventually appear in the system [1–3]. 4. Application to a cogeneration system 4.1. Description of the generic system The optimization procedure and the reliability analysis method presented in the previous sections will be applied for the synthesis, design and operation optimization of a cogeneration system, which supplies a production facility with electricity, high-pressure steam and low-pressure steam, all functions of time. The system consists of one or more cogeneration packages; each package contains a gas turbine unit, a heat recovery steam generator (HRSG) and related auxiliary equipment. Supplementary or backup heat (steam) is supplied by an auxiliary boiler, included in the system. A two-way connection to the electricity grid is envisaged, which provides supplementary or backup power, and absorbs any excess electricity produced by the system. The synthesis (configuration) of the system is not pre-specified, but it is the result of synthesis optimization. In order for the methods described in the preceding sections to be applicable, there is a need for a generic system (a super-configuration), which encompasses all the possible (under the particular assumptions and constraints) system configurations. In the present application, the generic system consists of an adequate number of cogeneration packages. The number of packages that will finally appear in the system is to be determined by the synthesis optimization. The generic system is depicted in Fig. 3. 4.2. Statement of the optimization problem Maximization of the net present value of the investment (NPV) is selected as the overall optimization objective function: maxNPV ⫽ ⫺C ⫹

冘 Ne

Fyr ⫹ SV (1 ⫹ r)yr yr ⫽ 1

(13)

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Fig. 3.

where Fyr ⫽ (1 ⫺ φ)·

Generic cogeneration system with Ncgmax cogeneration packages.

冘

ft·⌬tt ⫹ φCd,yr

(14)

t

ft ⫽ (C˙ op ⫹ C˙l)ncg ⫺ (C˙ op ⫹ C˙l)cg ⫹ B˙ el.

(15)

Eq. (14) is written for a certain taxation policy and can be modified accordingly. The complete optimization problem is solved by a two-level procedure: 앫 Level A: synthesis and design optimization, 앫 Level B: operation optimization. In order to handle the variation of conditions with time, the whole period of operation is divided into time intervals under two assumptions: (i) steady-state operation can be considered in each time interval; (ii) the conditions and operation in a time interval do not affect those of other time intervals. This type of decomposition results in a number of simpler operation optimization problems, equal to the number T of time intervals. The minimization of the operation cost of the cogeneration system after taxes is selected as the objective function for operation in each time interval: ˙ t(C˙op ⫹ C˙l)cg,t ⫹ φ·ft ⫹ Z˙cg,t ⫺ B˙el,t min Φ (16) It is noted that in the initial formulation of the IFA, the capital cost had been considered sunk for the operation optimization. This assumption is relaxed here for a more accurate and realistic solution, for the following reasoning. The equipment has a certain lifetime. If it is not used in a certain period of time, it is ‘saved’ and it will be available for use in another period of time, when the conditions may be more favorable. This fact can be taken into consideration during the optimization procedure, if the capital is considered as being consumed during the operation of the system. On the other hand, if the capital cost is ignored, then the marginal cost of the products

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is fictitiously low, which may result in wrong decisions regarding the optimal operating mode of the system. The independent decision variables for Level A optimization are of two types: integer binary numbers representing the existence or non-existence of a cogeneration package in the system (set of synthesis variables w) hi ⫽

再

1, unit i exits

冎

0, unit i does not exist

i ⫽ 1,2,…,Ncgmax,

(17)

and real numbers representing the nominal electric power output of each cogeneration package ˙ ni, i ⫽ 1,2,…,Ncg . (set of design variables z): W max The nominal capacity of each unit is restricted to be in certain limits (e.g., imposed by the available technology): ˙ lbⱕW ˙ niⱕW ˙ ub, W (18) while the total installed capacity must not exceed a certain limit (e.g., imposed by state regulations):

冘

˙ niⱕW ˙ max W

(19)

i

The independent variables of the Level B optimization (set of operation variables x) are also of two types. There are binary variables representing whether a unit operates or not qkt ⫽

再

1, unit k operates in time interval t 0, unit k does not operate in t

k ⫽ 1,2,…,Ncg

t ⫽ 1,2,…, T

(20)

and real variables representing the load factor of a cogeneration package: fL,kt. There may be technical limitation to the load factor: fL,lbⱕfL,ktⱕfL,ub

