1 GMC 2004

Compressor Station Optimization Using

Simulation-Based Optimization

Kirby S. Chapman Mohammad Abbaspour Prakash Kirishnaswami Professor and Director Associate Researcher Professor chapman@ksu.edu mabbas@ksu.edu prakash@ksu.edu

National Gas Machinery Laboratory, Kansas State University

245 Levee Dr., Manhattan, KS 66502 Phone: 785-532-2319, Fax: 785-532-3744

ABSTRACT Mathematical modeling is one of the most cost-effective tools that can be used to aid in design, operation, and optimization studies. The systems under consideration actually operate in an unsteady nature, and although much effort has been and continues to be spent on unsteady mathematical models, many over-simplifications are introduced that bring into question the simulation results. One of the primary concerns in the operation of a compressor station is minimization of fuel consumption while maintaining the desired throughput of natural gas. In practice, the station operator tries to achieve this by shutting down units or controlling individual unit speeds based on experience. This is generally a trial-and-error process without any guarantee of optimality. In this paper we present a robust structured solution process for tackling this problem using simulation-based optimization. The problem of optimizing the operation of a compressor station is formulated as a nonlinear programming problem (NLP) in which the design variables are the compressor unit speeds and the objective function to be minimized is the fuel consumption. A constraint was also placed on the minimum mass flow rate through the station to ensure that adequate flow is maintained while minimizing fuel consumption. This NLP was then solved using a sequential unconstrained minimization technique (SUMT) with a derivative-free grid search for handling the unconstrained minimizations. The simulation algorithm mentioned earlier is invoked whenever the optimization needs to evaluate the system response at a candidate operating point. Several representative numerical examples have been solved by the proposed approach. The results obtained indicate that the method is very effective in finding operating points that are optimal with respect to fuel consumption. The optimization can be done at the level of a single unit, a single compressor station,

a set of compressor stations, or an entire network. It should also be noted that the proposed solution approach is fully automated and requires no user involvement in the solution process. This work is a part of the DOE project “Virtual Pipeline System Testbed to Optimize the U.S. Natural Gas Transmission System.”

INTRODUCTION Natural gas enters the pipeline from a supply source, and then is transported to one or more delivery points. One of the most important collections of components in this system is the compressor station located about every 60 miles along the pipeline. The compressor station overcomes the gas pressure drop in the pipe. Consequently, detailed mathematical modeling of compressor stations is critical for optimizing and understanding the ability of the gas pipeline system to deliver natural gas to the end-user. Several investigators tried to simulate unsteady conditions for pipeline systems and some of them focused in compressor station modeling. Botros et al. [1,2] and Botros [3] presented a dynamic compressor station simulation that consisted of nonlinear partial differential equations describing the pipe flow together with nonlinear algebraic equations describing the quasi-steady flow through various valves, constrictions, and compressors. This model included mathematical descriptions of the control system, which consists of mixed algebraic and ordinary differential equations with some controller limits. Bryant [4] modeled compressor station control, which had some advantages such as the ability to set individual unit swing priority, the ability to try and meet multiple setpoints and the ability for units to automatically come on-line and off-line. The model used automatic linepack tuning instead of automatic pipeline roughness tuning. Stanley and Bohannan [5] discussed the application of dynamic simulation to centrifugal compressor control system

