Simulation
Modeling turbo-expander systems
Simulation: Transactions of the Society for Modeling and Simulation International 89(2) 234–248 Ó 2013 The Society for Modeling and Simulation International DOI: 10.1177/0037549712469661 sim.sagepub.com
Mehdi Taleshian Jelodar, Hasan Rastegar and Hossein Askarian Abyaneh
Abstract Turbo-expander systems have long been used instead of regulators, but they have recently received attention as a driving medium for power electrical generators. These systems typically replace the regulator valves that reduce the gas pressure in gas distribution networks. As a generator, however, its turbine generates mechanical power that can be considered as a new source for dispersed generation. To study the dynamics of turbo-expander systems, we first introduce a detailed model that incorporates components such as the preheater, reheater, nozzle angle, and the turbine. This model is validated using empirical data collected from thermo-dynamic analysis. To be specific, we also investigated the effects of using a variable-angle nozzle versus a fixed-angle nozzle turbine. As an application in distribution networks, we looked into the use of turbo-expander driven generators as highly efficient components. The dynamic behavior and transient stability of such generators are thoroughly analyzed for disturbances in either mechanical or electrical parts using the proposed model.
Keywords turbo-expander systems, variable nozzle angle, dispersed generation
1. Introduction The increasing growth of energy consumption is a direct consequence of the increasing industrial development. As a result, our dependence on fossil forms of energy has dramatically increased. Limited resources of fossil energy and increasing environmental pollution not only demand a search for renewable forms of energy, but also require us to improve the efficiency of industrial processes. Generally speaking, the efficiency can be improved by recovering wasted energy from energy-wasting processes. One such process is reducing the gas pressure in gas distribution networks. In these networks, the gas pressure is initially elevated before transmission to provide enough pressure down the distribution lines at the customer end. Since the size of transmission pipes is decreased at several stages in this network, the gas pressure is decreased using regulating valves to maintain the same level of pressure. This process simply wastes a significant amount of energy that was spent for compressing the gas in the first place. Turbo-expander systems could be an alternative for regulating valves to reduce the gas pressure, and can also be used to recover a considerable amount of the dispersed energy.1 These systems have been successfully exploited in liquefaction and chemical cooling processes in general, where they were used to generate electrical energy up to several megawatts of power.2 In order to investigate the
performance of such systems, it is important to build a complete model that incorporates detailed components for detailed analysis. Figure 1 demonstrates a model for turboexpander systems that can be used for reducing the gas pressure. In this model, high-pressure gas enters the system through two paths. One path passes through the pressure regulating valve, while the second path involves the preheater, expander turbine, reheater, and control valves, illustrated on the top row. While the regulating valve remains completely closed in the expander mode of operation, the turbo-expander path is kept open by setting the gas pressure of the operating point to some low threshold. To compensate for the temperature reduction caused by passing gas through the turbine, it is necessary to preheat the gas using a preheater to initially prevent vane corrosion. After the gas passes the turbine, the temperature of the delivered gas is increased to a certain temperature using a reheater. In order to control the opening and closing of the turbine Department of Electrical Engineering, Amirkabir University of Technology, Iran Corresponding author: Hasan Rastegar, Department of Electrical Engineering, Amirkabir University of Technology, No. 424 Hafez Avenue, Tehran 15875-4413, Iran. Email:
[email protected]
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Hot Steam Highpressure Gas
Hot Steam
Preheater Ball Valve
Reheater
Lowpressure Gas
STV TE
Cold Steam
Cold Steam G
Nozzle Control System
Gear Box
Figure 1. Schematic diagram of a turbo-expander system. The top row shows the control valves, heaters and expander turbine and the bottom row shows the regulating valve.