(21)

while the total annual efficiency of the system may have a lower limit, eg. imposed by state regulations in order for the system to be registered as a cogeneration system: ηcg,tⱖηmin

(22)

Many more equality and inequality constraints are derived by the thermoeconomic model of the system, which are not shown here. The general optimization problem is solved in two variations: 1. without reliability analysis (Variation 1), 2. with reliability analysis (Variation 2). 4.3. Modeling of the system 4.3.1. Thermoeconomic model Simulation of gas turbine performance is achieved by regression analysis and gray-modeling techniques applied on a number of commercial gas turbine performance data. Two sets of corre-

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Fig. 4. Nominal electric efficiency of a turbine-generator as a function of the nominal electric power output (simplecycle gas turbines are considered).

lations have been developed: one to express the nominal performance characteristics as functions of the size of a unit ˙ n), (23) Yn ⫽ Yn(W and another one for the simulation of performance under partial load ˙ n, Ta, fL) Y ⫽ Y(W

(24)

where Y is any one of hel, Tg, m ˙ g. Examples are given in Figs. 4 and 5. The type of HRSG used in this cogeneration system is a double-pressure exhaust gas boiler with parallel heat exchangers [11,12]. The economic model of the system consists of equipment cost functions, simulations of electricity, fuel and water tariffs, and operating costs. The installed unit cost of a gas turbine set is expressed as a function of the nominal electric output:

Fig. 5. Effect of load factor and ambient temperature on the temperature of exhaust gases of a gas turbine with nominal electric power output of 4835 kW.

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˙ n) cu,GT ⫽ cu,GT(W

323

(25)

The result of the regression analysis is shown in Fig. 6. Similar modeling schemes are applied for the heat recovery steam generators and the auxiliary boiler with their auxiliary equipment. The cost functions have been developed by regression analysis of market data. 4.3.2. Reliability model The number of states to be examined by the SSM, hence the size of the system of Eq. (9) to be solved, grows exponentially with the number of components. On the other hand, during the procedure there is a need to calculate the dependability characteristics of the equipment, i.e., the failure rate λ, the repair rate µ, and the failure upon demand rate γ. The direct assignment of failure and repair rates to every component of the system by the use of historical data would extremely complicate the problem. Instead, the system is considered as made up of major sub-systems: 앫 cogeneration packages, 앫 auxiliary boiler, 앫 grid connection equipment. The dependability characteristics of major components appearing in the three types of sub-systems are estimated by use of historical data provided by databases [13–15]. The dependability characteristics of each type of sub-system are then evaluated by the use of reliability block diagram analysis (RBD). The RBD analysis is a method for the evaluation of failure and repair rates of integrated systems using the individual failure and repair rates of their components. The sub-systems of the generic cogeneration system are modeled as a number of individual components connected in series and/or parallel. The pertinent equations for calculating the λ‘s and µ‘s are given in [8], while values for the γ‘s of subsystems are given in [15]. The dependability characteristics of the subsystems can be modeled as constant parameters or functions of time (deterministic or stochastic). In the present work the simple, yet most common, assumption of constant dependability characteristics is made. The SSM with IFA is applied in every step of the operation optimization procedure (Variation

Fig. 6.

Unit cost of a gas turbine-generator as a function of its nominal electric power output.

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2) in order to provide a more accurate assessment of each proposed solution at any instant of time. In order to have the SSM with IFA embedded in the optimization algorithm, an automated procedure has been developed [10]. In this procedure, it is considered that in a system with N units (sub-systems), M units are in passive redundancy. The configuration and the functional dependencies of the generic cogeneration system are taken into account in the procedure. An issue that arises when operating costs under partial failure are evaluated, is whether there will be a penalty (surcharge) in case the system cannot cover the loads. Such a penalty depends on the estimated economic loss of the production facility, due to power deficiency or loss. In the present work a production loss cost is applied, which is considered as a linear function of the (various forms of) power deficiency or loss: ˙ HP ⫹ cLP Q ˙ LP ˙ l ⫹ cHP Q C˙ l ⫽ cW W (26) l

