Solution of Power Flow With Automatic Load Frequency Control Devices Including Wind Farms

2186 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012

Solution of Power Flow With Auto-matic Load-Frequency Control Devices

Including Wind FarmsLuis M. Castro, Claudio R. Fuerte-Esquivel, Senior Member, IEEE, and J. Horacio Tovar-Hernández

Abstract—This paper proposes the integration of steady-statemodels of several types of wind generators into a power flow al-gorithm with automatic load-frequency control. Since the systemfrequency deviation is considered a state variable to be computedby the power flow solution, this formulation helps identify the op-erating point of wind generators after the action of the primary fre-quency control when power imbalances have occurred. The math-ematical formulation of fixed-speed wind generators is presentedbased on the steady-state representation of the induction gener-ator. Furthermore, as variable-speedwind generators keep gainingprominence in power systems, their potential contribution to fre-quency support is also analyzed herein. These models are formu-lated within the power flow approach by using a unified singleframe of reference and the Newton-Raphson algorithm. The pro-posed approach is then applied to the analysis of a three-machine,eight-bus system and the IEEE-14 bus test system.

Index Terms—Newton-Raphson algorithm, power flow analysis,primary frequency regulation, wind generator.

I. INTRODUCTION

W IND power generation is growing worldwide becauseof economic and environmental benefits. However,

from the technical standpoint, several challenges arise fromthe wind power penetration into electric power systems. In thiscontext, the service credibility is one of the primary goals toachieve even though wind power generation is stochastic innature.The system reliability can be degraded by power imbalances

between load and generation, which are inherent to the dailyoperation of power systems. In this context, the impact of apower imbalance to the system frequency can be quickly deter-mined by a power flow algorithm in which automatic load-fre-quency controllers are included [1]. This approach considersthe frequency deviation as a state variable, which is simultane-ously computed together with all magnitudes and phase anglesof nodal voltages in a unified frame of reference. Furthermore,

Manuscript received October 18, 2011; revised February 03, 2012 and April02, 2012; accepted April 04, 2012. Date of publication May 09, 2012; date ofcurrent version October 17, 2012. This work was supported by CONACYTMexico under the scholarship 209744 and under the research project 106198.The work of L. M. Castro was supported by the Consejo Nacional deCiencia y Tecnología (CONACYT) México, and the University of Michoacán(U.M.S.N.H). Paper no. TPWRS-00979-2011.L. M. Castro and C. R. Fuerte-Esquivel are with the Electrical Engineering

Faculty, Universidad Michoacana de San Nicolás de Hidalgo (UMSNH),Morelia, Michoacán, 58000, México (e-mail: [email protected];[email protected]).J. H. Tovar-Hernández is with the Instituto Tecnológico de Morelia, Morelia,

México (e-mail:[email protected]).Digital Object Identifier 10.1109/TPWRS.2012.2195231

the voltage and frequency dependence of loads can be taken intoaccount to represent their variation according to different oper-ating conditions. This methodology is rather attractive becauseit permits the estimation of the steady-state operating point afterthe action of the primary frequency regulation.Bearing the above mentioned in mind, suitable wind gener-

ator models must be developed and included into a power flowprogram with automatic load-frequency control (PFALFC) toassess the role of wind farms in frequency regulation. To achievethis goal, the preliminary work presented in [2] is expanded byintroducing two additional wind generator models to accom-plish a complete modeling idea of wind generators consideringthe frequency deviation of the electric network.The mathematical modeling of wind turbines developed

in this paper relies primarily on the power injection concept.For this purpose, a suitable representation of fixed-speed windgenerators (FSWG) is developed, which includes their activeand reactive powers in terms of the induction machine statorand rotor parameters as well as the system frequency deviation[3]. Furthermore, variable-speed wind generators (VSWG)are mathematically modeled based on both doubly-fed induc-tion generators (DFIG) and permanent magnet synchronousgenerators (PMSG). These models are developed with theunderstanding that they are able to provide primary frequencyregulation by maintaining some spinning reserve [4]–[10]. ThePFALFC reported in [1] is then expanded to incorporate the pro-posed models. In this context, these models are implemented insuch a way that the electrical network state variables, includingfrequency deviation and those related to wind generators,are solved simultaneously during the iterative process. Thissolution method is selected with the aim of taking advantage ofthe quadratical convergence of the Newton-Raphson method.The proposed approach is presented in detail as follows:

Section II addresses the basics of the potential contribution ofwind generators to frequency support. Section III reviews thePFALFC. The mathematical models of several types of windgenerators are presented in Section IV, while the initializationof their state variables is given in Section V. Study cases areprovided in Section VI; finally, Section VII points out theconclusions of this paper.

