Combustion Turbine And Combinedcycle Power Plants.pdf

chapter

13

Combustion Turbine and Combined-Cycle Power Plants

13.1 Introduction Two additional types of generating unit prime movers that are growing in importance are the combustion turbine and combined-cycle units. Combustion turbine units were once considered as generating additions that could be constructed quickly and were reliable units for rapid start duty. The early units were not large, limited to-about 10 MVA, but later units have become available in larger sizes and, in some cases, may be considered a reasonable alternative to steam turbine generating units. A more recent addition to the available types of generating units is the combined-cycle power plant, in which the prime mover duty is divided between a gas or combustion turbine and a heat recovery steam turbine, with each turbine powering its own generator. The dynamic response of combined-cycle power plants is different from that of conventional steam turbine units and they must be studied carefully in order to understand the dynamic performance of these generating units.

13.2 The Combustion Turbine Prime Mover Combustion turbines, often called gas turbines, are used in a wide variety of applications, perhaps most notably in powering jet aircraft. They are also widely used in industrial plants for driving pumps, compressors, and electric generators. In utility applications, the combustion turbine is widely used as fast-startup peaking units. Combustion turbines have many advantages as a part of the generation mix of an electric utility. They are relatively small in size, compared to steam turbines, and have a low cost per unit of output. They can be delivered new in a relatively short time and are quickly installed compared to the complex installations for large steam turbine units. Combustion turbines are quickly started, even by remote control, and can come up to synchronous speed, ready to accept load, in a short time. This makes these units desirable as peaking generating units. Moreover, they can operate on a rather wide range of liquid or gaseous fuels. They are also subjected to fewer environmental controls than other types of prime movers [1]. The major disadvantage of combustion turbines is their relatively low cycle efficiency, being dependent on the Brayton cycle, which makes combustion turbines undesirable as base-load generating units. Another disadvantage is their incompatibility with solid fuels. The combination of low capital cost and low efficiency dictates that combustion turbines are used primarily as peaking units.

513

514

Chapter 13

Combustion turbines can be provided in either one- or two-shaft designs. In the two-shaft design, the second shaft drives a low-pressure turbine that requires a lower speed. However, in practice the single-shaft design is the most common [1]. The combustion turbine model presented here represents the power response of a singleshaft combustion turbine generating unit [2]. The model is intended for the study of power system disturbances lasting up to a few minutes. The generator may be on a separate shaft, in some cases connected to the turbine shaft through a gear train. The model is intended to be valid over a frequency range of about 57 to 63 Hz and for voltage deviations from 50 to 1200/0 of rated voltage. These ranges are considered to be typical of frequency and voltage deviations likely to occur during a major system disturbance. It is assumed that the model is to be used in a computer simulation in which, to obtain economical computer execution times, the timestep of the model might be one second or longer. The model is a rather simple one, but it should be adequate for most studies since the combustion turbine responds rapidly for most disturbances. Figure 13.1 shows a simple schematic diagram of a single-shaft combustion turbine-generator system with its controls and significant auxiliaries [2]. The axial-flow compressor (C) and the generator are driven by a turbine (T). Air enters the compressor at point 1 and the combustion system at point 2. Hot gases enter the turbine at point 3 and are exhausted to the atmosphere at point 4. The control system develops and sends a fuel demand signal to the main turbine fuel system, which in tum, regulates fuel flow to the burner, based on the unit set point, the speed, load, and exhaust temperature inputs. Auxiliaries that could reduce unit power capability are the

AUXILIARY POWERBUS

AUXILIARY ATOMIZING AIR SYSTEM

AUXILIARY FUEL HANDLING SYSTEM

FUEL DEMAND EXHAUST TEMPERATURE

CONTROL SYSTEM AIR IN SPEED REFERENCE

BURNER

1

SPEED FEEDBACK

Fig. 13.1

SHAFT

3 ...-------. POWER GENERATOR ' " - - - - - - ' OUTPUT

Combustion turbine schematic diagram [2].

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Combustion Turbine and Combined-Cycle Power Plants

atomizing air and fuel handling systems shown in the figure. The atomizing air system provides compressed air through supplementary orifices in the fuel nozzles where the fuel is dispersed into a fine mist. The auxiliary fuel handling system transfers fuel oil from a storage tank to the gas turbine at the required pressure, temperature, and flow rate.