l

l

l

l

Thus at the optimal solution, increased costs for increased reliability and availability are balanced against reduced penalties for not covering energy needs. 4.3.3. Application of the IFA The functional diagram of the generic system appears in Fig. 7. In addition to the resources and products, penalties for not covering energy needs are shown in the diagram (represented by the functions y02.5, y0.6, y0.7). By applying the IFA, the values for the operation Lagrange multipliers are obtained, which are used with an automatic procedure in the form of a decision tree in order to determine the strategy for operation under partial failure. Next, the corresponding optimal economic and technical variables of the system at any partial failure state are calculated.

Fig. 7.

Functional diagram of the generic cogeneration system with Ncgmax cogeneration packages.

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It is revealed by the IFA that a rational lower bound for the unit costs cXl appearing in Eq. (26) is the marginal cost of covering energy needs by conventional equipment (electricity grid, auxiliary boiler). A value of cXl lower than this lower bound would mean that it is more profitable not to cover at least part of the energy demand. In such a case, the system for covering energy needs either with or without cogeneration would not be viable. 4.4. Optimization procedure For the numerical solution of the optimization problems described in Section 4.2, a genetic algorithm [16–17] coupled with a deterministic one, namely, GRG2 [18], is used. The optimization problems of both level A and B are mixed integer non linear programming (MINLP) problems. The genetic algorithm (GA) is used in order to find the optimum values of the integer variables and near optimum values of the real variables. The deterministic algorithm is used afterwards in order to find the optimum values of the real variables, using as an initial estimate the output of the GA. The decomposition of the overall optimization problem into two levels entails the iterative solution of the T subproblems of Level B for every step of the Level A solution. After the optimization operation of the fully operational state (FOS), the resulting optimal solution, which includes the determination of active and passive redundancy, is used by the SSM for reliability analysis. Then, a third set of optimization problems is solved by the IFA for operation under partial failure in each time interval. The whole optimization procedure consists of the following steps: A1. Specify the generic system. ˙ #n) by the GA. A2. Search for the synthesis and design near optimum solution (w∗,z#) ⫽ (h∗,W A3. Calculate the nominal performance characteristics by use of the thermo-economic model. Then, solve the operation optimization problem. For the first time interval (t=1)φ. B1. Search for the near optimum operating mode, x#t,fo ⫽ (q∗t ,f#Lt)fo for the fully operational state (FOS) of the system by the use of the GA. B2. Determine the optimum operating mode x∗t,fo ⫽ (q∗t ,f∗Lt)fo by the GRG2. B3. Determine the type of redundancy (active or passive) for each unit in the FOS. B4. Apply the SSM with IFA procedure. C1. Determine the optimum operating mode under each failure state and calculate the probability of each state. C2. Calculate the technical and economic variables in each failure state by the thermoeconomic model. C3. Calculate the expected values of all technical and economic variables of the system. Repeat steps B1–B4 for each time interval. ∗

1. A4. Determine the synthesis and design optimum solution (w∗,z∗) ⫽ (h∗,Wn ) by the GRG2. Steps B1–B4 are repeated here for each time interval.

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At the end of the procedure, the overall optimum (w∗, z∗, x∗), NPV∗ and all the technical and economic variables of the system are obtained. 4.5. Numerical results The synthesis-design-operation optimization problem of the cogeneration system is solved for given load conditions (Table 4) and a certain set of parameters (e.g., electricity and fuel tariffs) without and with reliability considerations. The main results are presented in Tables 5 and 6, respectively. With no reliability considerations, the optimum system consists of one cogeneration package of 8092 kWel. With reliability, two cogeneration packages of 5003 kWel and 2289 kWel are derived as the optimal configuration. Furthermore, the optimum value of the profit in the various time intervals is decreased by about 12%, while the optimum value of the overall objective, NPV, is decreased by 18.8%. With both considerations (with or without reliability), the optimum operation point of the system is the one corresponding to heat match (the co-generated heat is used and not rejected into the environment). One of the parametric studies that have been performed is the effect of the unit penalty for deficiency or loss of power, cXl, on the optimum solution. The results are depicted in Fig. 8. For an increase of cXl up to about 250% with respect to the lower bound mentioned in Subsection 4.3.3, there is no significant effect on the optimum power of each cogeneration package. A sudden increase in power occurs for an increase of cXl by about 300%. Further increase in cXl has an insignificant effect on the power of the packages. The increase in cXl results in a continuous increase in NPV∗, even though the capacity of the system may not increase significantly, which is explained by means of Eq. (15): an increase in the unit penalty has a much more significant effect on the related cost of the conventional system, C˙ l,ncg, than on the cost of the cogeneration system, C˙ l,cg, because of lack of redundancy in the conventional system. The optimization procedure may give a value for the power output of a gas turbine not available Table 4 Load profiles T