II. CONTRIBUTION OF WIND GENERATORSTO FREQUENCY SUPPORT

The increasing integration of wind power into power sys-tems has provided the momentum for exploring alternatives toendow wind turbines with frequency regulation capability. Inthe case of the primary frequency regulation, the key aspect is

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CASTRO et al.: SOLUTION OF POWER FLOWWITH AUTOMATIC LOAD-FREQUENCY CONTROL DEVICES INCLUDING WIND FARMS 2187

Fig. 1. Achieving a spinning reserve: (a) by using the blade pitch mechanism,(b) by increasing the mechanical rotor angular speed.

to attain a spinning reserve of active power in order to com-pensate active power imbalances. This goal can be achieved byforcing the wind generator to operate in a suboptimal poweroutput level, which is referred to as “deloading.” Thus, each de-loaded wind turbine will not extract the maximum power fromthe wind during normal conditions in order to maintain a spin-ning reserve, which will permit a release of power when a powerimbalance perturbation occurs [4]–[10].A technique employed to deload a wind turbine calls for the

action of the blade pitch mechanism, as can be seen in Fig. 1(a),[5], [7], [8]. In this case, the pitch angle is increased from toor (depending on the required level of deloading), resultingin different output powers obtained at the same rotor angularspeed . Therefore, the pitch angle controller acts like aconventional speed governor.On the other hand, by shifting the operating point to the

right of the wind turbine power curve, as shown in Fig. 1(b), awind turbine can attain a spinning reserve of active power tobe delivered in case of a power imbalance [4], [9]. However,a rotor overspeeding is presented with this deloadingtechnique which limits its application beyond the rated windspeed because of mechanical stress [6]. Lastly, both discussedapproaches can be simultaneously applied as proposed in [10].Note that shifting the operating point towards the left of the

power curve, shown in Fig. 1(b), may also deload the windturbine. However, when requiring frequency response, a rotorspeed increase becomes necessary. In such a situation, a portionof the energy extracted from the wind will accelerate the rotorthus reducing the amount of power delivered to the network [4].

III. POWER FLOWS WITH AUTOMATICLOAD-FREQUENCY CONTROL METHOD

The power flow formulation with automatic load-frequencycontrol devices must include the system frequency deviation asa state variable, which is calculated together with all the net-work’s nodal voltages [1]. Therefore, when a load or generationperturbation is simulated, the operating point obtained by thePFALFC method will be at an off-nominal frequency value.In this context, all wind generator models developed herein

can be readily included in the power flow formulation as shownin (1), where and represent the power balanceequation of each wind generator and its associated state vari-able, respectively. Power mismatch equations are denoted by

and at each network node, and is the system fre-quency deviation from its nominal value:

(1)

Furthermore, this formulation can manage multiple gener-ator regulators that respond to frequency changes as well asvoltage and frequency dependent load models, as briefly de-scribed below. Note that all equations are given in the per-unit(p.u.) system, unless otherwise stated.

A. Synchronous Generator Models

Two synchronous generator models from the four presentedin [1] are addressed in this paper. The first model represents agenerator regulator that adjusts its active power output andreactive power output according to (2)–(6), which are func-tions of the frequency deviation value estimated at each powerflow iteration:

(2)

(3)

(4)

(5)

(6)

where and are the specified active and reactivepowers of the generator, respectively, is the nominal activepower, is the speed droop of the corresponding generator,

and are the coefficients of the reactive power genera-tion, and the subscripts and represent maximum andminimum values. The second model used in this work corre-sponds to the conventional PV generator, where its generatedactive power and its terminal voltage magnitude are set tospecified values during the simulation process. Constant voltageoperation is only possible if the generator reactive power de-sign limits are not violated, i.e., . If thegenerator cannot provide the necessary reactive power supportto constrain the voltage magnitude at the specified value thenthe reactive power is fixed at the violated limit, and the voltagemagnitude is freed. In this case, the generated active powerand reactive power are specified, while nodal voltage mag-nitude and phase angle are computed.For the purpose of this research, only two generator models

are used, since this paper focuses on wind generator modelswithin the context of the PFALFC. Please, refer to [1] for detailsof the rest of existing synchronous generator models.