13.2. 1 Combustion turbine control Figure 13.2 shows a block diagram of a single-shaft combustion turbine-generator control system. The output of this model is the mechanical power output of the turbine. The input signal, AGCPS, is the power signal from the automatic generation control (AGC) system, in perunit power per second. The power is expressed in the system MVA base [2]. The governor speed changer position variable, noted in Figure 13.2 as GSCP, is the integral of the AGC input. An alternative input K M represents a manual input that is used if the generator is not under automatic generation control. The load demand signal shown in the diagram is the difference between the governor speed changer position and the frequency governing characteristic. The frequency governing characteristic is often characterized as a normal linear governor "droop" characteristic. Then the frequency error is divided by the per-unit regulation to determine the input demand. A nonlinear droop characteristic may be used in some cases. Typical data for the parameters shown in Figure 13.2 are provided in Table 13.1 [2]. The load demand upper power limit varies with ambient temperature according to the relation (13.1) where A = (the per-unit change in power output per per-unit change in ambient temperature) T = ambient temperature in °C T1 = reference temperature in °C

(Osys

-1

Linearor Nonlinear Frequency Governing Characteristics

f

Off-Nominal Voltage and Frequency

K3

AGCPS

Rate Limit

Effects on Power Output Nonwindup

Nonwindup Magnitude Limit

Magnitude Limit Governor

Speed Changer Position (GSCP) Fig. 13.2

Combustion turbine model block diagram [2].

Power

K3 ....O_ut~ PM

Chap~r

516

13

Table 13.1 Typical Combustion Turbine Model Parameters [2)

Constant

Description

Value

Manualrate, per-unitMW/s on given base Conversion, unit base/system base GSCPupper position temperature Combustion turbine time constant,s Normal regulation, per-unit freq/pu MVA Alternateregulation, see Figure 13.4

0.00278 0.11 0.25 0.04 0,01

According to (13.1), the turbine will provide 1.0 per-unit power at a reference ambient temperature of IS °C. The power limit is increased for temperatures below the reference and is decreased for ambient temperatures above the reference. The lower power limit corresponds approximately to the minimum fuel flow limit. This limit is necessary to prevent the blowing out of the flame and corresponds to zero electric power generated. There are three different off-nominal voltage and frequency effects. These are defined in the next section. Figure 13.3 shows the approximate computed response of a General Electric FS-5, Model N, single-shaft combustion turbine in response to a step change in setpoint from no load to full load, using liquid fuel [3] . The analytical model used to compute this response included the effects of the controls, the transport times, heat soak effect of turbine components in the hot gas path, and the thermocouple time constants. The turbine response will vary by several tenths of a second for other models or when using other fuels. Notice the fast response characteristic of the unit to its new power level.

1.0

-------

0.8 ..... '2

::l .... 0.6
0.

.5 ....
0.4

0

0..

0.2

o

o

0. 1

0.2

0.3

0.4

0.5

Time in seconds Fig. 13.3

CT response to a step change in setpoint from no load to rated load [3).

Combustion Turbine and Combined-Cycle Power Plants

517

13.2.2 Off-nominal frequency and voltage eHects The power supply for the governor system is usually provided by the station battery that can provide power for at least 20 minutes and is, therefore, unaffected by the voltage and frequency of the ac power system [3]. The shaft-driven main fuel and lubrication oil systems can be consideredas unaffectedby ac system voltage deviations. If the power demand exceeds the power limit, the combustionturbine power output capability decreases as the frequency drops. A basic characteristic of the combustionturbine is that the air flow decreases with shaft speed and the fuel flow must also be decreasedto maintainthe firing temperature limit. The amount of the air flow decreaseis on the order of 2% in output capability for each 1% drop in frequency. This is shown in equation (13.2), which represents the limitingmultiplieron power demandwhen the unit is running on an exhaust temperature limitation. RPFE= I-B}(DPF)(wBP- W Sy s )

= Reducedpower frequency effect multiplier

(13.2)

where 0 when power demand < power limit B I = { 1 when power demand> power limit DPF = per-unit change in unit output per-unit change in frequency