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design. The simulation studies resulted in design recommendations concerning the number and location of recycles required, sizing of recycle control valves, and setpoint, gain, and reset settings for control system instrumentation. This paper solves equations in ordinary form for compressors without considering pipe equations into the compressor station. Turner and Simonson [6,7] developed a computer program for compressor stations that is added to SIROGAS, which is a program that solves a pipeline network for steady state and transient mode. Schultz [8] derived the real-gas equations of polytropic analysis to show their application to centrifugal compressor testing and design. Odom [9] reviewed the theory of centrifugal compressor performance, and also presented a set of polynomial equations for centrifugal compressor maps, which used constant coefficients for these equations for different compressors. Carter [10] presented a hybrid mixed-integer-nonlinear programming method that is capable of efficiently computing exact solutions to a restricted class of compressor models, and attempted to place station optimization in context with regard to simulation. Letniowski [11] presented an overview of the design process for a compressor station model that is part of a network model. Jenicek and Kralik [12] developed optimized control of a generalized compressor station. The model described an algorithm for optimizing the operation of the compressor station with fixed configuration. Botros [13] presented a numerical study of gas recycling during surge control, and furnished a basic understanding of the thermodynamic point of view and showed the variation of gas pressure, temperature and flow. Boyd et al. [14] considered the fuel cost minimization problem in the steady-state gas pipeline networks by using mathematical model over compressor station. Carter [15] developed a nonsequential Dynamic Programming (DP) algorithm to handle looped networks when the mass flow rate variables are fixed. The main advantages of DP are that a global optimum is guaranteed to be found and that nonlinearity can be easily handled. Wu et al. [16] presented a two-model relaxation, one in the compressor domain and another in the fuel cost function, and derived a lower bounding scheme. The empirical evidence has been presented that showed the effectiveness of the lower bounding scheme. Cobos-Zaleta and Rios-Mercado [17] used a MINLP model for the problem of minimizing the fuel consumption in a pipeline network. A computational experience was presented by evaluating an outer approximation with equality relaxation and augmented penalty method. Siregar et al. [18] developed a mathematical model, which in turn solved analytically and numerically the optimum pipeline diameter and routing. Edgar et al. [19] presented a computer simulation to optimize the design of a gas transmission network, which considered the number of compressor stations, the diameter and length of pipeline segments, and the operating conditions of each compressor station. Two solution methods were used. Osiadacz [20] described the algorithm for optimal control of a gas network based upon hierarchical control and decomposition of the network. This work is concerned with the

minimization of operating costs for high-pressure gas networks under transient conditions. The work presented in this paper is an important advance over current methods in the accurate simulation of transient non-isothermal behavior in natural gas pipelines, and extends the knowledge found in the literature by demonstrating the impact of varying boundary conditions on compressor station components. In addition, it also shows how this type of detailed simulation can be used for optimizing the operation of a compressor station to minimize fuel consumption while maintaining desired throughput.

Nomenclature A Cross- sectional area of pipe (m2)

1b - 6b Coefficients for centrifugal compressor map (-)

pC Specific heat at constant pressure (J/kg. K)

D Pipe diameter (m) f Friction factor (-)

g Gravitational acceleration (m/s2)

h Specific enthalpy (J/kg) Head Isentropic head (kJ/kg) LHV Low heating value (kJ/kg) m& Mass flow rate (kg/s) N Number of node (-)

rN Speed (rpm) P Pressure of the gas (Pa) Q Capacity (m3/hr) R Specific gas constant (kJ/kg) t Time (s) T Temperature (K) v Velocity of the gas directed along the axis of the

pipe (m/s)

wV Isentropic wave speed (m/s)

W Frictional force per unit length of pipe and per unit time (N/m)

x Distance along the pipe (m) Z Compressibility factor (-) η Efficiency (-)

gγ Specific gravity (-)

θ Angle of inclination of pipe to the horizontal (radian)

σ Isentropic exponent (-) ρ Density of the gas (kg/m3)

Ω Heat flow (J/ms) Subscripts ac Actual condition

d Discharge dr Driver f Fuel

is Isentropic

mech Mechanical

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s Suction

Sc Standard condition

Governing equations

The non-isothermal compressible flow of natural gas in a compressor station is governed by the time-dependent continuity, momentum, and energy equations, and an equation of state for homogeneous, geometrically one-dimensional pipe flow. The compressors within the compressor station are modeled using centrifugal compressor map-based polynomial equations.

