research article piecewise model and parameter obtainment

Research ArticlePiecewise Model and Parameter Obtainment ofGovernor Actuator in Turbine

Jie Zhao Li Wang Dichen Liu and Jun Wang

School of Electrical Engineering Wuhan University Wuhan 430072 China

Correspondence should be addressed to Jie Zhao jiez whuwhueducn

Received 13 December 2014 Revised 10 March 2015 Accepted 10 April 2015

Academic Editor Georgios Sirakoulis

Copyright copy 2015 Jie Zhao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The governor actuators in some heat-engine plants have nonlinear valves This nonlinearity of valves may lead to the inaccuracyof the opening and closing time constants calculated based on the whole segment fully open and fully close experimental testcurves of the valve An improved mathematical model of the turbine governor actuator is proposed to reflect the nonlinearity ofthe valve in which the main and auxiliary piecewise opening and closing time constants instead of the fixed oil motive opening andclosing time constants are adopted to describe the characteristics of the actuator The main opening and closing time constants areobtained from the linear segments of thewhole fully open and close curvesTheparameters of proportional integral derivative (PID)controller are identified based on the small disturbance experimental tests of the valveThen the auxiliary opening and closing timeconstants and the piecewise opening and closing valve points are determined by the fully openclose experimental tests Severaltesting functions are selected to compare genetic algorithm and particle swarm optimization algorithm (GA-PSO) with other basicintelligence algorithms The effectiveness of the piecewise linear model and its parameters are validated by practical power plantcase studies

1 Introduction

Power system analysis is essential for the system security andstability For medium and long term stability analysis suchas the frequency control the mathematical model of primemover and its governor and the accuracy of the parametersare critical [1ndash4] After the blackout accidents occurred inthe Western United States in 1996 the World ElectronicCircuits Council (WECC) in 2001 conducted two large unitstripped disturbance tests in the Western United States gridand the study found that the frequency response of originalspeed governor model did not match with the measuredresponse process thus a new thermal governor systemmodelwas established to improve the simulation accuracy [5] Toidentify parameters and simulate the characteristics of thespeed regulation system based on experimental test data isa fundamental work for the power system Therefore thestudy of prime mover and its governor simulation modeland parameters identification for the power system is ofimportance to the theoretical and practical value

Studies on parameter identification methods of steamturbine governor for different simulation cycles can be found

in [6ndash11] The fixed oil motive opening and closing time con-stants are adopted in the existing literatures and power systemsimulation software settings [9ndash12] but they are difficult tobe applied to the nonlinear characteristics of some thermalpower plant actuator valversquos fully open or close test As forthe nonlinear valve of hydroturbine certain existing literatureuses several sets of proportional integral derivative (PID)parameters to meet the different operating conditions [13]However PID parameters tuning based on intelligent algo-rithm is relatively complex so a compromised parameteridentification method under different working conditions isrequired to avoid the difficulty caused by the large number ofPID parameters For parameter identification methods theclassical methods [14] are only applicable to linear systemwhereas the intelligent algorithms [13 15ndash19] are widely useddue to robustness in presence of the input disturbance signalssuch as input noises Nevertheless the measured data can beeasily polluted by the noise and the parameters need to beidentified in time

For different plants the speed limit value of the oil motivemovement is different through valve fully open and fully

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 709272 9 pageshttpdxdoiorg1011552015709272

2 Journal of Applied Mathematics

1

s

Ki

s

Valueinstruction +

+

+

+minusPCV

Kp

Kds

SPIDmax

SPIDmin VELclose

VELopen1Tc

1To

Pmax

PminPGV

LVDT

Valveopening

s

1

1 + T2

Figure 1 Servo and actuator system model

1

s

Valueinstruction +

+

+

+

minusPCV

Kp

Kds

SPIDmax

SPIDmin VELclose

VELopen

1To

1Tc

Pmax

Pmin

PGV

LVDT

Valve

Piecewise

Open

Close

opencloseselection

opening

s

1

1 + T2

1To1

1Tc1

Ki

s

Figure 2 Improved model of servo and actuator system

close test so that to get the speed limit value is the keyto simulate the characteristics of governor actuator withhigh precision since speed limit is related to the oil motiveopening and closing time constants For some thermal powerplants the valve is nonlinear As a result piecewise speed limitis adopted not only themaximumandminimum speed limitThe governor actuator piecewise linear model is establishedbased on the movement principle of the oil motive

The paper is organized as follows In Section 2 a governoractuator piecewise linear model is established (hereinafterreferred to as the improved model of servo and actuatorsystem) using piecewise opening and closing time constantsincluding the main opening and closing time constants andthe auxiliary opening and closing time constants The effec-tiveness of genetic algorithm-particle swarm optimization(GA-PSO) algorithm is compared with other basic intelli-gence algorithms through selection of multiple test func-tions in Section 3 Aimed at the nonlinear characteristics ofactuator valve the calculation method of main opening andclosing time constants and their influence on PI parameteridentification are discussed in Section 4 and the results anddiscussion are presented in this section as well Conclusionsare finally drawn in Section 5