path, a ball valve and a safety trip valve (STV) are used to intercept the turbine path compulsively.3 The performance of turbo-expander systems has been analyzed using operational models validated from smallscale experimental setups.4–6 These studies usually simulate the turbine in an operating condition, while no particular model is introduced for the expander turbine. Turboexpander systems, on the other hand, have been independently studied for producing electrical energy,7,8 where they exploit compressed air as the operating fluid. Since small turbines that generate power less than 1 kW rotate at several thousand rounds per minute (note that it is not economic to use gearboxes for reducing the rotation speed in these small systems), a permanent magnet generator is used to produce the electrical energy. The variation in turbine parameters, such as torque, is investigated on the efficiency of the produced power. Small-scale turbo-expander systems have been studied by analyzing the generator’s temperature with finite elements,9 in which a permanent magnet generator was similarly used as a result of highspeed rotation. Expander turbines with a 32.7 kW nominal power were used as a replacement for pressure regulator valves in airconditioner systems.10,11 The empirical analysis of this system demonstrates that tuning turbine parameters, such as velocity ratio, pressure ratio and output torque, greatly affects the produced power and efficiency. Semi-empirical analysis of turbo-expander systems uses equations of continuity and momentum from the fundamentals of thermo-dynamic demonstrate energy recovery in the form of mechanical energy from small heat sources in Organic Rankine Cycles (ORCs).12 Due to complication and non-linearity of ORCs, it is difficult to extend a dynamic model for other apparatus of the system. Assuming the operating fluid to be natural gas, simple models have been developed for turbo-expander systems
that replace the pressure regulator valve.13 The natural gas is assumed to be ideal, with a constant pressure ratio across the turbine. Using this model, the effect of variations of the mass flow rate on the turbine’s efficiency has been studied. Using expander turbines for generating electrical energy is investigated under various operational conditions for synchronous generators.14,15 The turbo-expander models in these studies are similar to Maddaloni and Rowe;13 however, details related to the shaft and gearbox have been added to investigate the effects of transient stability in the system. The gearbox is used to reduce the rotation speed of the turbine to comply with the generator’s nominal speed. In this paper, we initially describe different components of the turbo-expander system in detail. The models that we present have the advantage of being simple with fewer complexities, which facilitates their use in conjunction with electric power system studies. These simpler yet plausible models are introduced to replace the models that are based on the thermo-dynamic and momentum rules and considering the energy preservation role of the turboexpander system. As a result, these models are applied to general models of power systems that can be used to maintain performance. We also investigated the effect of variations in the nozzle angle of the turbine, which has been typically ignored in previous studies. We verify the proposed model using empirical data collected from tests applied on similar expander systems, where we also demonstrate the importance of having a variable-angle versus fixed-angle nozzle turbine in the system. In the second part of the paper, the proposed model of the turboexpander system is connected to an electrical distribution network to drive an induction generator. The objective was to investigate the effects of an expander system on the electrical generator and the distribution network. We examined the stability of the whole system as a response to disturbances introduced in mechanical and electrical parts.
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In Section 2, a proper model for turbo-expander systems is introduced, while the verification of this model is provided in Section 3. The proposed model is studied as coupled with a distribution network in Section 4. Conclusions are given in Section 5.
2. A model for turbo-expander systems The main components of a turbo-expander system include the preheater, expander turbine, reheater, and valves. High-pressure gas at approximately 19 bar enters the turbo-expander system by initially passing through the preheater. The preheater increases the gas temperature from approximately 288 up to 403 K using a steam reservoir. The heated gas then enters the turbine after passing through a filter that absorbs the gas impurities. The expander turbine decreases the gas pressure to 5.2 bar. Although the low-temperature gas coming out of the turbine is reheated using a heater before being delivered, this heater does not affect the expander system, and thus is not included in the proposed model. If the temperature does not reach the required minimum level, a bypass valve referred to as the temperature regulator valve (TRV) is used to pass some high-temperature gas directly to the consumer end. For simplicity, we assume that the system’s operational parameters remain under control in their permissible ranges and there is no need to operate the TRV; therefore, neither the TRV nor the STV are modeled.
2.1. Preheater The role of the preheater is to increase the generated power in a turbo-expander system and prevent vane corrosion, hydration, or the formation of solid crystals that can tribulate the operation of nozzles. The preheater regulates the temperature of the gas by controlling the amount of steam that enters the double-wall reservoir. The temperature of the output gas, Tout , has be described by the following differential equation:16,17 dTout = k2 ðk1 m_ steam + m_ in ðTin Tout + B1 Þ + B2 Þ dt
ð1Þ
The preheater model is illustrated in Figure 2. The outlet pressure of the boiler can be described using principles of fluid dynamics. In the steam side, the pressure loss due to change in flow velocity is prevailing; thus, the pressure loss model can be captured based on the steam areas effect, where the mass of accumulative volume is described as dM = m_ in m_ out dt
ð2Þ
The total accumulative action for any control volume in the boiler consists of the accumulative action of steam, hot
Figure 2. Output gas temperature of the preheater.
liquid, and metal parts. The differential equation for the total accumulative action can be written as ∂Msp ∂Mwp ∂Mmp ∂M = + + ∂P ∂P ∂P ∂P
ð3Þ
Since the effect of the second and third terms on the righthand side of (3) are negligible compared with the first term, we can simply express the total accumulative action as ∂Msp ∂M = ∂P ∂P
ð4Þ
In a Benson-type sliding pressure boiler, the pressure differences are the driving forces for mass flow through the components. The change of steam mass due to the pressure changes is given as dM d 1 = V: dt dP v(P)
ð5Þ
Assuming the value of the polytrophic exponent to be approximately 1.0 (p:v = p0 :v0 ), for dM dt > 0 and dP > 0, from (2)–(5) the outlet pressure is estimated as17 dp P0 = ðm_ in m_ out Þ dt τ:mv
ð6Þ
This equation can be used to describe the pressure drop between different turbine stages. A model for the mass flow responding to steam pressure variations has been proposed, where the swing of main steam flow strictly relies on the change in steam pressure:18 d m_ out dp m_ out = dt 2(Pin0 Pout0 ) dt
The final diagram is shown in Figure 3.