Pipe Equations

Chapman and Abbaspour [21,22,23] developed the basic equations for one-dimensional, unsteady, compressible flow that include the effects of wall friction and heat transfer. The continuity, momentum, and energy equations as functions of pressure, temperature and mass flow rate are written as:

2

1 1w

T

VP ZRT P Z Pt PA ZRT Z P x

m ∂ ∂ ∂ + − − ∂ ∂ ∂

&

2 1 11w

P

V m T Z TA x T Z T x

mm

∂ ∂ ∂ + + + ∂ ∂ ∂

&&&

2

21w

P

V T Z ZRT wCpT Z T A PA

m

∂ Ω= + +∂

& (1)

21

T

ZRT ZRT P ZPAP t PA x Z P

m m m m ∂ ∂ ∂ + − − ∂ ∂ ∂

& & & &

1P

P ZRT P P T Zt PA x T Z T

m m ∂ ∂ ∂ × + + + ∂ ∂ ∂

& &

ZRT P

P x

T ZRT Tt PA x

m ∂+

∂∂ ∂ × + ∂ ∂

&

sinwZRT

gPA

θ= − − (2)

2

21w

P

VT T Z ZRTP

t Cp Z T AP xm ∂ ∂ ∂ + + − ∂ ∂ ∂

&

1T

P Z P ZRTZ P x PA

mm ∂ ∂ − + ∂ ∂

&&

22

1 1w

P

V T Z TCpT Z T x

∂ ∂ × + + ∂ ∂

2

1w

T

V P ZCpP Z P

∂ = − ∂ 2

mZRTw

A PAΩ +

& (3)

The wave speed

wV is:

2

1 1w

T P

ZRTV

P Z P T ZZ P CpT Z Tρ

= ∂ ∂ − − + ∂ ∂

(4)

The parameter Z is the compressibility factor (Dranchuck

[24]):

2 33 5 621 43 31

A A AAZ A r A r r

Tr Tr Tr Trρ ρ ρ = + + + + + +

(5)

To simulate the compressor station, the following equations are used to describe the performance of a centrifugal compressor. Compressor head is determined by:

0.28704 1s s d

g s

T Z PHead

P

σ

σγ

= −

(6)

and the relationship between the flow rate for standard conditions and the actual mass flow rate is :

397.67 10

acsc

m RQ

−=

×& (7)

The power required by the compressor for these conditions is:

mechis

acHead mPower

η η×

=&

(8)

For the purpose of inputting centrifugal compressor characteristics into a pipeline simulation model, it is suggested that the entire head versus capacity map be digitized and stored as a table. However, a simplified but still accurate representation of the head versus capacity curve can be obtained through the use of the normalized characteristics. Figure 1 shows a sample compressor map.

Fig.1 Compressor Map [9]

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Three normalized parameters are necessary to describe a

compressor map, 2rHead N ,

ac rQ N , isη . Using standard polynomial curve-fit procedures for each centrifugal compressor, the relationship between these parameters is:

2

2 1 2 3ac ac

r rr

Q QHeadb b b

N NN+ +=

(9)

and:

2

54 6r r

ac acis

Q Qb b b

N Nη +

= +

(10)

The coefficients 1b , 2b , 3b , 4b , 5b , 6b that make Eqs. 9 and 10 fully characterize the specific centrifugal compressor map. With the coefficients for Eqs. 9 and 10 stored in the computer, knowing the isentropic head and inlet volumetric flow allow computation of compressor speed and isentropic efficiency. The fuel consumption for the compressor driver is currently obtained by:

drf

Power

LHVm

η=

×& (11)

The gas discharge temperature is obtained by:

1100s d

sdsis

PTT T

P

σ

η = + −

(12)

and the mass balance for suction and discharge of the compressor is:

, ,ac s a c d fm mm = +& && (13)