2 Governor Actuator Mathematical Model andParameter Identification Steps

21 Governor Actuator Mathematical Model Electrohy-draulic servo and actuator are essential one of the most

important factors influencing the dynamic characteristics ofthe speed control system and the ability of primary frequencyspeed control [20 21] The actuator amplifies and convertscontrol signal from the regulator to the opening of the valveto control steam flow into the turbine The mathematicalmodel is shown in Figure 1 where119870119901119870119894 and119870119889 are the pro-portional integral and derivative multiple coefficients of theintegrated amplifiermodule respectively which is commonlyproportional or proportional plus integral part 119878PIDmax and119878PIDmin are the upper and low limits of the integrated amplifiermodule respectively VELopen and VELclose denote the rapidopening and rapid closing coefficient respectively 119879119900 and 119879119888denote the oil motive opening and closing time constantsrespectively 119875max and 119875min are the maximum and minimumvalve opening value respectively 1198792 is the oil motive strokefeedback link time constant usually taken as 002 seconds

When the valve moves sharply the servo valve outputreaches its limit position and the oil motive has the maxi-mum opening or closing regulating speed The output valuedemonstrates linearity However nonlinear valve cases existand the model in Figure 1 is no longer applicable Thus animproved model of servo and actuator system is shown inFigure 2 where 1198791199001 and 1198791198881 also are the opening and closingtime constants respectively and for the distinction119879119900 and119879119888are defined as the main opening and closing time constantsrespectively 1198791199001 and 1198791198881 are defined as the auxiliary openingand closing time constants respectively The piecewise open-ing and closing valve points are determined according to theactual condition

Journal of Applied Mathematics 3

0 05 1 15 2 25 30

102030405060708090

100

Time (s)

13

2PG

V(

)

(1) The whole(2) The front(3) The middle

Figure 3 Opening time constant calculation diagram

As the model shown in Figure 2 the oil motive operatesby applying the main opening and closing time constantwhen the valve responds to a small disturbance underbig disturbances the oil motive soon reaches its maximumadjusting speed In other words oil motive firstly operateslinearly by the main opening and closing time constantWhen it is close to the steady state it acts with the auxiliaryopening and closing time constant So the model can beapplied to both linear and nonlinear valves Parameters thatneed to be identified in this model are the main and auxiliarypiecewise opening and closing time constants of the oilmotive PI parameters namely 119870119901 and119870119894

22 Parameter Identification Steps The current existingmethod to obtain the oil motive opening and closing timeconstants is based on the fully open or close test curves of theactuator [22] When the valve operates nonlinearly takingthe fully open test as an example the diagram using differentsegmented curves to obtain the oil motive opening timeconstant is shown in Figure 3

(a) Thewhole segment the valvemoves from fully closedto fully open

(b) The front segment the valve moves from fully closedto about 75 percent opening

(c) The middle segment the valve moves from about 25percent opening to 75 percent opening

The blue curve in Figure 3 is the measured actuator fullyopen curve and the dotted line uses the piecewise openingtime constants If the main opening time constant is obtainedfrom the front segment curve data of fully open test theexpression is shown as

119879119900 =(1205821119875max minus 119875min)

Δ1199051

(1)

If the main closing time constant is obtained from the frontsegment curve data of fully close test the calculation methodis

119879119888 =(119875max minus 1205822119875max)

Δ1199052

(2)

where 1205821 and 1205822 are constant coefficients The whole ormiddle segment curve of the fully openclose test can also beapplied to calculate 119879119900 and 119879119888 and the expressions are similarto (1)-(2)

Assuming that the piecewise opening and closing valvepoints are 119884119900 and 119884119888 1198791199001 and 1198791198881 are generally defined as

1198791199001 =(119875max minus 119884119900)

Δ1199053

1198791198881 =(119884119888 minus 119875min)

Δ1199054

(3)

whereΔ1199051Δ1199052Δ1199053 andΔ1199054 in (1) to (3) are the correspondingtime of the curves in the valve opening change process

The procedure to obtain parameters of the modifiedactuator model is described as follows

Step 1 Process the selected input valve instruction andthe output valve opening data with the method of waveletdenoising

Step 2 Calculate the main opening and closing time con-stants through the linear segment of the fully open or closetest curve

Step 3 Identify 119870119901 and 119870119894 using small disturbance experi-mental tests data of the valve

Step 4 Determine the piecewise opening and closing valvepoints and the auxiliary opening and closing time constantsthrough fully open or close test curve

Step 5 Check actuator parameters

The actuator parameter identification process of servoand actuator is shown in Figure 4

3 The Parameter Identification MethodBased on GA and PSO Algorithm

31The Parameter IdentificationMethod An improved parti-cle swarm optimization (IPSO) algorithm and GA-PSO algo-rithm are presented to identify parameters Particle swarmoptimization [17ndash19] is a kind of bionic algorithm to solveoptimization problems The location and speed of particleshould be constantly updated The concept of inertia weightis introduced to revise the speed update equation to improvesearch in the global scope To further enhance the searchability the learning factors 1198881 and 1198882 are linearly changed andthe convergence factor is introduced

120582 =2

10038161003816100381610038161003816100381610038162 minus 1198881 minus 1198882 minus

radic(1198881 + 1198882)2minus 4 (1198881 + 1198882)

1003816100381610038161003816100381610038161003816

(4)


Data preprocessing

Calculate the main openingclosing time constants

PI parameters identificationwith valve disturbance test

Determine the auxiliary

Simulation verificationand correction

The end

openingclosing time constants

Figure 4 The parameter identification process of servo and actua-tor

Then the location update equation is

119909119896+1

119894119863= 119909119896

119894119863+ 120582 sdot V119896+1

119894119863 (5)

To improve the performance of the initial solution chaosmethod is used to generate initial particle population Thetwo-dimensional chaotic map is