B. Load Model

Most of the loads in power systems are voltage and fre-quency dependent, and their demanded power varies accordingto changes in these variables. The static representation of theiractive power and reactive power is described by thefollowing [1]:


Fig. 2. Steady-state equivalent model of the induction machine taking into ac-count the system frequency deviation.

Fig. 3. Response of directly grid-connected wind generators to frequency vari-ations in the network.

(7)

(8)

where and are the specified active and reactivepowers drawn by the load, respectively, is the nominal op-erating voltage of the node where the load is connected,and are the coefficients of the load-frequency characteristic,whereas , , , , and are the coefficients of theload-voltage characteristic.

IV. MATHEMATICAL MODELING OF WIND GENERATORSAND THEIR INCLUSION IN THE PFALFC

The steady-state representation of the induction generatorshown in Fig. 2 is a function of its slip given by (9). Thisimplies that active and reactive powers generated by a directlygrid-connected wind generator will vary according to changesin the system frequency and in the mechanical angular speed ofthe rotor , as shown in Fig. 3

(9)

Based on the above mentioned, the stall regulated and pitchregulated fixed-speed wind generator models can be suitablyderived as a function of the frequency deviation by using thepower injection concept. The mechanical angular speed of thegenerator rotor is employed as the wind generator state variable

within the iterative process. Hence, the power converted fromthe mechanical to electrical form , the generated powersand , the rotor current and the stator current are

expressed as follows:

(10)

(11)

(12)

(13)

(14)

where the coefficients are defined as,

, ,, ,

, , ,, ,

, ,, ,

.

A. Stall Regulated FSWG

The extracted power from the wind of a stall regulated FSWG(SR-FSWG) is calculated by using (15)–(17). Clearly, it is afunction of the rotor angular speed among other variables [11]:

(15)

(16)

(17)

where is the mechanical power (W), is the air density( ), is the swept area of the blades ( ), is the windspeed (m/s), is the radius of the rotor (m), is the gearboxratio, is the pitch angle (degrees), is the mechanical an-gular speed of the generator (rad/s) and the constants toare the parameters of the wind turbine design. In this case, theonly unknown variable is the mechanical angular speed of thegenerator, as seen from (15). Consequently, this parameter actsas the state variable within the iterative process which enablesto achieve the internal equilibrium point in the SR-FSWG given


by the conversion process of mechanical power into electricalpower.In the framework of the PFALFC method, the set of

power flow mismatch equations that has to be solved when aSR-FSWG is connected at the node of the system is

(18)

(19)

(20)

where and represent the active and reactive powersdrawn by the load at bus , respectively, and andrepresent the powers injected at bus . They are given by

(21)

(22)

Lastly, the set of linearized equations (23) is integrated intothe set of linearized power mismatch equations of the whole net-work (1) to compute a new equilibrium point through a powerflow solution. In this case, the generator angular speed is up-dated after the th iteration as given by (24):

(23)

(24)

B. Pitch Regulated FSWG

A pitch regulated FSWG (PR-FSWG) has practically thesame mechanical construction as the SR-FSWG; however, thePR-FSWG has a blade pitch angle mechanism which operatesfor wind speeds beyond a rated value to limit the power ex-tracted from the wind [12]. A typical pitch angle controller isshown in Fig. 4, which takes the power output of the windgenerator as the input signal and regulates the pitch anglethrough a PID controller.For an operating condition where the generated active power

is below the specified value , the mathematical equationsrepresenting the SR-FSWG model are also valid to representthe steady-state operation of a PR-FSWG. On the other hand,

Fig. 4. Typical blade pitch angle controller.

when the output power surpasses , the blade pitch anglecontroller is activated. In this case, the controller sets at theconstant value , whereas the reactive power has to becalculated within the iterative process. Thus, by neglecting thecore losses in the induction machine, the mechanical power canbe computed by