The fully implicit method consists of transforming Eqs. 1, 2, and 3 from partial differential equations to algebraic equations by using finite difference approximations for the partial derivatives. These equations are nonlinear and the Newton-Raphson method is applied to solve these equations for the compressible, non-isothermal transient flows through a pipe. Quasi-steady flow can be assumed at each time step of the numerical solution for the centrifugal and reciprocating compressor equations. Formulation of the Optimization Problem In order to optimize the operation of a pipeline network, we first formulate the problem at hand in the format of a standard nonlinear programming problem (NLP). This standard form is developed as: Find the values of the design variables: [b1,b2,….,br]

T to minimize an objective function: f(b)

Subject to the constraints: hj(b) = 0 , j = 1,…, m and gj(b) ≤ 0 , j = m+1,…,n The formulation of the network operation problem in the standard NLP form must be done carefully, making sure that the NLP formulation captures all the relevant aspects of the associated network problem. Let have the following assumptions: N Number of compressor stations in the pipeline network NCj Number of compressors in station j nik Speed of compressor k in station i. nminik Minimum speed of compressor k in station i nmaxik Maximum speed of compressor k in station i mfi Fuel consumption rate of station i mi Mass flow rate at station i and and let the specified mmini Minimum allowable mass flow rate at station Then, the set of design variables is defined by nik, i = 1,…,N; k = 1, …NCi while the objective function is given by f = Σ(mfi) , i = 1,…,N and the constraints are nminik≤ nik ≤ nmaxik, i = 1,…,N; k = 1, …NCi mmin ≤ mi , i = 1,…,N. Solution of the Optimization Problem Once the network operation problem has been formulated as an optimization problem as outlined above, it can be solved using any of a variety of available methods. In this work, we used the sequential unconstrained minimization technique (SUMT) with an exterior penalty function. A directed grid search method was used for the unconstrained minimization that is required by the SUMT approach.

RESULT AND DISCUSSION

In this section we consider two parts, first a baseline simulation and second optimization.

Simulation Figure 2 shows the schematic of a compressor station, including the boundary conditions and geometry of the compressor station. Note that the numbers of compressors and boundary conditions could be different for different stations.

A

C

1 2 3 4

D= 23.62 in / 0.6 mL=62.14 mile / 100 km

D= 11.81 in / 0.3 mL=328 ft/ 100 m

P1=896.91Psi / 6.184 MpaT1=599.67 R / 333.18 K

P4= 740 Psi / 5.11 Mpa

B

D= 23.62 in / 0.6 mL=62.14 mile / 100 km

Fig. 2 Schematic of compressor station

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The compressor station is located between two long pipes (100 km/ 62.14 mile). For this simulation 50 nodes were used to dis cretize the compressor station for the inlet and outlet pipes and five nodes were used for the internal pipes in the compressor station. Three different types of centrifugal compressors are used for this simulation, i.e., each compressor has a different map and different parameters. The panhandle equation is used to initialize the mass flow rate and pressure drop in the pipe to start the simulations. At the first step time, the mass flow rates at each node adjust to the value that is calculated from the transient equation that is different from the initial value. These values show the transient behavior of flow inside the pipe. A constant pressure boundary condition was applied at the head of the inlet pipe that enters the compressor station, and the end of the outlet pipe that exit the compressor station, as shown in Figure 2. Another boundary condition for this simulation is constant speed for each compressor as follows:

13000rAN = rpm

12000rBN = rpm

11000rCN = rpm The formulation presented earlier was used to simulate the transient performance of this station. Figure 3 shows the variation of mass flow rate for inlet and outlet pipes at the compressor station for different nodes. As shown in this figure, between 50 and 100 minutes is required for the mass flow rate to become uniform throughout the pipeline segment. At this time the flow considered to be at steady- state.

Time (min)

Mas

sflo

wra

te

0 50 100 150 200100

120

140

160

180

200

220

240

Node 50

Node 1

x 1 kg/sx 3.4434 MMSCFD

Time (min)

Mas

sflo

wra

te

0 50 100 150 200100

120

140

160

180

200

220

240

Node 50

Node 1

x 1 kg/sx 3.4434 MMSCFD

(a) (b) Fig. 3 Variation of mass flow rate for the compressor station inlet (a) and outlet (b) the pipes In the same manner, we can explain the results for the outlet pipe as shown in Figure 3-b. Because different types of compressors were used, the mass flow rate through each compressor will be different as show in Figure 4-a.