= 0 if data not available, bypasses the multipliereffect wsys = system frequency WBP = system frequency when unit exceeds its power limit The RPFE is one of the possible limiting effects noted by the limitationblock on the righthand side of Figure 13.2The invocationof this limitation depends on the initial power level of the generating unit and the change in frequency during the transient. For example, if the frequency declines 3 Hz or 5% on a 60 Hz system, then the power capabilityof the unit will be reduced by 2% for each 1% reduction in speed after the power limit is exceeded. A unit operating initially at full load would reach the power limit immediately and the output of the unit would be decreasedby 10%. Off-nominal voltage and frequency both have an effect on the system auxiliaries, such as the fuel system, heaters, and air handling equipment. These effects vary depending on the unit design,the particularinstallation limitations, the utility practice, and the site variables. This represents another limiting function that is referred to in the literature as the auxiliary equipment voltage effect, or AEVE [2]: AEVE:= 1 - max[DPV(VBP - Vr), 0]

(13.3)

where DPV:= per-unit change in unit output per unit change in voltage VBP = voltage level above which there is no reduction in unit output Vt == generatorterminalvoltage

Anotherunit limitationis based on a reductionin system frequency. This limit in definedas [2] AEFE == Auxiliary equipmentfrequency effect == I-max[DPA(wBP- wsys ) , 0]

(13.4)

where DPA is the per-unit change in unit output due to a per-unit change in frequency from the base point frequency WBp.

Chapter 13

518 Aeo, pu

ro, pu

R2

----r--+-------t -- --1 -------L----l- ~-t>

- - - 1.0

RI

I I

o

I I Po

Rl

R2

R2

p

I

I I I

Fig. 13.4 Nonlinear governor droop characteristic [1].

All of the foregoing limiting functions apply to the limiter block on the right-hand side of Figure 13.2.

13.2.3 Nonlinear governor droop characteristic In some cases, it is desirable to include in simulations a nonlinear governor droop characteristic rather than the simple 4% or 5% linear droop characteristic often assumed . This might be necessary, for example, in providing an accurate model of the speed governor characteristic, which is not linear over a wide range, but tends to saturate for large excursions in speed or power. An example of a nonlinear droop characteristic is shown in Figure 13.4 [1, 3]. This is only one type of droop characteristic that might be examined . For example, it is not entirely clear that the slopes labeled R2 need to be equal in the high- and low-frequency ranges, nor is it clear that the center frequency in the R1 range should be exactly at the center between WI and Wz. Given adequate data, one might devise a continuous nonlinear curve to represent a range of frequencies and power responses. However, lacking better data, the droop characteristic of Figure 13.4 probably represents an improvement over the single droop characteristic so often used . Finally, it should be noted that the nonlinear droop characteristic was suggested as one device for improving the system response to very large disturbances, which create large upsets in power plants as well as loads. Some studies are not intended to accurately represent the power system under such extreme conditions, in which case the single droop characteristic may be adequate .

13.3 The Combined-Cycle Prime Mover There are a number of ways in which a combination of power cycles can be used in the generation of electricity, and power plants that use a combination of power cycles can have higher efficiencies that those dependent on a single power cycle. One typical combined-cycle turbine model is shown in Figure 13.5. This system utilizes a combination of a gas turbine Brayton cycle and a steam turbine using a Rankine cycle. The gas exhausted from the gas tur-

Combustion Turbine and Combined-Cycle Power Plants

Gas Turbine

519

Gene rator

Hot Gases Air

Generator

Condenser

Fig. 13.5

A typical combined-cycle power plant arrangement [3].

bine contains a significant amount of sensible heat and a portion of this heat is recovered in a steam generator, which in tum provides the working fluid for the steam turbine . Many combined-cycle power plants are more complex than that shown in Figure 13.5, which shows only the basic components. More practical systems are described below, but all systems can be conceptually reduced to the configuration of Figure 13.5. Figure 13.6 shows the schematic diagram for a combined-cycle power plant with a heat recovery boiler (HRG) [1]. In some designs, the steam turbine may have a lower rating than the gas turbine . In some large-system designs, supplementary firing is used, which may cause the steam turbine to achieve a rating greater than that of the gas turbine. Moreover, there may be more than one HRG, which could significantly increase the steam supply and therefore the power production of the steam subsystem. A descriptive technical paper on combined-cycle power plants has been prepared by the IEEE Working Group on Prime Mover and Energy Supply Models for System Dynamic Performance Studies [6]. Their detailed model of the combined-cycle unit is shown in Figure 13.7. Figure 13.8 shows the interactions among the subsystems of the combined-cycle system [6], and identifies the input and output variables of each subsystem and the coupling among these submodels . This structure is convenient for mathematical modeling of the combined-cycle power plant, which is described in greater detail below . The speed and load controls are described in block diagram form in Figure 13.9. The inputs are the load = \demand, VL> and the speed deviation, AN. The output is the fuel demand signal, F D'

Chapter 13

520

Combu stion Chamber

~~ Gas _

r Air Compressor ~============ll

T ur bime

----r

~ !

