Optimal design of a cogeneration plant for power and desalination taking equipment reliability into consideration

Desalination 229 (2008) 21–32

Optimal design of a cogeneration plant for power anddesalination taking equipment reliability into consideration

Ali M. El-Nashar22 Ahmed Gharbo St., Apt. #703, Zizinia, Alexandria, EgyptTel. +20 (012) 382-5263; email: [email protected]

Received 2 April 2007; accepted revised 9 July 2007

Abstract

This paper presents a method for incorporating equipment reliability considerations into the optimal design ofcogeneration systems for power and desalination. Design optimization is carried out using thermoeconomic theoryusing exergy as the transferable material while equipment reliability is carried out using the state–space methodthat uses the Markov process. Procedure for cost allocation of power and water is described. The procedure isapplied in an example cogeneration plant using a simple gas turbine, heat recovery steam generator and MSFseawater desalination plant.

Keywords: Cogeneration; Desalination; Thermoeconomic modeling; Reliability

1. Introduction

In the past two decades, reliability methodshave found widespread application in many in-dustries and they are becoming more popular inthe field of reliability assessment of chemical re-fineries. The main objective of any cogenerationutility in a competitive environment is to supplycustomers with electrical energy and water as eco-nomically as possible and with a high degree ofreliability and quality. The utility companies havebeen making every effort to achieve this objec-tive. To increase competitiveness and market valueof cogeneration systems, it is important to ana-

lyze the influence of equipment reliability on theresulting cost of power and water.

The reliability and economics of a cogenera-tion supply system have always been conflictingparameters. These parameters can be dealt withby establishing quantitative links between them.Such links can best be established by using proba-bilistic criteria which consider the stochastic na-ture of component outages, customer demands,etc. System managers and planners strive to ob-tain the highest possible reliability within the so-cioeconomic constraints. Many researchers andutilities are now using probabilistic approachesto relate to the overall cost to society of provid-

doi:10.1016/j.desal.2007.07.0240011-9164/08/$– See front matter © 2008 P ublished by Elsevier B.V.

22 A.M. El-Nashar / Desalination 229 (2008) 21–32

ing quality and continuity of electric and watersupply to the societal worth (benefit) of havingthat level of reliability. This paper is concernedwith probabilistic reliability evaluation of cogen-eration systems.

2. Design optimization

Design optimization implies finding the opti-mum technical characteristics (specifications) ofthe components and the properties of the sub-stances entering and leaving each componentwhen the system operates at full load (designpoint). The word ‘design’ is used here to implythe technical characteristics (specifications) of thecomponents and the properties of the substancesentering and leaving each component at the nomi-nal load of the system (design condition). Thenominal load is usually called the ‘design point’of the system. The design optimization proceduredescribed here is fully described by El-Sayed [1].

2.1. The general optimization problem

The search for an optimal design involves asearch over alternative system configurations andfor a given configuration over its alternative de-sign points. The number of feasible design pointsis generated by the large number of decision vari-ables that represent the degrees of design free-dom of a given configuration.

In engineering, the objective function is usu-ally a multi-criteria function. Some criteria canbe quantified in money such as fuel, equipmentand maintenance costs. Others involve non-uniqueassumptions such as simplicity, reliability, safety,and health hazards. In the design phase of an en-ergy system, however, concern peaks around twocriteria: fuel and equipment without violating otherdesired criteria. The objective function focuseson fuel and equipment by specifying two resourcesrequired by energy-conversion systems: resourcesto make the devices and resources to operate it.The leading item of the making resources is the

capital cost. The capital cost of a device can beexpressed as

ai iZ c A k= ⋅ +∑ (1)

where cai is a coefficient and Ai is a characteristicarea that specifies the size (or capacity) of com-ponent i. Ai is a function of the design capacity ofthe unit and its efficiency. The capital cost rate Z($/h) can be expressed as the capital cost timesthe capital recovery rate cz

z z aZ c Z c c A k ′= = + (2)

where k′ is a constant.The leading item of the operating resources

are the fuel resource and maintenance expenses.The objective function Ji of device i to minimizeat the device level is:

i zi ai i di iJ c c A c D= ⋅ ⋅ + ⋅ (3)

where both Ai and Di are functions of the capacityand efficiency of the device, tending to increasewith capacity and decrease with efficiency. Therates of operating resources that do not go to theproducts are directly quantified by the rates ofexergy destruction D. In monetary units the oper-ating (fuel) cost rate can be expressed as