˙ D (kW) W

˙ HP (kW) Q

˙ LP (kW) Q

1 2 3 4 5 6 7 8 9 10 11 12

9700 8900 7600 3500 9700 8200 7600 3700 9900 9000 8000 3600

13650 10800 7500 3150 11819 9720 7200 4200 10920 8400 6000 2640

3094 2958 2720 1360 2758 2460 2460 1200 2820 2640 2550 1350

⌬t (hours) 1376 688 280 568 1360 680 264 560 1184 592 224 448

Tamb (°C) 25 20 25 20 10 5 10 5 30 25 30 25

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Table 5 Optimization results without reliability considerations Generic system

Cogeneration packages

1 Optimal synthesis 1 ˙ ∗n , kW 8092 Optimal designW Optimal operation t 1 2 3 4 5 6 7 8 9 10 11 12 Overall objective function: NPV∗

2 0 –

3 0 –

4 0 –

5 0 –

f∗L1 Profit f∗t (106GRD) 1.000 77.29 1.000 39.27 0.705 9.13 0.000 0.00 1.000 87.16 0.791 33.71 0.671 9.29 0.000 0.00 1.000 59.91 0.773 22.17 0.602 4.82 0.000 0.00 = 1271.28앫106 GRD (1 Euro=340.75 GRD)

Table 6 Optimization results with reliability considerations Generic system

Cogeneration packages

1 Optimal synthesis 1 ˙ ∗n , kW 5003 Optimal design W

2 1 2289

f∗L1fo 1 1 1.000 2 1.000 3 1.000 4 0.000 5 1.000 6 1.000 7 1.000 8 0.000 9 1.000 10 1.000 11 0.944 12 0.000 Overall objective function: NPV∗=1032.05앫106 GRD

Optimal Operation

t

3 0 –

4 0 –

5 0 –

f∗L2fo 2 1.000 0.814 0.000 0.880 1.000 0.000 0.000 1.000 0.876 0.000 0.000 0.743

Profit f∗t (106 GRD) FOS Expected 76.77 69.26 34.76 30.33 10.01 8.24 8.22 6.75 78.04 68.59 28.50 27.59 10.23 8.42 10.40 8.54 57.71 50.26 21.84 21.17 6.82 4.72 5.10 4.19

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Fig. 8. Effect of unit penalty for power deficiency or loss on the optimum solution.

in the market. Selecting the closest unit available in the market affects the net present value (NPV) of the system. The results of a sensitivity analysis, which are given in Table 7, demonstrate that the NPV does not change significantly. 5. Conclusion Reliability considerations have been successfully included in the analysis and optimization of synthesis-design-operation of a cogeneration system. The numerical example has shown that the introduction of reliability leads to an entirely different optimum solution for each one of the three levels (synthesis-design-operation). Furthermore, the example has shown that profits are overestimated when reliability aspects are ignored. Table 7 Effect of the unit standardization on the net present value of the system Unit

Reliability considerations Yes ˙ (kW) W

Available (1) 7918 Optimum 8092 8132 Available (1) (1) Units available in the market. They from below and above.

No NPV (106 GRD)

˙ 1 (kW) W

˙ 2 (kW) W

NPV (106 GRD)

1267.37 5046 2206 1021.07 1271.28 5003 2289 1032.05 1270.70 5089 2400 992.81 were selected so that the total power output is the closest to the optimum

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