Air

-

Generator I

Optional Fuel --~--l Supplementary Firing System

HRB Legend: SU =Superheater B = Boiler EC = Economizer

To Stack

~S

Steam

Drum r--~---+--r U

~\.I'-~---+-l. Heat A A AY ~ B Recovery

fA

Y

.

Boiler

E

·C

)~-I Steam

i.>::= Steam Turbine

Generator 2

Condenser

.........

.... A

Feedwater Heater

......_ _J

Fig. 13.6 Schematic flow diagram ofa combined-cycle heat-recovery boiler [I] .

521

Combustion Turbine and Combined-Cycle Power Plants

Steam Turbine Generation

Heat Recovery Steam Generator

From Other HRSGsII r-

-t--

- -' From OthLT

HRSGs

Fuel

Condensate Pump

To Other Gas Turbines

Gas Turbine Generation

Fig. 13.7 Two-pressure nonreheatrecovery feedwaterheatingsteam cycle generatingunit(HRSGwithinternal deaerator evaporator) (6).

Inlet Temperature

Speed

I

Fue) Demand

L oad Retlerence

FD Speed/Load Control

Speed Dev iation

Speed Deviation

t

Air Flow

Fuel and Air Fuel Controls Flow

Gas Turbi ne

Gas Turbine

Mechan ical Powe r

t

Gas Turbine Flow Rate

Exhaust Temperature

HRS G and Steam Turbine Fig.

13.8

Subsystems ofthe combined-cycle power plant (6).

Stea m Turb'me Mecha nical Power

522

Chapter 13 MAX

!1N Fig. 13.9

13.3.1

MIN Combined -cycle speed and control [6].

Fuel and Air Controls

The gas turbine fuel and air controls are show in block diagram form in Figure 13.10 [6]. In this control scheme, the inlet guide vanes are modulated to vary the air flow, and are active over a limited range . This allows maintaining high turbine exhaust temperatures, improving the steam cycle efficiency at reduced load. The fuel and guide vanes are controlled over the load range to maintain constant gas turbine inlet temperature. This is accomplished by scheduling air flow with the load demand F D and setting the turbine exhaust temperature reference TNto a value that is calculated to result in the desired load with the scheduled air flow at constant turbine inlet temperature. The exhaust temperature reference is calculated from the following basic gas turbine thermodynamic relations (taken from reference [6]).

( 13.5)

TE

1 + J;s

~

Exhaust Temp

T,s

+

1.05

- --

1+ TRls

1.0

7;

>"

N 1.0

TR

Wo

Input Temp

F.D Fuel Demand Signal N Speed Fig. 13.10

Gas turbine fuel and air flow controls [6].

Combustion Turbine and Combined-Cycle Power Plants

523

where TR = reference exhaust temperature per unit of the absolute firing temperature at rated conditions Also (13.6) where PRO =

design cycle pressure ratio

= PRO W = isentropic cycle pressure ratio 'Y = ratio of specific heats = cp/cv

PR

We also define the following

W = design air flow per unit TIT = turbine efficiency Tf = turbine inlet temperature per unit of design absolute firing temperature

Then the per-unit flow required to produce a specified power generation at the given gas turbine inlet temperature r:r is given by the turbine power balance equation

iw«,

(13.7)

where kW is the design output in per unit. Also 3413 . kW Ko = - - - - o WgOTfoCp

(13.8)

and where we define

kWo = base net output per unit WgO = base net flow per unit

Tj'o = turbine inlet temperature per unit of design absolute firing temperature Cp = average specific heat T, = compressor inlet temperature per unit of design absolute firing temperature TIc = compressor efficiency