1

n

F Di i Fi

C c D c F=

= =∑ (4)

where cDi is the cost of exergy destruction of de-vice i and cF is the cost of primary fuel supply atthe system boundary and F is the fuel rate. Theobjective function at the system level given a siz-ing parameter for the production rate and assum-ing one fueling resource is:

1

1

Minimize n

s F zi ii

n

Di i zi ai ii

J c F c Z

c D c c A

=

=

= ⋅ + ⋅

= ⋅ + ⋅ ⋅

∑

∑ ∑(5)

A.M. El-Nashar / Desalination 229 (2008) 21–32 23

In the above equation the device cost, Zi, isexpressed in terms of a characterizing dimension(surface), Ai which can in turn be expressed interms of capacity (or duty) and efficiency param-eters as follows:

31 2 41 2 3 4

nn n ni iA k x x x x= ⋅ ⋅ ⋅ ⋅ (6)

where k is a constant, x1, x2, x3 and x4 are capacityand efficiency device design parameters and n1,n2, n3, n4 are constant exponents. The cost indexescF and cz in Eq. (5) can be expressed by the fol-lowing simple model given by El-Sayed [1]:

( )( ) ( )( )( )

exp 1 /

/ 1 exp

F Fo f f

z d d

c c n i i n

c i n i

= ⋅ − ⋅

= − − ⋅(7)

where n is the number of years, if is the inflationrate, id is the discount rate and cFo is the initialfuel price.

The minimum exergy destruction for a deviceis formulated in terms of the available area of thatdevice:

mind

i di iD k A= (8)

where kd and d are constants. For a device i theobjective function can be written as:

( )Minimize di i zi ai i di di iJ A c c A c k A= + (9)

This equation gives an optimum device areaand device cost given by:

( ) ( )

( )

1/ 1

opt

min opt opt

dz a

id d

di zi ai i di di i

c cA

c k d

J c c A c k A

−⎡ ⎤−⎢ ⎥=⎢ ⎥⎣ ⎦

= +(10)

The minimum operating cost of the whole sys-tem is the sum of the device minimums:

min mins ii

J J=∑ (11)

The proof of converging local optimizationswith respect to local decision variables to con-verge to a system minimum was shown by El-Sayed [1] by quoting his statement “what is goodfor a process is good for the system”.

2.2. Production cost allocation

The production cost is allocated between thetwo products (electricity and desalted water) asfollows:

(1)The capital cost is divided into three parts, onepart covers the system devices that are servingproduct #1 (electricity) ,Z1, and the second partcovers devices that are serving product #2(water), Z2, and the third part covers the costof common devices serving both products, Z12.The third part is split between electricity andwater in the same way as the capital allocatedto each. Thus the capital allocated to eachproduct can be expressed as:

11

1 1 121 2

22

2 2 121 2

i

i

i

i

ZZ Z ZZ Z

ZZ Z ZZ Z

= ++

= ++

∑

∑(12)

(2)The primary fuel supply, F, is divided betweenproduct #1 (electricity) (F1) and product #2(water) (F2) according to:

• Exergy dissipation in each product devicegroup

• Share of exergy dissipation in commondevices

• Exergy lost (wasted) by product• Exergy contained in product.

1 2F F F= +


1

2

112 1 1

1 121 21

1 2 12 1 2 1 21 2 12

212 2 2

2 121 22

1 2 12 1 1 2 21 2 12

i

i

pi j p

i ip p

i i i j j p pi i i

pi j p

i ip p

i i i p j j pi i i

EF D D E E

E EF

D D D E E E E

EF D D E E

E EF

D D D E E E E

⎡ ⎤+ + +⎢ ⎥

+⎢ ⎥⎣ ⎦=⎡ ⎤

+ + + + + +⎢ ⎥⎣ ⎦

⎡ ⎤+ + +⎢ ⎥

+⎢ ⎥⎣ ⎦=⎡ ⎤

+ + + + + +⎢ ⎥⎣ ⎦

∑ ∑

∑ ∑ ∑

∑ ∑

∑ ∑ ∑

(13)

where Di1 is the dissipation of product #1 device,Di2 is the dissipation of product #2 device, Di12 isthe dissipation of a device common to both prod-ucts, Ep1 is the exergy of product #1, Ep2 is theexergy of product #2, Ej1 is the exergy of thewasted (dumped) stream from product #1 and Ej2is the exergy of the wasted (dumped) stream fromproduct #2. As can be seen from the equationsabove, the dissipation of the common devices issplit between the two products according the frac-tion of exergy carried by each product. The nu-merator in the expressions for F1 and F2 representthe total fuel exergy supplied to the system.