(25)where and are the three-phase stator and rotorpower losses, respectively.Assuming that the PR-FSWG is connected at node , the

values of the network state variables and the wind generatorthat satisfy the mismatch (26)–(28) are obtained by solving (29).This last equation has to be accommodated with the rest of thelinearized power mismatch equations of the network for a uni-fied power flow solution:

(26)

(27)

(28)

(29)

where the state variable is updated after each iteration ac-cording to (24), and represents the power balanceinside the induction machine obtained by equating (10) and(25). In summary, the next conditions must be considered whena PR-FSWG is included into the PFALFC program: if

, then (15)–(24) must be used. If , then (21),(22), and (24)–(29) must be used instead, setting .From the software design perspective, the simulation starts

by employing (15)–(24). The active power generated by thePR-FSWG is checked at the end of the second iteration becausethe first iteration generally provides an inaccurate approxima-tion to the power flow solution. If happens to be equal or


Fig. 5. Automatic load-frequency control of a DFIG.

greater than , the next iteration is executed using (21),(22), and (24)–(29). Note that even when a high wind speed isselected and was meant to surpass its maximum limit, thedrop in the system frequency can force to be less than .In such a case the (15)–(24) will continue being used during theiterative process.

C. DFIG With Primary Frequency Regulation

The deloading of the wind turbine allows a DFIG to providefrequency support. Its response to frequency deviations will bedependent on its speed droop characteristic and its power gen-eration (see Fig. 5) [7], which is calculated according tothe level of deloading of its wind turbine, .Hence, the active power generated by the DFIG is computed

by using (30)–(31), where is the initial active power ob-tained from the power curve of the wind generator for any givenwind speed:

(30)

(31)

One of the most attractive features of this kind of wind gen-erator is its control capability of reactive power. The controlstrategy adopted to operate a DFIG defines if it will be absorbingor injecting reactive power. Therefore, depending on the reac-tive power control mode, a different set of power flow equationsmust be accommodated with the rest of the equations.1) Fixed Power Factor Control Mode: Given that the power

factor is specified, the reactive power generated by the windgenerator is computed according to ,where stands for the power factor angle. However, careshould be taken when choosing the value of the power factor soas not to exceed the reactive power limits, and ,imposed by the power electronic converter. Therefore, for aDFIG connected at node , the power flow mismatch equationsthat must be solved simultaneously with the ones correspondingto the rest of the network are

(32)

(33)

(34)

2) Constant Voltage Magnitude Control Mode: In this con-trol strategy, the voltage magnitude at the generator’s termi-nals is specified, and its active power output is obtained from(30)–(31). Therefore, the PFALFC formulation for a DFIG con-trolling the voltage magnitude at its terminal is given by

(35)

Fig. 6. PMSG-based wind generator model for power flow studies.

(36)

Even though the reactive power mismatch equation of thevoltage magnitude controlled node is not considered in the for-mulation, it is solved at each iterative step to assess whether ornot the generator reactive power is within limits. If a limit vio-lation occurs then the generated reactive power is fixed at thatlimit, and the voltage magnitude is freed. In this case, this modelis converted into the DFIG model operating with a fixed powerfactor.Note that the impact of the system’s frequency on the stator

electrical variables is considered to be negligible for bothDFIG models. This is a reasonable assumption if we regard theelectrical and mechanical rotor frequencies are decoupled: thevoltage source converter connected to the slip-rings of the rotorinjects a current with a variable frequency to compensate thedifference between the mechanical and electrical frequencies[11].

D. PMSG-Based Wind Generator With Primary FrequencyRegulation

Presently there is great interest in assessing how these windgenerators can participate in the primary frequency control sincethey have been gaining prominence in power systems [13]. Un-like a DFIG, the PMSG-based wind generator supplies all itsgenerated active power through its full-scale converter. Basedon the electric circuit shown in Fig. 6, a suitable model forpower flow studies can be derived by explicitly including therepresentation of the wind generator step-up transformer, where

stands for the output power obtained from the windgenerator power curve for a given wind speed, andare the voltages at the machine-side converter and grid-side con-verter terminal, respectively, and is the step-up transformerimpedance.Note that this model will also allow for direct voltage magni-

tude control at the transformer’s high-voltage side [14] for thecases where this reactive power control mode is selected.Assuming that the PMSG-basedwind generator has a primary

frequency controller represented by Fig. 5, its generated activepower will be determined by