The combustor pressure drop, specific heat changes, and the detailed treatment of cooling flows have been deleted for purposes of illustration of the general unit behavior. These performance effects have been incorporated into equivalent compressor and turbine efficiency values [6]. Equations (13.7) and (13.8) determine the air flow Wand pressure ratio parameter X for a given per-unit generated power in kW, and at a specified per-unit ambient temperature T; The reference exhaust temperature TR is given by (13.6) by setting 1j.= 1.0. The air flow must be subject to the control range limits. The block identified as A in Figure 13.10 represents the computation of the desired air flow WD and the reference exhaust temperature over the design range of air flow variation by means of vane control. Desired values of WD and TR are functions of FD (the desired values of turbine output from speed/load controls) and ambient temperature T; These are determined by the solution of (13.7) and (13.8) with appropriate limits on WD and TR . The vane control response is modeled with a time constant TR and with nonwindup limits corresponding to the vane control range. The actual air flow WA is shown as a product of desired air flow and shaft speed. The reference exhaust temperature TR is given by (13.6) with Tfset equal to unity.

524

Chapter 13

The measured exhaust temperature TE is compared with the limiting value TR and the error acts on the temperature controller. Normally, T£ is less than TR, which causes the temperature controller to be at the maximum limit of about 1.1 per unit. If TE should exceed TR, the controller will come off limit and integrate to the point where the its output takes over as the demand signal for fuel Vee through the low-select (LS) block. The fuel valve positioner and the fuel control are represented as given in [7], giving a fuel flow signal Wfas another input to the gas turbine model.

13.3.2 The gas turbine power generation A block diagram of the computat ion of gas turbine mechanical power PM G and the exhaust temperature TE is shown in Figure 13.11. The equations used in the development of the gas turbine mechanical power PMG are shown in Figure 13.11. The gas turbine output is a function of the computed turbine inlet temperature Tfo which is a function of the turbine air flow Wf '

1<2 Tf = TCD + -W'W

=

X-I] T; [ 1 + - +W-j(2 TIc

(13.9)

W

where

K2 =

TdT = per-unit combustor temperature rise fO

TCD = compressor discharge temperature per unit of absolute firing temperature Wr= design air flow per unit The gas turbine exhaust temperature TE is determined by equation (13.6), substituting TE for TR and using (13.7) for the computation of X. The mechanical power PM G is a function of the turbine inlet temperature and the flow rate of combustion products Wa + Wf

K

K+ - ' •

Fig. 13.11

1+

I;s

Gas turbine mechanical power and exhaust temperature model [6].

T,'E

Combustion Turbine and Combined-Cycle Power Plants

525

13.3.3 The steam turbine power generation The heat recovery steam generator (HRSG) system responds to changes in the exhaust flow from the gas turbine W and its exhaust temperature Te. This heat is delivered to the high- and low-pressure steam generators, which can be approximated. The exhaust gas and steam absorption temperatures through the HRSG are indicated in Figure 13.12. The transient heat flux to the high- and low-pressure steam generation sections can be approximated using the relations for constant gas side effectiveness , and are computed as follows [6]. 7]g l =

Tex - T'

(13.10)

t; - r: T' - T"

7]g2 =

(13.11)

T' - T m2

where T' and T" are the gas pinch points shown in Figure 13.12. Temperatures Tml and Tm2 are the average metal temperatures in the HP and IP evaporators, respectively. The gas heat absorption by the HRSG section can be computed as follows [6]. (13.12) (13.13) where (13.14) (econ2 = ~ w + 7JeciT" - Ti n)

(13.15)

and where Qeconl, Qecon2, and Q 'econl are the HP and IP economizer heat fluxes.

HP Superheater

HP Evaporator HP Economizer Qed

IP Evaporator IP Evaporator

STEAM ENERGY

Qec2

Heat Absorption, % Fig. 13. 12 Steam energy exhaust gas tempe rature versus heat absorption (6).