Time (min)

Mas

sflo

wra

te

0 50 100 150 20050

55

60

65

70

75

80

Comp. No. A

Comp. No. BComp. No. C

x 1 kg/sx 3.4434 MMSCFD

Time (min)0 50 100 150 2000.11

0.12

0.13

0.14

0.15

0.16

0.17

Comp. No. A

Comp. No. B

Comp. No. C

Fuel Consumption

x 1 kg/sx 3.4434 MMSCFD

(a) (b) Fig. 4 Variation of mass flow rate (a) and fuel consumption (b) for different types of compressor in compressor station

The fuel consumption for this compressor station is found by using the difference in the mass flow rate between the inlet and outlet of the compressor station as shown in Figure 4-b. Figure 5 shows the temperature variation with respect to time in the inlet and outlet pipes of the compressor station for different nodes. The temperature at the head of the inlet pipe is a constant boundary condition. Because of the heat transfer between the pipe and surroundings, the gas temperature will decrease in the flow direction and after about 22 km the gas temperature will be constant (surrounding temperature).

Time (min)

Tem

per

atur

e(D

imen

sion

less

)

0 50 100 150 200290

300

310

320

330

340

0 km

2 km

6 km

10 km18 km

>22 km

x 1 Kx 1.8 o R

Time (min)

Tem

per

atur

e(D

imen

sion

less

)

0 50 100 150 200290

300

310

320

330

340

3500 km

2 km

6 km

10 km

18 km>22 km

x 1 Kx 1.8 oR

(a) (b) Fig. 5 Temperature distribution for inlet (a) and outlet (b) pipe at compressor station Figure 6-a shows the change in power with respect to time for the three centrifugal compressors at the compressor station. As shown in this figure, after a while the power will reach constant value, which is the steady state condition. Fig. 6-b shows the variation of isentropic efficiency with respect to time for each compressor.

Time (min)0 50 100 150 2002400

2600

2800

3000

3200

3400

3600

Comp. No. A

Comp. No. B

Comp. No. C

Power

x 1 KWx 1.341 HP

Time (min)0 50 100 150 20068

70

72

74

76

78

80 Comp. No. A

Comp. No. B

Comp. No. C

Effeciency

(a) (b) Fig. 6 Variation of power (a) and efficiency (b) for different types of compressor in compressor station Figure 7 shows the variation of head with respect to flow rate for each compressor and we can see that all compressors reach the same value for head because of maintaining the discharge pressure, but the flow rate in each one becomes different.

Flow rate

Hea

d

0.8 0.9 1 1.1 1.2 1.3 1.4

25

30

35

40

45Comp. No. AComp. No. BComp. No. C

x 1 m3/s

x 2118.88 cfm

x 1 kj/kgx 334.553 ft.lbf/lbm

Fig. 7 Variation of head with respect to flow rate As shown in this section, the model has an ability to accept

different boundary and initial conditions using different types

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and numbers of compressors that show the behavior of operating parameters for different compressors.

Optimization The optimization examples presented in this paper have

been carefully selected to illustrate specific points. In first example, we consider a compressor station with three dissimilar compressors; this is a common situation in practice and the optimum operating condition is difficult to find by other means. The second example consists of a fourteen compressor station, and is designed to illustrate the application to situations where unit shutdowns have to be taken into account. Example 1:

The system considered here is a single compressor station with three dissimilar compressors as shown in Figure 2. In this case, the three units are not identical and they each have different compressor maps. The compressor speed limits for this case are given in Table 1, and the goal of the optimization is to minimize the total fuel consumption while maintaining a station throughput of 170 kg/s (585.38 MMSCFD).