3. Reliability analysis in energy systems

Cogeneration plants are made up of a largenumber of components, with multiple interactionsand functional dependencies. Failure of a com-ponent may result in failure of a sub-system or ofthe whole system with various detrimental con-sequences: loss of power may result in loss ofproduction, in damage of production equipmentand it may cause accidents. Therefore, reliabilityhas to be considered in the design and implemen-tation of energy systems.

Many reliability analysis methods have beendeveloped throughout the years, that can be groupedinto qualitative and quantitative methods [2–4].

In the present work the State–Space Method hasbeen selected for the following reasons: it is ap-propriate for quantitative analysis of availability,reliability and maintainability of systems; it canbe used with large, complex systems; it is not onlyuseful, but often irreplaceable, for assessing re-pairable systems. For the reader’s convenience, itis described in brief in the following. The methodconsists of three steps:

3.1. Step 1— Identification of all functional andfailure modes of the system my making an inven-tory of all possible sytates

A Markovian process with discrete states anddiscrete time is a Markovian chain. For such aprocess it is convenient to consider the momentsof time t1, t2,… when the system S can change itsstate, as successive steps in the process, from theinitial state S(0) to the states S(1), S(2),….,S(k),…The event {S(k) = si} that the system is found inthe state si immediately after the kth step (i =1,2,…) is a random event. Therefore, the sequenceof states S(0), S(1), S(2),…,S(k)….can be viewedas random events.

Let P(k) is the probability that the system S isin the state si, i = 1,2…,n after taking the kth stepand prior to the (k + 1)th step. The set of prob-abilities Pi(k) is the probabilities of the states of aMarkovian Chain. For any k we must have:


1( ) 1 0,1,2...

n

ii

P k k=

= =∑ (14)

For a system of K components the number ofpossible states is

2KI = (15)

3.2. Step 2 — Establishment of all rules for tran-sition between states and formulation of the tran-sition rate matrix (TRM)

The probability of transition of a system fromstate i to state j, Pij , is a conditional probabilitythat the system S will be found in the state sj aftertaking the kth step, provided that it was in state siafter the previous (k – 1)th step was taken. Thiscan be expressed as:

( ) ( ){ }| 1j i ijP S k s S k s P= − = = (16)

The first index i is the state of the system atthe earlier moment of time and the second j thelater moment. A system having n states have atransition probability matrix given by:

11 12 1 1

21 22 2 2

1 2

1 2

... ...

... ...... ... ... ... ... ...[ ]

... ...

... ...

j n

j n

ij

i i ij in

n n nj nn

P P P PP P P P

PP P P PP P P P

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(17)

( )1

1 0,1,2...n

ijj

P k i=

= =∑ (18)

The knowledge of the initial probability Pi(0)and the transition matrix [Pij] enables one to evalu-ate the probability Pi(k) at any step k. This can bedone on the basis of the recursion formula:

( ) ( )1

1 , 1,2,...n

i j jij

P k P k P i j n=

= − ⋅ =∑ (19)

A Markovian process with discrete states butcontinuous time is a continuous Markov chain.For such a process, the probability of transitionfrom state si to state sj , Pij is replaced by the tran-sition probability density λij

The probability of state i, at any time t, Pi(t),as given by Kolmogorov [3] can be expressed by:

( ) ( ) ( )1

d

d 1, 2, ...,

ni

ji j i ijj

P tP t P t

ti n

=

= λ − λ

=

∑ ∑ (20)

where ( )1

1n

ii

P t=

=∑

For a two-component system we have fourstates as shown in Fig. 1. Applying Kolmogo-rov’equation for each state we get [5–7]:

( )( )

( )( )

1 2 1 2

1 2 1 2

2 1 2 1

2 1 1 2

1 01 0

0 10 1

ijP

⎡ − λ + λ λ λ ⎤⎢ ⎥μ − λ + μ λ⎢ ⎥⎡ ⎤ =⎣ ⎦ ⎢ ⎥μ − λ + μ λ⎢ ⎥

μ μ − μ + μ⎢ ⎥⎢ ⎥

(21)


State 1 #1 Up #2 Up

State 2 #1 Down #2 Up

State 3 #1 Up #2 Down

State 4 #1 Down #2 Down

μ1

λ1

μ1

λ2 μ2 λ2 μ2

λ1

Fig. 1. Example state-space graph of a two-component system.