119909119899+1 = (119909119899 + 119910119899) mod 1

119910119899+1 = (119909119899 + 2119910119899) mod 1(6)

Genetic algorithm (GA) [9] is also a kind of optimizationmethod simulating natural biological evolution mechanismapplied to many optimization problems However it has itslimitations such as low efficiency and premature convergenceSo a combined GA-PSO algorithm is used to improve opti-mization performance and the algorithm process is shownin Figure 5

The algorithm process of GA-PSO is divided into fourparts Part (1) for particle initialization Part (2) for geneticalgorithm initialization Part (3) for extremum comparisonPart (4) for the particle position and speed update Individualfitness value is determined by objective function and thefollowing criterion function is used as fitness function

119891 =

(sum119873

119894=1(119910 (119894) minus 1199100 (119894))

2)

119873 (7)

where 119873 is the number of sampling points 119910(119894) is the 119894thsimulation output value and 1199100(119894) is the 119894th measured outputvalue

32 The Characteristic of Different Algorithms The test func-tions of Griewank Ackley Schwefel Shubert and Schaffer[23 24] are used to compare the optimizing performanceof four algorithms (ie the basic PSO algorithm IPSO

The start

Initialize particle Set control parameters

Code and generate initialpopulation

Calculate individual

Select cross over mutate andgenerate a new population

Update particle speed

fitness valueCalculate particle

Upgrade individual and

Termination criterion

Termination criterionis satisfied

is satisfied

Compare and updatePSO extreme

The end

The end

Yes

Yes

No

No

group extremes

and position

Calculate particle

Look for individualand group extremes

(1) (2)

(3) (4)

fitness value

fitness value

position and speed

Figure 5 The algorithm process of GA-PSO

algorithm GA algorithm and GA-PSO algorithm) The fivetest functions are named as 1198911 to 1198915 respectively

For each test function the algebra of four algorithms is70 and the maximum iteration step is 100 For the functionswith the extreme point of zero the tolerance of error is setwithin 10minus10 and the optimization of four algorithms is testedfor 50 times For each algorithm the optimal value obtainedby per optimization program running has little differenceThe average runtime on optimization program costs averageoptimal value and standard deviation of four algorithms areshown in Table 1

In Table 1 the average optimal value is displayed insteadof the best and the worst optimal values of four algorithmsThe results show that GA algorithm performs better forcertain functions (eg 1198913 namely the Schwefel function) butthe running time is longer GA-PSO performs best overallwith shorter running time

Take the Schaffer function as an example to give evolvingoptimization curve Under the condition of given precisionthe fitness value curves of the four identification algorithmsare shown in Figure 6

The vertical axis in Figure 6 is the logarithm of fitnessvalue It shows that IPSO algorithm and GA-PSO algorithmcan obtain higher accuracy with less algebra


Table 1 The average running time average optimal value and standard deviation of the four algorithms

Function name PSO IPSO GA GA-PSO Theory extreme

1198911

Average runtime (s) 0115 0092 0453 01990Average optimal value 00049 00040 00125 00019

Standard deviation 00071 00066 00188 00039

1198912

Average runtime (s) 0166 0115 0501 02300Average optimal value 738119890 minus 7 604119890 minus 11 0162 585119890 minus 11

Standard deviation 107119890 minus 6 649119890 minus 11 05932 619119890 minus 11

1198913

Runtime (s) 0164 0170 0487 0336minus8379Average optimal value minus82255 minus81428 minus83796 minus83441


1198914

Average runtime (s) 0114 0122 0508 0318minus1867Average optimal value minus18028 minus18346 minus18150 minus18448


1198915


Standard deviation 0007 0006 0034 0004649119890 minus 11 is equal to 649 times 10minus11

0 10 20 30 40 50 60 70 80 90 100

0

Iter

PSOIPSO

GAGA-PSO

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

log10

(err

)

Figure 6 Curves comparison of fitness value

By comparing several algorithms of five test functionoptimization results the optimizing performance of thesealgorithms ranks from good to bad as follows GA-PSOalgorithm IPSO algorithm GA algorithm and the basic PSOalgorithm

4 Results and Discussion

GA-PSO algorithm which has better parameters identifi-cation stability is used to identify the parameters of twodifferent turbine governor actuators with linear and nonlin-ear characteristics respectively and the governor actuatorpiecewise linear model is verified Two thermal power plants(ie Xiaolongtan power plant and Qujing power plant) withreheat steam turbine in Yunnan province of West China areadopted to test the improvedmodelThe rated power of thesetwo power plants is 300MW With the field static tests ofturbine actuator fully open or close wave record can be got

020 04 06 08 1 1250

0100

GV

1 (

)

050

0100

GV

2 (

)

01 02 03 04 05 06 07 08 09 1

020 04 06 08 1 1250

0100

GV

3 (

)

020 04 06 08 1 1250

0100

GV

4 (

)

020 04 06 08 1 1250

0100

GV

5 (

)

020 04 06 08 1 1250

0100

GV

6 (

)

Time (s)

Figure 7 Servo and actuator fully open wave record of Xiaolongtanpower plant

aswell as the input valve instruction and output valve openingdata under small disturbance experimental tests of the valve

41 The Verification of Piecewise Model Xiaolongtan powerplant with linear valve is studied to verify the improvedservo and actuator system model Fully open wave record ofXiaolongtan power plant is shown in Figure 7