ð7Þ
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237 variation of pressure ratio across the turbine versus the mass flow rate of the passing gas.22 These curves are generated from the empirical experiments performed on the turbines. By using these curves, the pressure ratio of a turbine can be determined from its mass flow rate accordingly. These curves depend on the rotating speed of the turbine, where the mass flow rate decreases as the speed of the turbine increases. The turbine in Figure 4 is coupled with the three-phase induction generator, where the deviation of the generated speed in operation modes is below 5%, and the turbine mass flow change is less than 1.45% in this range of speed variations, and thus can be neglected.23 As a result, the operating curves in the speed related to generator nominal speed would be valid for the model. Analytical closed-form formulations, such as the Stodola formulation, have been presented for the mass flow rate of the turbines.16,24,25 The Stodola formulation determines that the amount of mass flow rate passing through a nozzle in an isentropic expansion process for a compressible ideal gas. According to this formulation, the passing flow through the turbine is related to input temperature and the pressure of both sides of the turbine. Its proportion factor can be calculated by adjusting the parameters with the operation condition (OC) of the turbine and then this analytical formula can be used instead of the proposed curves:
Figure 3. Diagram of preheater pressure and flow rate.
CT pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m_ = pffiffiffiffiffiffi Pin 2 Pout 2 Tin
Figure 4. Variation of turbine mass flow rate versus its pressure ratio.22
2.2. Turbine model To deal with the complexity introduced by having different sizes of turbines in the proposed model, we introduce quasi-steady equations of turbine. These equations are helpful to accurately model different turbines and to reduce the complexity of the model. For simplification, we assume that the volume between components is negligible, and the input gas flow equals the output gas flow at any time instant.19 Thus, as far as the gas flow is concerned, the system is in a steady-state condition: m_ in ≈ m_ out
ð8Þ
In previously developed dynamic models for expander turbines, the pressure ratio of the turbine is assumed to be constant and independent from the mass flow rate.13–15 In operational turbines, however, this ratio varies by variations in the mass flow rate.20,21 Figure 4 shows the
ð9Þ
To calculate the amount of generated power and temperature of the outgoing gas, first the efficiency of the turbine should be estimated. The efficiency of the turbine varies with any change in flow rate, input gas pressure, or speed of the turbine. In a range up to 5% of normal speed variation, the amount of change in efficiency is below 0.75%.23 Therefore, the effect of speed variation on efficiency is neglected and the effects of the other two parameters will be investigated. Deviation of mass flow rate or gas pressure from its optimum point reduces the efficiency. So we define the OC such that the efficiency varying between two minimum and maximum values is considered. The total efficiency can be estimated accordingly as13–15,26 Prated Pin × 1 m_ rated m_ in O:C: = 1 m_ rated Prated ð10Þ ηtotal = O:C: × ðηub ηlb Þ + ηlb
ð11Þ
The complete adiabatic expansion process for a turbine is represented by the enthalpy–entropy diagram (Mollier diagram), shown in Figure 5. The ideal enthalpy change, that is, the ideal or reversible expansion, lies between the inlet and outlet pressure lines, but at a constant entropy (line 0–
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Figure 5. Enthalpy entropy diagram (hs-diagram) of expansion process in the turbines.7
1s–2s), whereas the real expansion occurs according to line 0–1–2. Assuming adiabatic flow through the turbine, the relation between gas pressure and temperature can be inferred as k1 T1 P2 k = T2s P1
Figure 6. Nozzle ring of the turbo-expander system installed in the Neka power plant.