(37)

(38)


On the other hand, assuming that , theactive and reactive powers flowing from the grid-side converterterminal to the th bus are

(39)

(40)

For the active and reactive powers flowing from bus to thegrid-side converter terminal, the subscripts and are ex-changed in (39) and (40).Regarding the adopted reactive power control strategy, a dif-

ferent set of power flow equations must be integrated with therest of the network equations as shown next.1) Fixed Power Factor Control Mode: For a given power

factor angle , the reactive power generated by this type of windgenerator is calculated as . Hence, havingdefined both generated powers, the set of power flow mismatchequations that must be solved when the generator is connectedat node is given by (41)–(45) shown at the bottom of the page.2) Constant Voltage Magnitude Control Mode: Under this

control strategy, the nodal voltage magnitude at the trans-former’s high-voltage side is maintained at a constant value byadjusting the generator’s reactive power. In this case, the initialoutput of active power is computed using (37) and (38). Thus,the NR-based power flow formulation for this control strategyis given by

(46)

(47)

(48)

(49)

Lastly, if the generator cannot provide the reactive powersupport to constrain its terminal voltage magnitude at a spec-ified value, the reactive power is fixed at the offending limit. Inthis case, the control strategy is changed to operate with a fixedpower factor.

V. INITIALIZATION OF WIND GENERATORS

When solving power flows by using Newton’s method, ade-quate initial conditions must be specified in order to achieve aniterative solution with quadratic convergence. In this context,the proposals to initialize the wind generator state variables arestated below.A good initial value of to execute power flow studies

with a network having SR-FSWGs and PR-FSWGs is given by, since the operating point of these generators is

normally found close to the nominal mechanical angular speed.The adopted strategy control must be considered first when

initializing VSWGs. If a constant power factor is specified, thenthe reactive power is computed using the initial active power ob-tained from the power curve and is kept constant during the iter-ative process. On the other hand, if a constant voltagemagnitudeis selected, the output reactive power is calculated within the it-erative process as stated in Section IV-CII. Lastly, the voltagemagnitudes are initialized at 1 p.u. at all uncontrolled voltagemagnitude nodes, while the controlled PV nodes are initializedat specified values that remain constant throughout the iterativesolution if no generator reactive power limits are violated. Theinitial voltage phase angles are selected to be 0 at all buses.

VI. CASE STUDIES

The suitability of the proposed approach to conduct a powerflow analysis with automatic load-frequency control is testedon a three-machine, eight-bus system and the IEEE 14-bus testsystem. Data for both systems are given in the Appendix A. For

(41)

(42)

(43)

(44)

(45)


Fig. 7. Power system used to accommodate a PR-FSWG.

the former case, only the PR-FSWG model is examined giventhat the analysis of the SR-FSWG was addressed in [2]. Thedesign of the case studies is given below.

A. Example I

As stated in Section IV-B, the blade pitch angle regulator actswhen the generated active power rises above . In this con-text, because the power output of the FSWG is frequency de-pendent as given in (12), the blade angle regulator can be ac-tivated or deactivated depending on the frequency deviationsin the network, and not only because of wind speeds beyond aspecified rated value. To validate this statement, this exampleconcerns the performance of the PR-FSWG model when usingthe PFALFC algorithm. The eight-bus test system, comprisingtwo synchronous generators (SG) and one PR-FSWG as shownin Fig. 7, is used to execute this study. The SG connected at node1 is considered the reference generator with automatic load-fre-quency control capability while the one connected at node 3 actsas a conventional PV generator controlling its terminal voltageat 1 p.u.The disturbances that deviate the frequency from its

nominal value are simulated by altering the active and reac-tive power demanded by all the system’s loads assuming aconstant power factor as shown in Table I. The base case corre-sponds to the power flow solution obtained by considering that

. For all simulations whose power flow solutions arereported in Table I for a power mismatch tolerance of ,the rated values of the active power and the wind speed of thewind generator are 2 MW and 15 m/s, respectively.Three key aspects can be drawn from the results presented

in Table I. Firstly, when simulating the PR-FSWG with a windspeed of 14 m/s, the generated active and reactive powers aredifferent for each value of . This is primarily because ofthe frequency deviations since they force the wind generator tofind another equilibrium point. Secondly, the blade pitch anglemechanism is activated when simulating a atrated wind speed. In this case, the generated active power hits

because of the increment in the system’s frequency.Lastly, when simulating the scenario where the wind speed isabove the rated one, i.e., 16 m/s, a deactivation of the pitchangle mechanism occurs because of the frequency drop causedby the increase of 10% in the total system load.The above conclusions can be physically inferred: a decrease

in the mechanical speed of the induction machine will lead to adecrement in the output power of the wind generator and viceversa.