The valve fully open curve is approximately linear inFigure 7 as well as the fully close test curve After analyzingeach valve movement speed from the fully open and closetests the valves with similar movement speed are selectedAs a result the main opening and closing time constants are119879119900 = 102 seconds and 119879119888 = 145 seconds and the piecewise


0 05 1 15 2 25 3 35 4049

05

051

052

053

054

055

056

Time (s)

Simulation curveMeasured curve

PG

V(p

u)

(a)

Time (s)


0 05 1 15 2 25044

045

046

047

048

049

05

051

PG

V(p

u)

(b)

Figure 8 Valve opening simulation compared with the measured curve

opening and closing valve points are approximate to be 0Therefore the auxiliary opening and closing time constantsare set as 1198791199001 = 102 seconds and 1198791198881 = 145 seconds

Take the average valve opening value of the six valve (ieGV1 to GV6) curves as the simulation compared valve open-ing valueThe identification results by GA-PSO algorithm are119870119901 = 15167 119870119894 = 0 under small disturbance experimentaltests with valve opening ranging from 50 to 55 and valveopening ranging from 50 to 45 The valve opening simu-lation comparedwith themeasured data is shown in Figure 8

Figure 8 shows that the results identified by GA-PSOalgorithm coincide with the measured data For Xiaolongtanpower plant since the piecewise opening and closing valvepoints are set to 0 the simulation results of the improvedservo and actuator system model (see Figure 2) are similarto the servo and actuator system model in Figure 1

42 The Influence of Different Calculated Main Opening TimeConstants on PI Parameter Identification Qujing power plantwith nonlinear valve is also studied to verify the improvedservo and actuator system model The actuator fully openwave record of Qujing power plant is shown in Figure 9

It can be seen from Figure 9 that the former period offully open test curve is approximately linear but it turns flatwhen being close to steady-state value The fully closed testcurve presents similar trend Given 1205821 = 07 1205822 = 03in (1) to (2) the front the middle and the whole segmentcurve data of valve fully open test are selected to calculatethe main opening time constant Similarly the main closingtime constant is obtained The results of different methods(see Figure 3) calculating the main opening and closing timeconstants are shown in Table 2

Table 2 shows that since the fully open test curve isnonlinear the main opening and closing time constant cal-culation results are different when using different curve data

The valve opening data ranging from 50 to 55 undersmall disturbance is selected to discuss the influence ofdifferent main opening and closing time constant calculationresults on PI parameter identification The identified PIparameters using GA-PSO algorithm are shown in Table 3

0 05 1 15 2 25 3

0 05 1 15 2 25 3

0 05 1 15 2 25 3

020406080

100

GV

1 (

)

020406080

100

GV

2 (

)

020406080

100

GV

3 (

)

Time (s)

Figure 9 Servo and actuator full open recorded diagram of Qujingpower plant

Table 2Main opening and closing time constant calculation results

Calculated The wholesegment

The frontsegment

The middlesegment

119879119900 254 189 214119879119888 235 158 191

119879119900 has been calculated out already As shown in Table 3the PI parameter identification results are varying as the cal-culated main opening time constant changes In conclusionthe PI parameters identified depend on the calculation resultsof themain opening and closing time constantThehigher thecalculated 119879119900 is the higher the identified119870119901 is

Taking the three groups of parameters in Table 3 intosimulation model the valve opening results are shown inFigure 10

In Figure 10 the parameters of simulated curve 1 are119870119901 =952 119870119894 = 1072 The parameters of simulated curve 2 are119870119901 = 683 119870119894 = 77 and the parameters of simulatedcurve 3 are 119870119901 = 8 119870119894 = 887 With small disturbance


Table 3 Identification results using GA-PSO algorithm

Parameters Range 119879119900 = 254 s 119879119900 = 189 s 119879119900 = 214 s119870119901 0sim100 952 683 8119870119894 0sim100 1073 77 887

0 05 1 15 2 25 3 35

05051052053054055056

Time (s)

Simulation curve 1 Simulation curve 2

Simulation curve 3 Measured curve

PG

V(p

u)

(a)



0 05 1 15 2 25 3 35 4 45 5

05051052053054055056

Time (s)

PG

V(p

u)

(b)

Figure 10 Valve opening simulation and measured contrast

0 05 1 15 2 25 3 35

0

02

04

06

08

1

Time (s)

Piecewise openOriginal modelMeasured

PG

V(p

u)

(a)


0 05 1 15 2 25 3 350

02

04

06

08

1

Time (s)

PG

V(p

u)

(b)

Figure 11 Valve fully open and close simulation comparison

experimental tests contrast the simulation results of threegroups of parameters are consistent with the measured data

43 Actuator Comparison under Large Disturbance Test Asshown in Figure 7 the measured fully open or close testcurve is nonlinear so a certain group of oil motive openingand closing time constants can not match with the measuredcurveThemain and auxiliary piecewise opening and closingtime constants are adopted The specific plan is as follows

(a) When valve fully opens the oil motive opening timeswitches from 189 seconds to 4 seconds at the valveopening of 064

(b) When valve fully closed the oil motive closing timeswitches from 158 seconds to 57 seconds at the valveopening of 03

For instance for Qujing power plant the parameters are119879119900 = 189 seconds 1198791199001 = 4 seconds 119879119888 = 158 seconds1198791198881 = 57 seconds 119884119900 = 064 and 119884119888 = 03 Actuator PIparameters are 119870119901 = 683 119870119894 = 77 The comparison ofmeasured and simulation data with and without piecewise oilmotive openingclosing time constants is shown in Figure 11