ð12Þ
Let us define the efficiency of the turbine as ηT =
T 1 T2 T1 T2s
ð13Þ
The temperature of output gas can be estimated as27–29 " T2 = T1 1
P2 1 P1
k1 ! # k :ηT
ð14Þ
The ideal work generated by the turbine is W = m_ × Dh. Assuming the gas to be ideal, Dh can be replaced with the temperature, and the work can be estimated as3,7,16 ! k1 P2 k W = m_ × η × CP × T1 × 1 P1
ð15Þ
2.3. Turbine nozzle system An important part of the turbine that has been less analyzed in the previous studies is the variable nozzle. Accurately studying the operation of the turbo-expanders requires consideration of the effect of variable nozzles from two aspects: firstly the nozzle’s operational mechanisms and dynamics and secondly the effects of this nozzle on the operation of the turbine. The variable nozzle ring of a turbo-expander system installed in the Neka power plant
Figure 7. Simple diagram of a nozzle system actuator.30
is demonstrated in Figure 6. The position of the nozzles is controlled by the ring that envelops them. The motion of the enveloped ring causes the angle of the nozzles to change, and by closing or opening these nozzles, the operational parameters of the turbine are affected, such as the pressure drop, flow rate, output temperature, and efficiency. The nozzles of turbo-expander systems usually have pneumatic actuators. According to Figure 7, this system has two separate chambers, which are isolated from each
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Figure 8. Mass flow rate variation versus pressure ratio for different nozzle angles.20
other by a diaphragm. Moving this diaphragm up or down causes the ring of nozzles to rotate via the lever, which is connected to this diaphragm, thus opening or closing the nozzles. The pressure of the lower chamber is equal to atmosphere pressure and the pressure of the upper chamber, which is controlled by a servo valve, is directly proportional to the desired nozzle angle.30 Closed-form equations of nozzle actuator system are presented elsewhere.30 The final transfer function, which relates the variations of the diaphragm level, xd, to the upper chamber pressure, can be expressed as θerror = θrefrence θactual
ð16Þ
Pact = k0 θerror
ð17Þ
Tr ðsÞ =
xd Ad = Pact md s2 + bd s + ksm
ð18Þ
θnozzle = k1 xd + k2
ð19Þ
It is very difficult to obtain analytical formulation for the effect of the nozzle on the specifications of the output gas. Since the analysis of the turbine is based on operational curves, the test results in different nozzle angles will be used as the basic curves for simulation of turbine operation for different nozzle angles. For nozzle angles other than the test angles, interpolation will be used. Figures 8 and 9 show the variations of mass flow rate versus the turbine pressure ratio, and efficiency versus its velocity ratio for different nozzle angles.20
2.4. Nozzle control logic modeling The controlling mechanism of turbo-expanders that relates the output pressure via turbine nozzles is studied here. Since the gas delivered to the consumer end at the turbine outlet should have a minimum pressure, a nozzle control system is used for regulating the output pressure. The control system, in fact, operates on the nozzle angle and
Figure 9. Efficiency variation versus velocity ratio for different nozzle angles.20
Desired Pressure Outlet Gas Pressure
+
∑ _
PID Controller
Output Signal to Nozzle Actuator
Figure 10. Diagram of turbo-expander control system. PID: proportional–integral–derivative.
therefore controls the output pressure. This mechanism also allows the output temperature and power generated by the electrical generator to be affected by the turbine. The TRV, which is parallel to the turbine, can compensate for the gas temperature reduction. A simple diagram of the desired turbo-expander control system is shown in Figure 10.
2.5. Shaft and gearbox Usually the rotation speed of turbo-expanders is very high. For example, the turbo-expander of the Neka power plant rotates at 23,500 rpm. At this rotation speed, the frequency of the generated voltage would be very high, such that it would require electronic power converters; alternatively, it could simply be reduced by using a gearbox. The proposed gearbox is considered to have transfer losses and low- and high-speed shafts are modeled by a two-inertia system. It should be mentioned that all dampings are neglected and this is the worst-case scenario in transient stability and torsional oscillations studies. Considering the above assumptions, we have14 n=
nL JH kH , JL = 2 , kL = 2 nH n n
ωH =
1 ðTin TH Þ JH S
ð20Þ ð21Þ
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ð22Þ ð23Þ ð24Þ
where S is the Laplace transform. Figure 11 illustrates the block diagram for the turbo-expander system coupled with the gearbox and generator.
+_
+_
+_
+_
Figure 11. Shaft and gearbox model for the coupled turboexpander, gearbox and generator.14
+ Vds
L´lr (ω-ωr)φ´ qr R´r + _
ωφ´qs Lls Rs _ + ids
_
i´dr
Lm d axis
+ Vqs
L´lr (ω-ω _ r)φ´+dr R´r
Rs ωφ´ds_ Lls + ids
_
Lm
i´qr
q axis
Figure 12. Electrical model of the induction generator.
2.6. Generator model The generator of this study is an induction three-phase generator with a squirrel cage rotor. The electrical model of this generator is a four-order state-space model and the mechanical part is modeled with a constant inertia and a damping factor,31 as demonstrated in Figure 12.
3. Validation of the proposed model The experimental tests in Cho et al.10 are used for verifying the presented model. In this empirical analysis, an expander turbine is applied in addition to the existing pressure regulator valve in an air-conditioner system. The turbine is coupled with a permanent magnet generator to recover the power in the form of electrical energy. The experimental setup consists of some apparatus, among which the nozzle, turbine, and generator are simulated. The nozzle angle in the experimental system is fixed at 65°. The flowing gas in this system is HFC134a and in normal operation it passes through the pressure regulating valve. After 994 seconds, the regulator valve closes completely and the gas is transmitted into the turbine path immediately. For model validation, the mass flow rate of the turbine is reconstructed from the experimental data. Original and reconstructed flow rates are shown in Figure 13, given this input. The model experiences a 0.25 kg/s step change in the mass flow rate. The input pressure and temperature of the gas entering the turbine are 14.7 bar and 320 K, respectively, and the turbine is working at 1.7% of the partial admission rate. Variations of input and output pressures are shown in Figure 14 for both the experimental and simulated systems. Figure 15 demonstrates the variations of the input/ output temperatures of the flow. The simulation results show a good compatibility between the measured and simulated pressures of output flow. In the experimental setup, there are two temperature sensors in the turbine outlet to measure the output gas temperature. From Figure 15, the measurements of these two sensors are almost equal. Simulation shows that the low-
Mass Flow Rate (kg/s)
0.75 Experimental Mass Flow Rate Simulated Mass Flow Rate 0.5
0.25
0 750
1000
1250
1500 Time (s)
Figure 13. Experimental and simulated mass flow rate.