The steady state probabilities were derived asfollows:

( )( )

( )( )

( )( )

( )( )

1 21

1 1 2 2

1 22

1 1 2 2

2 13

1 1 2 2

1 24

1 1 2 2

P

P

P

P

μ μ=

λ + μ λ + μ

λ μ=

λ + μ λ + μ

λ μ=

λ + μ λ + μ

λ λ=

λ + μ λ + μ

(22)

Consider the two components as representingthe gas turbine (GT) and the exhaust heat steamgenerator (EHSG) which are two components inseries. The probability of the two components inthe Up State is P1 as given by Eq. (23). If the twocomponents are represented by a single modelcharacterizing the failure (λs) and repair rates (μs)of the combined system, it can be shown that [5,8–10]:

1 2sλ = λ + λ

1 2

1 1 2 2 1 2 1 2/ / /( )sλ + λ

μ =λ μ + λ μ + λ λ μ μ

(23)

The expression for μs in the above equationcan be approximated by noting that the term λ1λ2/(μ1μ2) in the denominator of the second equationis much small that of the other two terms:

1 2

1 1 2 2/ /sλ + λ

μ =λ μ + λ μ (24)

In general for n-component series system wehave:

1

1

1/

n

n ii

n

ii

n n

i ii

=

=

=

λ = λ

λμ =

λ μ

∑

∑

∑

(25)


3.3. Step 3 — Evaluate the cost of products andnet profit

The expected value of product cost can beobtained using the state probabilities Pi’s asweights for every possible operating state, Pi,obtained in Step 2 above. The product costs canbe obtained from the equations

e i eii

w i wii

c Pc

c Pc

=

=

∑

∑(26)

4. Application to a cogeneration system forpower and desalination

As a demonstration of the application of thedesign optimization and reliability proceduresoutlined above and the resulting cost of the prod-ucts from a typical cogeneration plant for powerand desalination, the following example is given.

compressor expander

combustor

HRSG

Brine heater

MSF Plant Gas Turbine

air

fuel

generator

exhaust gas

steam

distillate

seawater

brine

Fig. 2. Schematic of the cogeneration plant.

4.1. Description of the example system

Fig. 2 shows a schematic of the cogenerationsystem considered. The example system is a co-generation plant for power and seawater desali-nation which consists of a simple cycle gas tur-bine (GT), a heat recovery steam generator(HRSG) and a multistage flash (MSF) plant ofthe brine recycle type. The gas turbine has nomi-nal power of 100 MW and a net power output of95 MW. The MSF unit has a rated capacity of7.7 MGD (1431.7 t/h) and has a top brine tem-perature of 115°C. Basic features of the gas tur-bine have a multistage axial compressor wheretip blade speed and axial air velocity are kept con-stant at 350 m/s and 150 m/s, respectively [1].Mass rate, pressure ratio and temperature rise perstage are varied and the number of stages, totalsurface of fixed and moving blades, adiabatic ef-ficiency and rotor speed are computed. The gasexpander is axial with un-cooled blades having atip speed of 240 m/s. The inlet gas temperature isfixed at 870°C. The total blade surface of the fixed


and moving blades is correlated in terms of gasrate, expansion ratio and efficiency parameterη/(1 – η).

The heat recovery steam generator (HRSG) isa single-pressure water tube boiler of the radianttype. The water boils in the tubes and the vaporformed is separated in an upper drum. Heat ex-change takes place by convection, conduction andradiation between the hot gas flowing in a ductand water inside the tubes. The surface of the walltubes is correlated in terms of the rate of heat trans-fer and the conventional logarithmic mean tem-perature difference.

Steam produced by the HRSG is first throttledbefore going in the brine heater of the MSF plantto reduce its pressure to about 1.5 bar. The MSFplant is a brine recycle-type and has a top brinetemperature of 115°C and a bottom design tem-perature of 35°C.