The simulation results using the actuator piecewise lin-ear model with main and auxiliary piecewise opening andclosing time constants coincide with the measured curve(see Figure 11) In conclusion when the valve is nonlinearadopting main and auxiliary piecewise opening and closingtime constants is not able to influence the simulation of valveunder small disturbance At the same time simulation underlarge disturbance ismore consistent with themeasured curveThus it can solve the problem of nonlinearity of the valveBut using fixed oil motive openingclosing time constants


the difference between the simulation and the measuredcurve under large disturbance is significant

5 Conclusion

The GA-PSO parameter identification algorithm is studiedand the effectiveness of GA-PSO algorithm is validated bymultiple test functions and simulation results Aimed atactuators in certain power plants whose valve fully open andfully close tests are nonlinear three segmented curves (iethe whole segment curve the front segment curve and themiddle segment curve) are selected to calculate the mainopening and closing time constants and their effect on the PIparameter identification is studiedThe conclusion is that thehigher the calculated value of the main opening and closingtime constant is the higher the proportion link of actuator is

Although the existing servo and actuator system modelcan meet the simulation need under small disturbance of thevalve it has large error with tests under large disturbanceThe actuator piecewise linear model with main and auxiliarypiecewise opening and closing time constants increases thefeasibility of thismodel which can fit the cases when the valveis nonlinearThe actual power plant cases verify the reliabilityof the established governor actuator piecewise linear modeland its parameters are effective

Conflict of Interests

The authors have declared that no conflict of interests exists

Acknowledgments

This work is funded by the National Natural Science Foun-dation of China (51307123) and China Postdoctoral ScienceFoundation (2013M531736)

References

[1] J L Dineley and P J Fenwick ldquoThe effects of prime-moverand excitation control on the stability of large steam turbinegeneratorsrdquo IEEETransactions on PowerApparatus and Systemsvol 93 no 5 pp 1613ndash1623 1974

[2] L Pereira D N Kosterev D Davies and S PattersonldquoNew thermal governor model selection and validation in theWECCrdquo IEEE Transactions on Power Systems vol 19 no 1 pp517ndash523 2004

[3] L Pereira ldquoNew thermal governor model development itsimpact on operation and planning studies on the WesternInterconnectionrdquo IEEE Power and Energy Magazine vol 3 no3 pp 62ndash70 2005

[4] W Zhang F Xu W Hu M Li W Ge and Z Wang ldquoResearchof coordination control system between nonlinear robust exci-tation control and governor power system stabilizer in multi-machine power systemrdquo in Proceedings of the IEEE InternationalConference on Power System Technology (POWERCON rsquo12) pp1ndash5 October-November 2012

[5] L Pereira J Undrill D Kosterev D Davies and S Patterson ldquoAnew thermal governor modeling approach in theWECCrdquo IEEETransactions on Power Systems vol 18 no 2 pp 819ndash829 2003

[6] IEEE Committee ldquoDynamic models for steam and hydroturbines in power system studiesrdquo IEEE Transactions on PowerApparatus and Systems vol 92 no 6 pp 1904ndash1914 1973

[7] V Kola A Bose and P M Anderson ldquoPower plant modelsfor operator training simulatorsrdquo IEEE Transactions on PowerSystems vol 4 no 2 pp 559ndash565 1989

[8] W Shinohara and D E Koditschek ldquoA simplified model basedsupercritical power plant controllerrdquo in Proceedings of the 35thIEEE Conference on Decision and Control vol 4 pp 4486ndash4491Kobe Japan December 1996

[9] G K Stefopoulos P S Georgilakis N D Hatziargyriou and AP SMeliopoulos ldquoA genetic algorithm solution to the governor-turbine dynamic model identification in multi-machine powersystemsrdquo in Proceedings of the 44th IEEE Conference on Decisionand Control and the European Control Conference (CDC-ECCrsquo05) pp 1288ndash1294 December 2005

[10] T Inoue and H Amano ldquoA thermal power plant model fordynamic simulation of load frequency controlrdquo in Proceedingsof the IEEE PES Power Systems Conference and Exposition (PSCErsquo06) pp 1442ndash1447 November 2006

[11] B Vahidi M R B Tavakoli and W Gawlik ldquoDeterminingparameters of turbinersquos model using heat balance data of steampower unit for educational purposesrdquo IEEE Transactions onPower Systems vol 22 no 4 pp 1547ndash1553 2007

[12] S Jiang Q-J Chen and W-Y Cai ldquoSimulation optimizingstrategy for parameters of hydro-governorrdquo Proceedings of theChinese Society of Electrical Engineering vol 28 no 3 pp 102ndash106 2008

[13] C Jiang and P Yan ldquoNeural network PID control system forthe FTGSrdquo in Proceedings of the 2nd International Conference onIntelligent Computation Technology and Automation (ICICTArsquo09) pp 87ndash90 October 2009

[14] D J GMorrell andBWHogg ldquoIdentification and validation ofturbo-generator modelsrdquo Automatic vol 26 no 1 pp 135ndash1561990

[15] C Alippi and V Piuri ldquoIdentification of non-linear dynamicsystems in power plantsrdquo in Proceedings of the 38th MidwestSymposium on Circuits and Systems vol 1 pp 493ndash496 August1995