1750
2000
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20
7000 Expander Speed (rpm)
Measured and Simulated Input Pressure
Pressure (bar)
15
Simulated Output Pressure
10
5000 4000 3000 2000
5 Measured Output Pressure 0 750
6000
1000
1250
1500
1000
1750
0 750
2000
1000
1250 1500 Time (s)
Time (s)
1750
2000
Figure 16. Measured and simulated speed of the turbinegenerator.
Figure 14. Measured and simulated input and output pressures.
700 330
Measured Voltage Simulated Voltage
600 500
320
Voltage (V)
Temperature (°K)
Measured and Simulated Input Temperature
Measured Output Temperature 310
(Two Measurments)
300 200
300
100
Simulated Output Temperature 290 750
400
1000
1250
1500
1750
2000
0 750
1000
1250
Time (s)
1500
1750
2000
Time (s)
Figure 15. Measured and simulated input and output temperatures.
Figure 17. Measured and simulated voltage of the generator.
pressure gas temperature is increased to real measured values, but at the primary seconds, it has more divergence from the sensor measurements. This is due to the fact that the gas temperature at the turbine output decreases more than in the situation in which it passes from the regulator valve. Therefore, when the turbine path is closed in the real system, the temperature of the regulator and the turbine common output path is about 305 K, and when output gas temperature decreases due to turbine operation, it takes some seconds for the sensors to clear the exact temperature of the flowing gas. In the simulated system, this thermal capacity of the system is not modeled and, hence, the sudden variation of output gas temperature can be seen in the turbine outlet. Another important parameter investigated in the experimental system is the variations of the turbine speed when the output power changes. Figures 16 and 17 demonstrate
variations of the generator speed and generated voltage during variations in the output power. The generator power is changed by an electrical load variation. Simulation results have a good compatibility with the experimental results, where the maximum deviation from the measured speed and voltage values is approximately 2.8%.
4. Studied system We simulated the complete system in MATLAB and SIMULINK software. This system is selected in accordance with the turbo-expander system installed in the Neka power plant and involves two identical parallel turbo-expander-generators. A diagram of this system is shown in Figure 18. The induction generator output voltage is 10 kV and connects via a 10/20 kV transformer to
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Preheater
Cold Steam Highpressure Gas
TE
PCC
Nozzle Control System
IG Gear Box 15.6/1
Pressure Feedback
Hot Steam
9MW 10KV
Lowpressure Gas
Distribution Network 90MVA, 20KV
Transformer 10KV/20KV 30 MVA
Preheater
Cold Steam
IM
TE Nozzle Control System Gear Box 15.6/1 Pressure Feedback
Transformer 20KV/6.6KV 15 MVA
IG
9MW 10KV
Lowpressure Gas
3.9 MW 6.6KV
Transformer 10KV/20KV 30 MVA
Figure 18. Diagram of the studied system.
a distribution network. The distribution network has a 90 MVA short circuit capacity with a ratio of X/R = 1.2. Detailed information of the whole system components are listed in Tables 1-8.
1.1 Power without Preheater and without Nozzle Control Power with Preheater and without Nozzle Control Power with Preheater and with Nozzle Control
1
4.1. System behavior during step change in mass flow rate A common change among the system variables is variations in the mass flow rate. To investigate these effects, a 5% reduction in nominal mass flow rate is assumed for three different cases. Case (1). System without preheater and nozzle-angle control: The nozzle control angle is fixed at 75°. In this case the steady-state mass flow rate, input pressure, and temperature are 59.1 kg/s, 19 bar and 341 K, respectively. At t = 4 s, the flow rate decreases to 56.145 kg/s. Case (2). System with preheater and without nozzle-angle control: Unlike the previous case, the effect of the preheater on the system should be studied. Similar to the previous case, the mass flow rate changes from 59.1 to 56.145 kg/s at t = 4 s and input pressure and temperature of the preheater are 19 bar and 341 K, respectively, but input pressure and temperature of the turbine are determined by the preheater behavior. The preheater imposes a time lag smaller than 0.1
Power (pu)
0.9
0.8
0.7
0.6
0.5
3
4
5
6
7 8 Time (s)
9
10
11
12
Figure 19. Generator output power when the mass flow rate decreases by 5%.