4.2. Thermoeconomic modeling of the system

The cost models provided by El-Sayed [1]were used for the main plant components and aregiven below:

Compressor:0.45 0.453389.4 Pr

25 (kg/s) 455; 5 Pr 15; 2.3 11.5

Z M eMe

= × × ×≤ ≤ ≤ ≤≤ ≤

(27)

Gas expander:0.5 0.857263 Pr

25 (kg/s) 455; 5 Pr 15; 4 19

Z M eMe

−= × × ×≤ ≤ ≤ ≤≤ ≤

(28)

Combustor:

0.5 0.24 0.75561.1 d 180 (kg/s) 410; 0.34 (MPa) 1.38; 0.01 (kPa) 0.3

Z M P PMPdP

−= × × ×≤ ≤≤ ≤≤ ≤

(29)

Heat recovery steam generator:1 0.33 0.26

o

10393.6 d d

25 (MW ) 55; 40 ( C) 110; 40 d (kPa) 90; 0.4 d (kPa) 3

m t s

t m

t s

Z Q T P P

Q TP P

− − −= × ×Δ × ×

≤ ≤ ≤ Δ ≤≤ ≤ ≤ ≤

(30)

Brine heater:0.7 0.08 0.04

ot

157.8 d d

40 (MW ) 185; 5 ( C) 15; 0.1 d (kPa); 0.001 d (kPa) 1.3

t t s

t

t s

Z Q T P P

Q TP P

− − −= × ×Δ × ×

≤ ≤ ≤ Δ ≤

≤ ≤ ≤

(31)

Multistage flash plant:0.75 0.5 0.1

ot

o

688 d

14 (MW ) 110; 1.7 ( C) 6;

1.7 ( ) 7; 13 d (kPa) 70

n t t

n

t t

Z Q T T P

Q T

T C P

− − −= × ×Δ ×Δ ×

≤ ≤ ≤ Δ ≤

≤ Δ ≤ ≤ ≤

(32)

where Z is the device capital cost in US$, M is themass flow rate in kg/s, Pr is the pressure ratio, Pis the pressure in MPa, dP is the pressure drop inkPa, with subscript t refers to tubes, subscript srefers to shell, ΔT is the temperature drop in °Cwith subscript m refers to log-mean temperaturedifference, n refers to flash drop per stage, Q re-fers to heat transfer rate in kW and e is the effi-ciency ratio e/(1 – e).

For the economic evaluation, the capital re-covery is assumed 10% and the number of plantoperating hours per year 8000.

5. Results

For repair times of r1 = 20 d, r2 = 5 d, and r3 =15 d for the GT plant, the HRSG and the MSFplant we can write:

μ1 = 0.05 repair/d, μ2 = 0.2 repair/d,μ3 = 0.067 repair/d

The availability of the system that consists ofthe three components (GT, HRSG and MSF plant)is the probability of the system being in an Up


State P0. Assuming identical failure rate for eachof the three components, λ1= λ2 = λ3 = λ, we get:

0

1 2 3

1 2 3

1 2 31 2 3

1 2 3

(20 5 15 )

( )(20 5 15 )

11 40

s

s s

P μ=λ + μ

λ + λ + λλ + λ + λ

=λ + λ + λ

+ λ + λ + λλ + λ + λ

=+ λ

(33)

1.881.9

1.921.941.961.98

22.022.042.062.08

0 0.2 0.4 0.6 0.8 1 1.2

System failure rate, failure/y

Cos

t of w

ater

, $/m

3

Fig. 3. Effect of system failure rate on the cost of water.

The cost allocated to water and power are cal-culated using Eq. (9) after the optimum (lowest)production cost (capital charge plus fuel cost plusO&M cost) has been achieved by the optimiza-tion program. The effect of cogeneration systemfailure rate on the cost of water and power isshown in Fig. 3 and Fig. 4, respectively. As shown,the cost of each increases with increasing systemfailure rate due to the increase in unexpected (un-planned) system downtime for repair action.

Fig. 5 shows the influence of the primary fuelcost on the cost of water with and without reli-

Fig. 4. Effect of system failure on the cost of power.