Table 1- The input data for optimization example 1

NrA NrB NrC Initial Value 13000 rpm 12000 rpm 11000 rpm Max. Value 15000 rpm 15000 rpm 15000 rpm Min. Value 10000 rpm 10000 rpm 10000 rpm Minimum Mass flow rate

170 kg/s – 585.38 MMSCFD

The results obtained by optimization are shown in Table 2.

It is seen that the optimal solution in this case gives us three different speeds for the three compressors (12650, 11650, and 10650 rpm), and the final mass flow rate is close to its minimum allowable value at 170.07 kg/s (585.62 MMSCFD). The optimization reduces the fuel consumption by 3.49×10-3 kg/s (11.94×10-3 MMSCFD), which is a savings of about 8.15%. At the optimum, none of the units is operating at a limiting speed.

Table 2- Final result for speed, fuel consumption, temperature

and efficiency for optimization example1

Initial Final NrA (rpm) 13000 12650 NrB (rpm) 12000 11650 NrC (rpm) 11000 10650

Fuel Consumption (kg/s- MMSCFD) ×103

42.86 – 147.59 39.39 - 135.65

Mass flow rate (kg/s- MMSCFD)

173.12- 596.13 170.07- 585.62

isAη 79.75 79.60

isBη 79.35 79.35

isCη 76.98 76.97

Outlet Temp. T3 (K) 342.67 340

Example 2: The system considered here is a single compressor station

with fourteen identical compressors as shown in Figure 8. The maximum and minimum values of speed for each compressor are 15000 and 10000 rpm respectably and the goal of the optimization is to minimize the total fuel consumption while maintaining a station throughput of 600 kg/s (2066.06 MMSCFD). In this case, we also consider the possibility that the optimum operating condition for this station may require the shutdown of one or more units.

2

3

4

5

6

9

10

11

12

13

A B C D

PA=6.183977MPaTA= 330.15 K

PD=5.10212MPa

D= 35.43 in / 0.9 mL=62.14 mile / 100 km

D= 39.37 in / 1 mL=62.14 mile / 100 km

D= 11.81 in / 0.3 mL=328 ft/ 100 m

D= 11.81 in / 0.3 mL=328 ft / 100 m

7

8

1

14

Fig. 8 Compressor station using fourteen compressors Accordingly, we run the optimization separately using 9,

10, 11, 12, 13, and 14 compressors as shown in Table 3. By comparing the optima thus obtained, we can see that the best solution is to operate twelve compressors, with five compressors running at 13,725 rpm and the remaining seven compressors running at 13,750 rpm.

Fig. 9 Compressor station optimization map

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Table 3. Final result for speed, isothermal efficiency, outlet temperature, and fuel consumption of optimization example 2

14 Compressors on-line 13 Compressors on-line 12 Compressors on-line Initial Final Initial Final Initial Final Nr1 (rpm) 15000 13475 15000 13537.5 15000 13725 Nr2 (rpm) 15000 13450 15000 13575 15000 13725 Nr3 (rpm) 15000 13475 15000 13575 15000 13725 Nr4 (rpm) 15000 13475 15000 13575 15000 13725 Nr5 (rpm) 15000 13475 15000 13575 15000 13725 Nr6 (rpm) 15000 13475 15000 13575 15000 13750 Nr7 (rpm) 15000 13475 15000 13575 15000 13750 Nr8 (rpm) 15000 13475 15000 13575 15000 13750 Nr9 (rpm) 15000 13475 15000 13575 15000 13750 Nr10 (rpm) 15000 13475 15000 13575 15000 13750 Nr11 (rpm) 15000 13475 15000 13575 15000 13750 Nr12 (rpm) 15000 13475 15000 13575 15000 13750 Nr13 (rpm) 15000 13475 15000 13575 0 0 Nr14 (rpm) 15000 13475 0 0 0 0