s on the system, which is not important when compared to the time lag of other parts of the system. Case (1). System with preheater and nozzle-angle control: Since the nozzle controls actions, the turbine output pressure does not change as it would in cases 1 and 2, where it remains almost constant at 5.2 bar. Figure 19 shows the generator power in the three cases. Two main conclusions can be drawn from this figure:
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Output Pressure without Preheater and without Nozzle Control Output Pressure with Preheater and without Nozzle Control Output Pressure with Preheater and with Nozzle Control
8
TE Power without Preheater and without Nozzle Control TE Power with Preheater and without Nozzle Control TE Power with Preheater and with Nozzle Control
1.3
1.2
7.5 TE Power (pu)
Turbine Output Pressure (bar)
8.5
7 6.5
1.1
1
6 0.9 5.5 0.8 3
4
5
6
7 8 Time (s)
9
10
11
12
Figure 20. Turbine output pressure when the mass flow rate decreases by 5%.
1. 2.
in cases 1 and 2, the output power has decreased about 30% due to a 5% decrease in mass flow rate; by adding the preheater to the system, the generated power has increased about 4% compared to case 1, where the flow rate is increased by 5%.
The first conclusion can be inferred from the turbine’s behavior. At the turbine’s operating point, the parameters of (9) are Pin = 19 bar, Tin = 341 K, CT = 59:67, and m_ nominal = 59:1 kg s ; therefore if m_ = 59:1
kg ) Pout = 5:1bar s
kg if m_ = 59:1 × 0:95 ) Pout = 7:7bar s
In the case with the preheater and a fixed nozzle angle, the turbine input gas temperature increases more than in the case without the preheater when the mass flow rate decreases and it causes the output pressure to be lower, about 7.5 bar (in accordance with (9)). The value of the output pressure causes the generated power to change significantly, as estimated by (15). Variations of the turboexpander pressure and generated power are demonstrated in Figures 20 and 21. When the nozzle-angle control system is active, the output pressure variation from its setting point (5.2 bar) is very small due to mass flow rate reduction, and according to Figure 20, the maximum variation is below 3.5%. Therefore, the decrease in the value of power is small in comparison to cases with a fixed nozzle angle. Variations of the nozzle angle in the three cases are shown in Figure 22. By adding the preheater to the system, the power increases by about 4%. When the gas flow rate decreases to 95% of its nominal value, the output gas temperature increases by about 3.7 K. This is due to the assumption of
3
4
5
6
7 8 Time (s)
9
10
11
12
Figure 21. Turbine output power when the mass flow rate decreases by 5%. TE: turbo-expander. 80 Nozzle Angle without Preheater and without Nozzle Control Nozzle Angle with Preheater and without Nozzle Control Nozzle Angle with Preheater and with Nozzle Control
79.5 79 78.5 Nozzle Angle (°)
5
78 77.5 77 76.5 76 75.5 75 74.5
3
4
5
6
7 8 Time (s)
9
10
11
12
Figure 22. Turbine nozzle angle when mass flow rate decreases by 5%.
constant steam flow for the preheater operation, which causes the turbo-expander and generator power to increase. Figure 23 demonstrates variation of the turbo-expander input and output temperature. In all three cases, the system voltage is in a normal range and the motor operation is stable. The voltage of the generator terminal is shown in Figure 24. Due to the higher powers generated in the case with an active nozzle control, the generator draws more reactive power from the network. As a result, the terminal voltage in this case is a little smaller than in the other two cases.
4.2. System behavior during momentary reduction in input gas pressure The input gas pressure may have some uncertain variations determined by the consumer, system control, or protection
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Input Temperatur with Preheater and without Nozzle Control Input Temperatur with Preheater and with Nozzle Control
20
345
340
3
4
5
6
7 8 Time (s)
9
10
11
12
320
Turbine Output Temperature (°K)
Input Pressure Output Pressure
Input Temperatur without Preheater and without Nozzle Control
Output Temperatur without Preheater and without Nozzle Control Output Temperatur with Preheater and without Nozzle Control Output Temperatur with Preheater and with Nozzle Control
300
Turbine Pressure (bar)
Turbine Input Temperature (°K)
350
20
15
15 10
10
5 0
4.7
4.71
4.72
5 280
260
3
4
5
6
7 8 Time (s)
9
10
11
0 3
12
Figure 23. Turbine input and output temperature when mass flow rate decreases by 5%.
4
5
6
7 8 Time (s)
9
10
11
12
Figure 25. Turbine input and output pressure for a momentary decrease in input gas pressure.