0.092

0.0930.094

0.0950.096

0.0970.098

0.0990.1

0.101

0 0.2 0.4 0.6 0.8 1 1.2

System failure rate, failure/y

Cos

t of p

ower

, $/k

Wh


ability considerations. The results shown here areapplicable to an assumed equipment failure rate˜ = 0.0027 failure/d (equivalent to 1 failure/y)and the repair rates assumed above. Based on thesevalues we have P0 = 0.9025.

As can be seen from this figure, the effect ofthe inclusion of equipment reliability is to increasethe water cost due to unexpected equipment down-time resulting from failure and subsequent equip-ment repair. The same trend applies for the costof power as shown in Fig. 6.

The sensitivity of the cost of water and elec-tricity to changes in equipment capital cost is

00.5

11.5

22.5

33.5

4

Cos

t of w

ater

, $/m

3

0.03 0.04 0.05 0.06 0.07

Cost of primary fuel, $/kWh

cost of water (no reliability), $/m3 cost of water (with reliability), $/m3

Fig. 5. Cost of water with and without reliability considerations (λ = 0.0027, μ1 = 0.05, μ2= 0.2, μ3= 0.067).

Fig. 6. Cost of power with and without reliability considerations (λ = 0.0027, μ1 = 0.05, μ2= 0.2, μ3= 0.067).

0

0.04

0.08

0.12

0.16

0.2

Cos

t of p

ower

, $/k

Wh

0.03 0.04 0.05 0.06 0.07


cost of power (no reliability) $/kWh cost of power (with reliability) $/kWh

shown in Fig. 7 and Fig. 8, respectively. As canbe seen, a 50% increase in capital cost results inapproximately 15% increase in water cost and a5% in power cost.

The sensitivity of the optimal performanceratio (PR) and number of stages of the MSF plantto variation in the primary fuel cost is shown inFig. 9. The PR is seen to increase slightly as thefuel cost increases but soon reaches a plateauwhere it reaches a limiting value of about 10. Thenumber of stages increases from 24 at a fuel costof 0.03 $/kWh to 27 at high fuel costs.


1.7

1.8

1.9

2

2.1

2.2

2.3

Cos

t of w

ater

, $/m

3

0.9 1 1.1 1.2 1.3 1.4 1.5

Capital cost multiplication factor

Fig. 7. Sensitivity of cost of water to variation in capital cost of equipment (cost of primary fuel = 0.03 $/kWh).

0.090.0910.0920.0930.0940.0950.0960.0970.0980.099

Cos

t of p

ower

, $/k

Wh

0.9 1 1.1 1.2 1.3 1.4 1.5

Capital cost multiplication factor

Fig. 8. Sensitivity of cost of power to variation in capital cost of equipment (cost of primary fuel = 0.03 $/kWh).

0

5

10

15

20

25

30

PR

and

num

ber o

f sta

ges

0.03 0.04 0.05 0.06 0.07


no. of MSF stages performance ratio

Fig. 9. Optimal performance ratio and number of stages of MSF plant.


6. Conclusions

Reliability considerations have been success-fully incorporated in the design optimization ofcogeneration systems for power and desalination.In the present work the State-Space Method hasbeen selected for the reliability analysis. Designoptimization using the field of thermoeconomicsand second law (exergy) analysis. The numericalexample given has shown that the introduction ofreliability leads to higher product costs due to re-duced plant uptime. The example has shown thatreliability considerations are important and shouldbe carried out in any cogeneration system design.

Symbols

A — Area, m2

cF — Cost of fuel, $/kWhtD — Exergy destruction, kWdP — Pressure differerence, kPadT — Temperature difference, oCe — Efficiency ratio, η/(1 – η)E — Exergy, kWF — Primary fuel rate, kWid — Discount rateif — Inflation rateI — Number of statesk — Time stepK — Number of componentsM — Mass flow rate, kg/sn — Number of yearsP — Pressure, MPa, state probabilityP0 — Initial state probabilityPr — Pressure ratioQ — Heat rate, kWJ — Objective functionT — Temperature, Kt — Time, hx — Design parameterZ — Capital cost of equipment, US$

Greek

η — Efficiencyλ — Failure rate, failure/d

λij — Transition rate, transition/dμ — Repair rate, repair/d

Subscripts

1,2,3— System components or design param-eters

e — Electrical poweri, j — State indexesp1 — Product #1p2 — Product #2s — Shell side, systemt — Tube sidew — Desalted water

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Optimal design of a cogeneration plant for power and desalination taking equipment reliability into consideration