100

Chapter 13

526

The economizer heat absorption is approximated using the constant effectiveness expressions, as follows [6]: Q econ2 = T/eciT" - Tf W)mLP Qeconl = T/ecl(T' - tecon2)mllP Q;conl = T/eciT" - Trw)mJIP tecon2 = tfw + T/ec2(T" - Tf n)

(13.16)

Then equations (13.11) through (13.17) are solved to find the temperature and heat flux profiles . The steam flows, mHP and mLP are computed by the pressure/flow relationship at the throttle and admission points as follows: mHP=KrPHP mlfp

+ m/p =

(13.17)

K 'P/p

where K r = throttle valve flow coefficient K' = admission point flow coefficient Steam pressures P HP and PLP are found by integrating the transient energy equations, which are given as DllP?HP = QgHP - hhpmllP + hjWmHP + hjWmllPjW DLP?LP = Q gl.P - hLPmLP + hjWmLPjW

(13.18)

The HP and LP metal temperatures Tm l and Tm2 are determined by integration of the gas and steam side heat flux as shown in Figure 13.13. The steam turbine power in kilowatts is computed as

kWg -

mIlP . AEIlP + mLP . AELP

3413

(13.19)

HP Evap . Metal Temp

T.III

Exhaust Flow

IP Evap. Meta l Temp

IP Admiss ion Pressure Fig. 13.13 Steam system model.

Combustion Turbine and Combined-Cycle Power Plants

1

527

PMS

(1 + sTM Xl + st; ) Fig.13.14

A simplified steam power response model [6].

where AE1f P and AELP are the steam actual available energies [6]. The dynamic relations for the HRSG and steam turbine are shown in Figure 13.13. Note that the heat transferred from the high pressure boiler QGI is a function of the exhaust gas temperature TE , the HP evaporator metal temperature TMI' and the IP evaporator metal temperature TM2' It is noted in reference [6] that the total contribution to mechanical power from the two pressure boilers can be approximated with a simple two-time constant model. The gain between the gas turbine exhaust energy and the steam turbine output will, in general, be a nonlinear function that can be derived from steady-state measurements through the load range, or from design heat balance calculations for rated and partial load conditions. These simplifications will result in a low-order model as shown in Figure 13.14 [6]. Such a low-order model would be very simple to implement in a computer simulation, and may be quite satisfactory for may types of studies, especially studies in which the major disturbance of interest is far removed from the combined cycle power plant. Moreover, this simple model could be "tuned" by comparing it against the more detailed model of Figure 13.13. The detailed model should be considered for studies of disturbances in the vicinity of the combined-cycle plant. From [6] the values of the time constants for this simrlified model are given as

TM = 5 S

TB = 20 s

Problems 13.1

The combustion turbine presented in Figure 13.1 is a single-shaft design. Other combustion turbines are designed to employ two different shafts. Sketch how such a two-shaft unit might be configured and compare with the single-shaft design. What are the advantages of a two-shaft design? Hint: Consult the references at the end of the chapter, if needed. 13.2 The single-shaft combustion turbine shown in Figure 13.1 is called a "direct open cycle" design since it exhausts its hot exhaust to the atmosphere. A different design is called a "closed-cycle" system, which recycles the exhaust back to the air input port. Make a sketch of how such a closed-cycle system might be configured. 13.3 It has been noted that the ideal cycle for the gas turbine is the Brayton cycle. Explore this cycle using appropriate references on thermodynamic cycles and sketch both the P-V and the T-S diagrams for this cycle.

References 1. EI-Wakil, M. M., Powerplant Technology, McGraw-Hill, New York, New York, 1984. 2. Turner, A. E. and R. P. Schulz, Long Term Power System Dynamics, Research Project 764-2, User's Guide to the LOTDYS Program, Final Report, Electric Power Research Institute, Palo Alto, CA, April 1978.

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Chapter 13

3. Bailie, R. C., Energy Conversion Engineering, Addison-Wesley, Reading, MA, 1978. 4. Pier, J. B. and S. Bednarski, "A simplified single shaft gas turbine model for use in transient system analysis," General Electric Company Report, 72-EU-2099, 1972. 5. Schulz, R. P., A. E. Turner, and D. N. Ewart, Long Term Power System Dynamics, volume 1, Summary and Technical Report, EPRI Report 90-7-0 Final Report, June 1974. 6. IEEE Working Group on Prime Mover and Energy Supply Models for System Dynamic Performance, F. P. deMello, Chairman, "Dynamic models for combined cycle plants in power system studies," IEEE Transactions Power Systems, 9, 3, August 1994, p. 1698. 7. Rowen, W. I., "Simplified mathematical representations of heavy-duty gas turbines," Trans. ASME, 105 (1),1983, Journal ofEngineeringfor Power, Series A, October 1983, pp. 865-869.