100
0.935 Gen. Terminal Voltage without Preheater and without Nozzle Control Gen. Terminal Voltage with Preheater and without Nozzle Control
80
0.925 Nozzle Angle (°)
Generator Terminal Voltage (pu)
90
Gen. Terminal Voltage with Preheater and with Nozzle Control
0.93
0.92
0.915
70 100
60 80
50
60
0.91 40 0.905
3
4
5
6
7 8 Time (s)
9
10
11
12
Figure 24. Generator terminal voltage when the mass flow rate decreases by 5%.
equipments. One of these variations is a momentary decrease in the input pressure due to either control or protection reactions. To test the outcome of this variation, a short-time reduction in the input gas pressure is simulated and at t = 4 s, where the gas pressure decreases from 19 to 8.54 bar for 0.7 s. In this simulation, a preheater and a nozzle control system are included. According to Figure 25, when input pressure decreases, the output pressure reaches a low level about zero and it comes back to the nominal value when the input pressure is restored. The output pressure has some transient variations due to the nozzle control system response. According to Figure 26, when the input pressure is restored, the nozzle angle changes from 35° to 95° in 0.0003 seconds, where it has some swings before reaching the final value in a short time duration of about 0.02 seconds. During these swings, the output pressure spikes whenever the nozzle angle has a low value. This
40 4.7
30
3
4
5
6
7 8 Time (s)
4.71
9
10
4.72
11
12
Figure 26. Turbine nozzle angle for a momentary decrease in input gas pressure.
behavior causes similar swings in the turbine-generated power. The simulation results in Figure 27 show the unstable behavior of the generator and motor in the system. There are two reasons for this instability. First is the increase of the turbo-expander output power to approximately 2.8 pu when input gas pressure decreases. In fact, when the input pressure is decreased, the output pressure will also decrease to a low value. Because of the non-linear behavior of the turbine, a higher level of generated power is achieved. Secondly, after the input pressure is restored, the transient response of the nozzle control system causes the output pressure to change such that the output power remains above its nominal value for 0.68 s. These high levels of turbo-expander output power cause the generator to pass its critical slip and, hence, it is not able to come back
245 0.95
1.6
0.85
1.2 1
3
4
5
6
7 8 Time (s)
9
10
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12
2 0
0.8 0.75 0.7 0.65
-2
0.6
-4
0.55
-6
0.5
3
4
5
6
7 8 Time (s)
9
10
11
12
Generator Speed (pu)
5 4.5
5
4
4
3.5
3
3
2
0
4.71
4.72
1.5 1 0.5 0
3
4
5
6
7 8 Time (s)
9
10
11
6
7 8 Time (s)
9
10
11
12
1.2 1.1
3
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6
7 8 Time (s)
9
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7 8 Time (s)
9
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12
5 Motor Speed (pu)
4.7
5
1.3
1
2
4
1.4
1
2.5
3
Figure 29. System voltages for a momentary decrease in input gas pressure. PCC: point of common coupling.
Figure 27. Generator and motor speed for a momentary decrease in input gas pressure.
TE Power (pu)
Generator Terminal Voltage PCC Voltage
0.9
1.4
System Voltage (pu)
Motor Speed (pu)
Generator Speed (pu)
Jelodar et al.
12
0
-5
-10
Figure 28. Turbo-expander (TE) output power for a momentary decrease in input gas pressure.
Figure 30. Unstable speed of the generator and motor due to a three-phase fault on the generator terminal.
to its operating point. As a result, the generator draws a considerable amount of reactive power from the network, which causes voltage collapse at the point of common coupling (PCC) and also could destabilize the motor. Figures 28 and 29 demonstrate the turbo-expander output power and system voltages, respectively.
duration is selected regarding four cycles (80 ms) of opening time of the circuit breakers at the 20 kV level. Since generator output power decreases during the fault time, it accelerates until it passes through its critical speed corresponding to the maximum torque and becomes unstable. The simulation results shown in Figure 30 show an unstable speed for both the turbo-expander-generator and the motor connected at the PCC. Figure 31 demonstrates the voltage of the generator terminal and the PCC. These results imply the necessity of equipment to improve the power quality for this system.
4.3. System behavior during short circuit fault on turbo-expander-generator terminals To investigate the turbo-expander system behavior during abnormal conditions, a three-phase fault is simulated on the generator terminals in one turbo-expander system. The fault occurs at t = 4 s and the faulted expander-generator is disconnected from the network at t = 4.085 s. This fault
5. Conclusion In this study, a comprehensive model of the turboexpander system is presented and a detailed model of its
246
Simulation: Transactions of the Society for Modeling and Simulation International 89(2) Table 4. 15 MVA transformer.
1.1 Generator Terminal Voltage PCC Voltage
Generator Terminal Voltage (pu)
1 0.9 0.8 0.7 0.6
Symbol
Quantity
Value
Connection Sn V1/V2 Z1 Z2 Zm
Mega watt Kilo volt Per unit Second Per unit
Ynd 15 20/6.6 0.0024 + j0.07 0.0024 + j0.07 500 + j500
0.5
Table 5. Turbine.