Fuel Consumption (kg/s- MMSCFD)

2.06 – 7.09 1.57– 5.39 2.00 – 6.91 1.55– 5.35 1.94 – 6.68 1.55 – 5.32

Mass flow rate (kg/s- MMSCFD)

579.12 – 1994.15

550.51– 1895.64

577.53– 1988.68

550.51– 1895.64

574.60 – 1978.58

550.51 – 1895.64

1isη 78.47 78.88 79.382 79.43 79.90 79.94

2isη 78.47 78.50 79.382 79.62 79.90 79.94

3isη 78.47 78.88 79.382 79.62 79.90 79.94

4isη 78.47 78.88 79.382 79.62 79.90 79.94

5isη 78.47 78.88 79.382 79.62 79.90 79.94

6isη 78.47 78.88 79.382 79.62 79.90 79.95

7isη 78.47 78.88 79.382 79.62 79.90 79.95

8isη 78.47 78.88 79.382 79.62 79.90 79.95

9isη 78.47 78.88 79.382 79.62 79.90 79.95

10i sη 78.47 78.88 79.382 79.62 79.90 79.95

11isη 78.47 78.88 79.382 79.62 79.90 79.95

12isη 78.47 78.88 79.382 79.62 79.90 79.95

13i sη 78.47 78.88 79.382 79.62 0 0

14isη 78.47 78.88 0 0 0 0

Discharge Temp. Tc (K)

358.94 347.39 357.75 347.09 356.33 347.04

This example shows by using numerical optimization we

can find solutions that may be more fuel efficient than other solutions. It can also be seen from Table 3 that at the optimum (i.e. with only 12 compressors running at their optimized speeds), the efficiency of each unit is about 79.95%. It is also seen from Table 3 that in this case, the outlet temperature dropped from 356.33 K to 347.04 K. Most importantly, the total fuel consumption is reduced from 1.94 kg/s (6.68 MMSCFD) at the initial speeds for the 12-compressor case to 1.55 kg/s (5.32 MMSCFD) at the optimum. It should also be noted that the second best solution is that obtained using 11 or

13 compressors, followed by the solutions obtained using 10 or 14 compressors; the worst optimum is the one for the 9-compressor case.

Figure 9 shows the comp ressor station optimization map for the fourteen-compressor example. This map expresses the number of compressors on the horizontal axis and the station-level fuel consumption on the vertical axis. The families of curves on the map correspond to (1) total gas throughput and (2) compressor speed. Interesting, for this compressor station configuration, the optimal number of operating compressors is consistently 12. the vertical line extending from 12

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compressors highlight the optima. The intersection of the vertical line and each gas throughput line identifies the

compressor speed for optimal operation.

Table 3- Continue

11 Compressors on-line 10 Compressors on-line 9 Compressors on-line Initial Final Initial Final Initial Final Nr1 (rpm) 15000 14000 15000 14400 15000 15000 Nr2 (rpm) 15000 14000 15000 14400 15000 15000 Nr3 (rpm) 15000 14000 15000 14400 15000 15000 Nr4 (rpm) 15000 14000 15000 14400 15000 15000 Nr5 (rpm) 15000 14000 15000 14400 15000 15000 Nr6 (rpm) 15000 14000 15000 14400 15000 15000 Nr7 (rpm) 15000 14000 15000 14400 15000 15000 Nr8 (rpm) 15000 14000 15000 14400 15000 15000 Nr9 (rpm) 15000 14000 15000 14400 15000 15000 Nr10 (rpm) 15000 14025 15000 14425 0 0 Nr11 (rpm) 15000 14025 0 0 0 0 Nr12 (rpm) 0 0 0 0 0 0 Nr13 (rpm) 0 0 0 0 0 0 Nr14 (rpm) 0 0 0 0 0 0

Fuel Consumption (kg/s- MMSCFD)

1.86– 6.40 1.553– 5.35 1.75 – 6.03 1.57 – 5.42 1.61 – 5.55 1.61 – 5.55

Mass flow rate (kg/s- MMSCFD)