[16] S Lu and B W Hogg ldquoDynamic nonlinear modelling of powerplant by physical principles and neural networksrdquo InternationalJournal of Electrical Power and Energy System vol 22 no 1 pp67ndash78 2000

[17] A N Abdalla K Li and S J Cheng ldquoParameter identifica-tion of governor-turbine system based on a PSO algorithmrdquoDynamics of Continuous Discrete and Impulsive Systems-SeriesBmdashApplications amp Algorithms vol 14 no 3 pp 1ndash6 2007

[18] S Chen T Mei M Z Luo and X Q Yang ldquoIdentification ofnonlinear system based on a new hybrid gradient-based PSOalgorithmrdquo in Proceedings of the International Conference onInformation Acquisition (ICIA rsquo07) pp 265ndash268 Seogwipo-siRepublic of Korea July 2007

[19] I Ghatuari N Mishra and B K Sahu ldquoPerformance analysisof automatic generation control of a two area interconnectedthermal system with nonlinear governor using PSO and dealgorithmrdquo in Proceedings of the International Conference onEnergy Efficient Technologies for Sustainability (ICEETS 13) pp1262ndash1266 Nagercoil India April 2013

[20] M Dulau and D Bica ldquoSimulation of speed steam turbinecontrol systemrdquo Procedia Technology vol 12 pp 716ndash722 2014


[21] M Azubalis V Azubalis A Jonaitis and R Ponelis ldquoIdentifi-cation of model parameters of steam turbine and governorrdquoOilShale vol 26 no 3 pp 254ndash268 2009

[22] Working Group on Prime Mover and Energy Supply Modelsfor System Dynamic Performance Studies ldquoDynamic modelsfor fossil fueled steam units in power system studiesrdquo IEEETransactions on Power Systems vol 6 no 2 pp 753ndash761 1991

[23] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[24] X Xu and L Lei ldquoThe research of advances in adaptive geneticalgorithmrdquo in Proceedings of the IEEE International Conferenceon Signal Processing Communications and Computing pp 1ndash6Xirsquoan China September 2011

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1

s

Ki

s

Valueinstruction +

+

+

+minusPCV

Kp

Kds

SPIDmax

SPIDmin VELclose

VELopen1Tc

1To

Pmax

PminPGV

LVDT

Valveopening

s

1

1 + T2

Figure 1 Servo and actuator system model

1

s

Valueinstruction +

+

+

+

minusPCV

Kp

Kds

SPIDmax

SPIDmin VELclose

VELopen

1To

1Tc

Pmax

Pmin

PGV

LVDT

Valve

Piecewise

Open

Close

opencloseselection

opening

s

1

1 + T2

1To1

1Tc1

Ki

s

Figure 2 Improved model of servo and actuator system

close test so that to get the speed limit value is the keyto simulate the characteristics of governor actuator withhigh precision since speed limit is related to the oil motiveopening and closing time constants For some thermal powerplants the valve is nonlinear As a result piecewise speed limitis adopted not only themaximumandminimum speed limitThe governor actuator piecewise linear model is establishedbased on the movement principle of the oil motive

The paper is organized as follows In Section 2 a governoractuator piecewise linear model is established (hereinafterreferred to as the improved model of servo and actuatorsystem) using piecewise opening and closing time constantsincluding the main opening and closing time constants andthe auxiliary opening and closing time constants The effec-tiveness of genetic algorithm-particle swarm optimization(GA-PSO) algorithm is compared with other basic intelli-gence algorithms through selection of multiple test func-tions in Section 3 Aimed at the nonlinear characteristics ofactuator valve the calculation method of main opening andclosing time constants and their influence on PI parameteridentification are discussed in Section 4 and the results anddiscussion are presented in this section as well Conclusionsare finally drawn in Section 5

2 Governor Actuator Mathematical Model andParameter Identification Steps

21 Governor Actuator Mathematical Model Electrohy-draulic servo and actuator are essential one of the most

important factors influencing the dynamic characteristics ofthe speed control system and the ability of primary frequencyspeed control [20 21] The actuator amplifies and convertscontrol signal from the regulator to the opening of the valveto control steam flow into the turbine The mathematicalmodel is shown in Figure 1 where119870119901119870119894 and119870119889 are the pro-portional integral and derivative multiple coefficients of theintegrated amplifiermodule respectively which is commonlyproportional or proportional plus integral part 119878PIDmax and119878PIDmin are the upper and low limits of the integrated amplifiermodule respectively VELopen and VELclose denote the rapidopening and rapid closing coefficient respectively 119879119900 and 119879119888denote the oil motive opening and closing time constantsrespectively 119875max and 119875min are the maximum and minimumvalve opening value respectively 1198792 is the oil motive strokefeedback link time constant usually taken as 002 seconds

When the valve moves sharply the servo valve outputreaches its limit position and the oil motive has the maxi-mum opening or closing regulating speed The output valuedemonstrates linearity However nonlinear valve cases existand the model in Figure 1 is no longer applicable Thus animproved model of servo and actuator system is shown inFigure 2 where 1198791199001 and 1198791198881 also are the opening and closingtime constants respectively and for the distinction119879119900 and119879119888are defined as the main opening and closing time constantsrespectively 1198791199001 and 1198791198881 are defined as the auxiliary openingand closing time constants respectively The piecewise open-ing and closing valve points are determined according to theactual condition


0 05 1 15 2 25 30

102030405060708090

100

Time (s)

13

2PG

V(

)









119879119900 =(1205821119875max minus 119875min)

Δ1199051

(1)