0.4 0.3
3
4
5
6
7 8 Time (s)
9
10
11
12
Figure 31. System voltages for a three-phase fault on the generator terminal. PCC: point of common coupling. Table 1. Induction generator data. Symbol
Quantity
Value
Pn ZS ZR Xm H F Pole pairs
Mega watt Per unit Per unit Per unit Second Per unit
9 0.005 + j0.1014 0.0178 + j0.1521 3.894 3.5 0.2 2
Table 2. Induction motor data. Symbol
Quantity
Value
Pn ZS ZR Xm H F Pole pairs
Mega watt Per unit Per unit Per unit Second Per unit
3.903 0.020 + j0.1085 0.0092 + j0.0723 3.411 0.304 0.1 2
Table 3. 30 MVA transformer. Symbol
Quantity
Value
Connection Sn V1/V2 Z1 Z2 Zm
Mega watt Kilo volt Per unit Second Per unit
Ynd 30 20/10 0.0027 + j0.08 0.0027 + j0.08 500 + j500
components, such as the nozzle variable angle, nozzle control system, and preheater, are added. By considering a shaft, gearbox, and an induction generator coupled with the system, we have completed the model. To validate the
Symbol
Quantity
Value
Pn Nominal speed Input pressure Output pressure ηlb ηub Input temperature m_
Mega watt Round per minute Bar Bar % % K Kilogram per second
10 23,400 19 5.2 70 85 341 59.1
Table 6. Shaft and gearbox. Symbol
Quantity
Value
JH Gear ratio KH BH Efficiency
kg m2
4 1/15.6 30 800 0.95
Newton per radian Newton second per radian
Table 7. Nozzle system. Symbol
Quantity
Value
Ad md bd ksm k0 k1 k2
m2 Kilogram Newton second per meter Newton per radian Radian per meter Radian Bar/radian
0.03 0.63 206 12,850 2146.8 0 2000
Table 8. Preheater. Symbol
Value
B1 B2 K1 K2 K3 K4
0.1706 6.2893 3120 0.1 1.84e-5 5.44e5
Jelodar et al. proposed model, empirical data from a parallel experimental setup of an expander and regulator is used. Simulation results show a good compatibility between measured and estimated variables. We have used the proposed model to investigate the overall performance of the turbo-expander-generator system under different modes of operation. We have concluded that the existence of a preheater has some effects on the turbine input temperature, but its time lag does not have significant importance in comparison with other system equipment. Neglecting the preheater could decrease the accuracy of estimations in the system. Also, these results demonstrate the importance of a variable-angle nozzle and its control system on the turbine’s operation and the output gas pressure. In addition, the turbine output power depends on the nozzle-angle variations too. We have shown that the nozzle-angle control system is very sensitive and can cause the whole system to be unstable due to interruption in the momentary flow pressure. Finally, in voltage stability studies, it has become clear that the turbo-expander system is vulnerable against typical short circuit faults on the generator terminal. It seems that using power quality improvement equipment is necessary in these systems to save the network and also to protect sensitive loads against these distortions. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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Indices act d H in L lb out sm ub V
actuator diaphragm high speed input low speed lower band output spring ( + membrane) upper band constant volume
Author biographies
Nomenclature A B b CP CT h J k m m_ n O:C: P T T2s Tr ðsÞ W xd
area constant damping constant specific heat capacity (kJ/kg K) turbine constant enthalpy inertia spring constant mass mass flow rate gearbox ratio operating condition pressure temperature, torque temperature in isentropic expansion transfer function work diaphragm displacement
ηT D y t v(P)
turbine efficiency delta angle time constant (s) specific volume (m3/kg)
Mehdi Taleshian Jelodar received BSc and MSc degrees in electrical engineering from Amirkabir University of Technology, Tehran, Iran, in 1999 and 2001, respectively. Currently, he is a PhD student at Amirkabir University of Technology. His research interests include power system dynamic analysis and also power system protection. Hasan Rastegar received BSc, MSc, and PhD degrees in electrical engineering from the Amirkabir University of Technology, Tehran, Iran, in 1987, 1989, and 1998, respectively. Currently, he is an associate professor at Amirkabir University of Technology. He has published many papers in journals and conferences. His research interests include power system control, the application of computational intelligence in power systems, simulation and analysis of power systems, and renewable energy. Hossein Askarian Abyaneh (SM’09) was born in Abyaneh, Isfahan, on 20 March 1953. He received a BS from the Iran University of Science and Technology in 1976 and an MS from Tehran University, Tehran, Iran, in 1982. He also received another MS and PhD in electrical power system engineering from the University of Manchester Institute of Science and Technology, Manchester, UK, in 1985 and 1988, respectively. Currently, he is a professor and chairman of the department of electrical engineering, Amirkabir University of Technology, Tehran, Iran, working in the area of relay protection and power quality. He has published many scientific papers in international journals and conferences.