569.74– 1961.84

550.51– 1895.64

532.08 – 1935.49

550.51– 1895.64

550.64 – 1895.64

550.51– 1895.64

1isη 79.83 79.74 78.95 78.80 76.95 76.95

2isη 79.83 79.74 78.95 78.80 76.95 76.95

3isη 79.83 79.74 78.95 78.80 76.95 76.95

4isη 79.83 79.74 78.95 78.80 76.95 76.95

5isη 79.83 79.74 78.95 78.80 76.95 76.95

6isη 79.83 79.74 78.95 78.80 76.95 76.95

7isη 79.83 79.74 78.95 78.80 76.95 76.95

8isη 79.83 79.74 78.95 78.80 76.95 76.95

9isη 79.83 79.74 78.95 78.80 76.95 76.95

10isη 79.83 79.70 78.95 78.80 0 0

11isη 79.83 79.70 0 0 0 0

12isη 0 0 0 0 0 0

13isη 0 0 0 0 0 0

14isη 0 0 0 0 0 0

Discharge Temp. Tc (K)

354.59 347.37 352.42 348.17 349.6 349.6

For example, to transport 1835 MMSCFD, the best

operation condition is 12 compressors with 13000 rpm speed, and for this condition fuel consumption becomes 4.5 MMSCFD.

While this particular compressor station map is specifies to this compressor station, the methodology suggests that one can

create a compressor station map to any compressor station configuration.

CONCLUSION This study used a fully implicit finite difference method to

analyze transient and non-isothermal flow within a pipe and a quasi-steady flow assumed at each time step of the numerical

9 GMC 2004

solution for centrifugal compressor equations to simulate compressor stations under non-isothermal conditions. The results show that:

• The simulation approach that is developed here is adequate for supporting numerical optimization.

• Numerical optimization is an effective tool for optimizing compressor speeds, and can yield significant reductions in fuel consumption. This, in turn, will increase throughout.

• Determination of the optimal number of compressors to shutdown in a compressor station and selection of optimal speeds for the remaining compressors can be done simultaneously using the methods developed herein.

• The result is a compressor station map that can be used to optimize the fuel consumption as a function of mass flow rate through a compressor station.

ACKNOWLEDGMENTS The authors graciously acknowledge the funding of this

project, “ Virtual Pipeline System Testbed to Optimize the U.S. Natural Gas Transmission Pipeline System” from the U.S. Department of Energy, National Energy Technology Laboratory, Grant No.DE-FC26-01NT41322.

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13. Botros, K.K., 1990,” Thermodynamic Aspects of Gas Recycling During Compressor Surge Control”, ASME Proceeding Pipeline Engineering Symposium, New Orleans, LA. PP. 57-65.

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10 GMC 2004

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Biographies of Authors

Dr. Kirby S. Chapman, a professor and director of the NGML, researches methods and technologies to monitor and reduce pollutant emissions in turbochargers and internal combustion engines. He serves as advisor to the GMC, teaches turbomachinery courses to industry professionals and K-State students, and mentors graduate students. He earned his Ph.D. in mechanical engineering at Purdue University under GRI guidance.

Dr. Prakash Krishnaswami is a professor of Mechanical and Nuclear Engineering at Kansas State University, his areas of expertise include computer-aided mechanical design and optimization of systems. He received his undergraduate degree from the Indian Institute of Technology, Madras, his Master's degree from the State University of New York at Stony Brook. and his doctorate from theUniversity of Iowa

Mohammad Abbaspour, a Ph.D. candidate and associate researcher at the NGML, his areas of expertise include single and two phase gas pipeline simulation, compressor station analysis, thermal system simulation and optimization. He received his B.S. and M.S. in mechanical engineering from the Isfahan University of Technology (IUT) in IRAN and now he is Ph.D. student in mechanical engineering at Kansas State University.