TABLE IELECTRICAL RESPONSE OF A PR-FSWG FOR DIFFERENT VALUES

Fig. 8. Modified IEEE 14-bus test system.

The number of iterations required to obtain the solution ineach case is reported in Table I. Note that the proposed ap-proach arrives at the solution with local quadratic convergenceexcept when the pitch angle mechanism is activated as describedabove; in this case, one or two additional iterations are neededto obtain the power flow solution.

B. Example II

In order to examine the joint operation of the wind gener-ator models proposed in the previous section, the IEEE 14-bustest system [15] is modified according to Fig. 8 to accommo-date four wind farms. Wind farms I and II are composed offive SR-FSWGs and five PR-FSWGs, respectively, whereas theremaining are VSWG-based wind farms comprised of fifteenDFIGs and fifteen PMSGs, respectively. Generators connectedat buses 1 and 2 are assumed to possess automatic load-fre-quency control capability. Lastly, the generator 1 is consideredas the reference generator.In this example, the active and reactive powers generated by

each wind farm as well as the system frequency are reported fortwo different load perturbations in the system, that is,

and . In addition, two scenarios were simulatedto demonstrate the power regulation capability of the VSWG-based wind farms by assuming a deloading of 5% and 10% for


TABLE IIPOWER GENERATION, VOLTAGES OF WIND FARMS AND SYSTEM FREQUENCY FOR DIFFERENT VALUES OF AND

each wind generator composing these wind farms. The powerflow results of these simulations are given in Table II.For the case of the FSWG-based wind farms, simulations

show that their active and reactive powers are different fromone scenario to another because of their frequency and voltagedependency. Additionally, the PMSG-based wind farm controlsits terminal voltage at 1 p.u., as specified in all simulations, byinjecting into the grid the required reactive power.The results presented in Table II also reveal that a load in-

crease of 15% leads to a system frequency beyond 49.5 Hz whenthe VSWG-based wind farms are not providing frequency sup-port. However, when they are participating in the frequency con-trol, all wind turbines use all their spinning reserve regardless ofthe level of deloading, resulting in a system frequency of 49.526Hz and 49.563 Hz for and 10%, respectively. In thiscase, the wind power penetration accounts for 30% of the totalsystem load. The VSWG-based wind farms’ contribution to thefrequency regulation corresponds to 3 MW and 6 MW of activepower that was used as spinning reserve for and 10%,respectively.If frequency response is provided by wind farms then the

power system is clearly strengthened and acquires more flex-ibility when facing abnormal conditions. The importance of thedeloading of wind generators lies in the fact that the system willcount on other distributed slack generators which will help tomitigate active power imbalances.

VII. CONCLUSIONS

This paper has proposed the mathematical modeling ofseveral types of wind generators taking into account theirdependence with respect to system frequency deviations. Thesemodels were implemented in a Newton-based power flowalgorithm with frequency control devices to estimate theirelectrical response after the action of the primary frequencyregulation. Through numerical simulations, the PFALFC algo-rithm developed is shown to maintain its quadratic convergencecharacteristic, and the solutions obtained illustrate the contri-butions to the frequency support of VSWG-based wind farms.

APPENDIX A

Parameters of the SR-FSWG: Generator data [16]:, , , ,

; 2 MW, 690 V, 50 Hz, 2 pole pairs. Fixed capac-itor of 0.6 MVAr. Wind turbine data [16]: , ,

; , , , ,, , , , , .

Parameters of the PR-FSWG: Generator data [16]: thesame as the SR-FSWG. Fixed capacitor of 0.6 MVAr. Windturbine data [17]: , , ;

, , , , ,, , , .