119879119888 =(119875max minus 1205822119875max)

Δ1199052

(2)



1198791199001 =(119875max minus 119884119900)

Δ1199053

1198791198881 =(119884119888 minus 119875min)

Δ1199054

(3)











120582 =2


radic(1198881 + 1198882)2minus 4 (1198881 + 1198882)

1003816100381610038161003816100381610038161003816

(4)


Data preprocessing





The end




119909119896+1

119894119863= 119909119896

119894119863+ 120582 sdot V119896+1

119894119863 (5)


119909119899+1 = (119909119899 + 119910119899) mod 1

119910119899+1 = (119909119899 + 2119910119899) mod 1(6)



119891 =

(sum119873

119894=1(119910 (119894) minus 1199100 (119894))

2)

119873 (7)



The start










is satisfied


The end

The end

Yes

Yes

No

No

group extremes

and position

Calculate particle


(1) (2)

(3) (4)

fitness value

fitness value

position and speed










1198911



1198912



1198913



1198914



1198915



0 10 20 30 40 50 60 70 80 90 100

0

Iter

PSOIPSO

GAGA-PSO

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

log10

(err

)





020 04 06 08 1 1250

0100

GV

1 (

)

050

0100

GV

2 (

)

01 02 03 04 05 06 07 08 09 1

020 04 06 08 1 1250

0100

GV

3 (

)

020 04 06 08 1 1250

0100

GV

4 (

)

020 04 06 08 1 1250

0100

GV

5 (

)

020 04 06 08 1 1250

0100

GV

6 (

)

Time (s)






0 05 1 15 2 25 3 35 4049

05

051

052

053

054

055

056

Time (s)


PG

V(p

u)

(a)

Time (s)


0 05 1 15 2 25044

045

046

047

048

049

05

051

PG

V(p

u)

(b)









0 05 1 15 2 25 3

0 05 1 15 2 25 3

0 05 1 15 2 25 3

020406080

100

GV

1 (

)

020406080

100

GV

2 (

)

020406080

100

GV

3 (

)

Time (s)




The frontsegment

The middlesegment

119879119900 254 189 214119879119888 235 158 191







0 05 1 15 2 25 3 35

05051052053054055056

Time (s)



PG

V(p

u)

(a)



0 05 1 15 2 25 3 35 4 45 5

05051052053054055056

Time (s)

PG

V(p

u)

(b)


0 05 1 15 2 25 3 35

0

02

04

06

08

1

Time (s)


PG

V(p

u)

(a)


0 05 1 15 2 25 3 350

02

04

06

08

1

Time (s)

PG

V(p

u)

(b)










5 Conclusion





Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 30

102030405060708090

100

Time (s)

13

2PG

V(

)









119879119900 =(1205821119875max minus 119875min)

Δ1199051

(1)


119879119888 =(119875max minus 1205822119875max)

Δ1199052

(2)



1198791199001 =(119875max minus 119884119900)

Δ1199053

1198791198881 =(119884119888 minus 119875min)

Δ1199054

(3)











120582 =2


radic(1198881 + 1198882)2minus 4 (1198881 + 1198882)

1003816100381610038161003816100381610038161003816

(4)


Data preprocessing





The end




119909119896+1

119894119863= 119909119896

119894119863+ 120582 sdot V119896+1

119894119863 (5)


119909119899+1 = (119909119899 + 119910119899) mod 1

119910119899+1 = (119909119899 + 2119910119899) mod 1(6)



119891 =

(sum119873

119894=1(119910 (119894) minus 1199100 (119894))

2)

119873 (7)



The start










is satisfied


The end

The end

Yes

Yes

No

No

group extremes

and position

Calculate particle


(1) (2)

(3) (4)

fitness value

fitness value

position and speed










1198911



1198912



1198913



1198914



1198915



0 10 20 30 40 50 60 70 80 90 100

0

Iter

PSOIPSO

GAGA-PSO

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

log10

(err

)





020 04 06 08 1 1250

0100

GV

1 (

)

050

0100

GV

2 (

)

01 02 03 04 05 06 07 08 09 1

020 04 06 08 1 1250

0100

GV

3 (

)

020 04 06 08 1 1250

0100

GV

4 (

)

020 04 06 08 1 1250

0100

GV

5 (

)

020 04 06 08 1 1250

0100

GV

6 (

)

Time (s)






0 05 1 15 2 25 3 35 4049

05

051

052

053

054

055

056

Time (s)


PG

V(p

u)

(a)

Time (s)


0 05 1 15 2 25044

045

046

047

048

049

05

051

PG

V(p

u)

(b)









0 05 1 15 2 25 3

0 05 1 15 2 25 3

0 05 1 15 2 25 3

020406080

100

GV

1 (

)

020406080

100

GV

2 (

)

020406080

100

GV

3 (

)

Time (s)




The frontsegment

The middlesegment

119879119900 254 189 214119879119888 235 158 191







0 05 1 15 2 25 3 35

05051052053054055056

Time (s)



PG

V(p

u)

(a)



0 05 1 15 2 25 3 35 4 45 5

05051052053054055056

Time (s)

PG

V(p

u)

(b)


0 05 1 15 2 25 3 35

0

02

04

06

08

1

Time (s)


PG

V(p

u)

(a)


0 05 1 15 2 25 3 350

02

04

06

08

1

Time (s)

PG

V(p

u)

(b)