Parameters of the Test System Used in Example I on aBase Power of 100 MVA: TL1, TL2, and TL3:

, . TL4: .TR1 and TR2: . TR3: . TR4:

. PL1: . PL2: . Thecoefficients for reactive power generation of the generator reg-ulator are and ; , .

Modified Parameters of the IEEE 14-Bus Test System Usedin Example II on a Base Power of 100 MVA: Step-up trans-former impedance for each wind farm: .The transformer impedance for each wind generator is

. The governor droop characteristic for eachDFIG and PMSG is 0.05 p.u. The DFIG and PMSG powercurve is shown in Fig. 9 [18]. The coefficients for reactivepower generation of the generator regulators are (see alsoTable III) and . The loads connected at nodes10, 11 and 12 are dependent on frequency and voltage withthe following parameters: , , ,

, , , and , whereasthe rest are constant.

APPENDIX B

The relevant Jacobian elements which are created and accom-modated with the original algorithm are the following:

; ; ;; ; ;

; ; ; ;


Fig. 9. DFIG and PMSG power curve.

TABLE IIIGENERATOR CHARACTERISTICS ACCORDING TO THE DELOADING

OF THE VSWG-BASED WIND FARMS

; ;; ;

; ;; ;

; ; ;;; .

Stall Regulated FSWG: See equation (B1)–(B11) at thebottom of the page.

Pitch Regulated FSWG: See equation (B12)–(B14) at thebottom of the next page.

DFIG:

(B15)

PMSG-Based Wind Generator:

(B16)

(B17)

(B18)

(B19)

(B1)

(B2)

(B3)

(B4)

(B5)

(B6)

(B7)

(B8)

(B9)

(B10)

(B11)


(B20)

(B21)

(B22)

(B23)

(B24)

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[14] L. M. Castro, C. R. Fuerte-Esquivel, E. Barrios, and C. Angeles-Ca-macho, “An integrated power flow solution of flexible AC transmissionsystems containing wind energy conversion systems,” in Proc. 2011IET Renewable Power Generation Conf., pp. 1–6.

[15] Power Systems Test Case Archive, Univ. Washington. [Online]. Avail-able: http://www.ee.washington.edu/research/pstca/,

[16] V. Akhmatov, “Analysis of dynamic behaviour of electric power sys-tems with large amount of wind power,” Ph.D. dissertation, TechnicalUniv. Denmark, Electric Power Engineering, Orsted-DTU, Lyngby,2003.

[17] J. G. Slootweg, H. Polinder, and W. L. Kling, “Representing wind tur-bine electrical generating systems in fundamental frequency simula-tions,” IEEE Trans. Energy Convers., vol. 18, no. 4, pp. 516–524, Dec.2003.

[18] S. Li and T. A. Haskew, “Energy capture conversion and control studyof DFIG wind turbine under Weibull wind distribution,” in Proc. 2009IEEE Power Eng. Soc. General Meeting, pp. 1–9.

Luis M. Castro received the B.Eng. degree from the Instituto Tecnológicode Morelia, Morelia, México, in 2006 and the M.S. degree from the InstitutoTecnológico de Morelia, Morelia, México, in 2008. He is currently pursuingthe Ph.D. degree at the Universidad Michoacana de San Nicolás de Hidalgo(UMSNH) in the area of wind power integration in power systems.

Claudio R. Fuerte-Esquivel (SM’07) received the B.Eng. (Hons.) degree fromthe Instituto Tecnológico deMorelia, Morelia, México, in 1990, theM.S. degree(summa cum laude) from the Instituto Politécnico Nacional, México, in 1993,and the Ph.D. degree from the University of Glasgow, Glasgow, U.K., in 1997.Currently, he is a Professor at the Universidad Michoacana de San Nicolás

de Hidalgo (UMSNH), Morelia, where his research interests lie in the dynamicand steady-state analysis of FACTS.

J. Horacio Tovar-Hernández received the B.Eng. degree from the InstitutoTecnológico de Morelia, Morelia, México, in 1984 and the M.S. and Ph.D. de-grees from the Instituto Politécnico Nacional, México, in 1989 and 1995, re-spectively.Currently, he is a Professor at the Instituto Tecnológico de Morelia, where his

research interests lie in power systems and electricity markets.

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Solution of Power Flow With Automatic Load Frequency Control Devices Including Wind Farms