5 Conclusion





Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Data preprocessing





The end




119909119896+1

119894119863= 119909119896

119894119863+ 120582 sdot V119896+1

119894119863 (5)


119909119899+1 = (119909119899 + 119910119899) mod 1

119910119899+1 = (119909119899 + 2119910119899) mod 1(6)



119891 =

(sum119873

119894=1(119910 (119894) minus 1199100 (119894))

2)

119873 (7)



The start










is satisfied


The end

The end

Yes

Yes

No

No

group extremes

and position

Calculate particle


(1) (2)

(3) (4)

fitness value

fitness value

position and speed










1198911



1198912



1198913



1198914



1198915



0 10 20 30 40 50 60 70 80 90 100

0

Iter

PSOIPSO

GAGA-PSO

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

log10

(err

)





020 04 06 08 1 1250

0100

GV

1 (

)

050

0100

GV

2 (

)

01 02 03 04 05 06 07 08 09 1

020 04 06 08 1 1250

0100

GV

3 (

)

020 04 06 08 1 1250

0100

GV

4 (

)

020 04 06 08 1 1250

0100

GV

5 (

)

020 04 06 08 1 1250

0100

GV

6 (

)

Time (s)






0 05 1 15 2 25 3 35 4049

05

051

052

053

054

055

056

Time (s)


PG

V(p

u)

(a)

Time (s)


0 05 1 15 2 25044

045

046

047

048

049

05

051

PG

V(p

u)

(b)









0 05 1 15 2 25 3

0 05 1 15 2 25 3

0 05 1 15 2 25 3

020406080

100

GV

1 (

)

020406080

100

GV

2 (

)

020406080

100

GV

3 (

)

Time (s)




The frontsegment

The middlesegment

119879119900 254 189 214119879119888 235 158 191







0 05 1 15 2 25 3 35

05051052053054055056

Time (s)



PG

V(p

u)

(a)



0 05 1 15 2 25 3 35 4 45 5

05051052053054055056

Time (s)

PG

V(p

u)

(b)


0 05 1 15 2 25 3 35

0

02

04

06

08

1

Time (s)


PG

V(p

u)

(a)


0 05 1 15 2 25 3 350

02

04

06

08

1

Time (s)

PG

V(p

u)

(b)










5 Conclusion





Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198911



1198912



1198913



1198914



1198915



0 10 20 30 40 50 60 70 80 90 100

0

Iter

PSOIPSO

GAGA-PSO

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

log10

(err

)





020 04 06 08 1 1250

0100

GV

1 (

)

050

0100

GV

2 (

)

01 02 03 04 05 06 07 08 09 1

020 04 06 08 1 1250

0100

GV

3 (

)

020 04 06 08 1 1250

0100

GV

4 (

)

020 04 06 08 1 1250

0100

GV

5 (

)

020 04 06 08 1 1250

0100

GV

6 (

)

Time (s)






0 05 1 15 2 25 3 35 4049

05

051

052

053

054

055

056

Time (s)


PG

V(p

u)

(a)

Time (s)


0 05 1 15 2 25044

045

046

047

048

049

05

051

PG

V(p

u)

(b)









0 05 1 15 2 25 3

0 05 1 15 2 25 3

0 05 1 15 2 25 3

020406080

100

GV

1 (

)

020406080

100

GV

2 (

)

020406080

100

GV

3 (

)

Time (s)




The frontsegment

The middlesegment

119879119900 254 189 214119879119888 235 158 191







0 05 1 15 2 25 3 35

05051052053054055056

Time (s)



PG

V(p

u)

(a)



0 05 1 15 2 25 3 35 4 45 5

05051052053054055056

Time (s)

PG

V(p

u)

(b)


0 05 1 15 2 25 3 35

0

02

04

06

08

1

Time (s)


PG

V(p

u)

(a)


0 05 1 15 2 25 3 350

02

04

06

08

1

Time (s)

PG

V(p

u)

(b)










5 Conclusion





Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 3 35 4049

05

051

052

053

054

055

056

Time (s)


PG

V(p

u)

(a)

Time (s)


0 05 1 15 2 25044

045

046

047

048

049

05

051

PG

V(p

u)

(b)









0 05 1 15 2 25 3

0 05 1 15 2 25 3

0 05 1 15 2 25 3

020406080

100

GV

1 (

)

020406080

100

GV

2 (

)

020406080

100

GV

3 (

)

Time (s)




The frontsegment

The middlesegment

119879119900 254 189 214119879119888 235 158 191







0 05 1 15 2 25 3 35

05051052053054055056

Time (s)



PG

V(p

u)

(a)



0 05 1 15 2 25 3 35 4 45 5

05051052053054055056

Time (s)

PG

V(p

u)

(b)


0 05 1 15 2 25 3 35

0

02

04

06

08

1

Time (s)


PG

V(p

u)

(a)


0 05 1 15 2 25 3 350

02

04

06

08

1

Time (s)

PG

V(p

u)

(b)










5 Conclusion





Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 05 1 15 2 25 3 35

05051052053054055056

Time (s)



PG

V(p

u)

(a)



0 05 1 15 2 25 3 35 4 45 5

05051052053054055056

Time (s)

PG

V(p

u)

(b)


0 05 1 15 2 25 3 35

0

02

04

06

08

1

Time (s)


PG

V(p

u)

(a)


0 05 1 15 2 25 3 350

02

04

06

08

1

Time (s)

PG

V(p

u)

(b)










5 Conclusion





Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusion





Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article piecewise model and parameter obtainment

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