EEEl Transactions on Energy Conversion, Vol. 10, No. 3, September 1995 577

A GENERAL SIMULATION ALGORITHM FOR THE ACCURATE ASSESSMENT OF ISOLATED DIESEL - WIND TURBINES SYSTEMS INTERACTION.

PART I: A GENERAL MULTIMACHINE POWER SYSTEM MODEL.

G. S . Stavrakakis G. N. Kariniotakis

Technical University of Crete Dept. of Electronic and Computer Engineering

73133 Chania, Crete, Greece.

Abstract - In the first part of this two-part paper, detailed dynamic equations for the power system and Wind Energy Conversion System (WECS) components and their synthesis to a unified model are presented. This model is the bask for creating a simulation software able to perform transient stability analysis of isolated Diesel-wind turbines power systems for accurate assessment of their interaction. Approximations In the various component models, when necessary, remain between limits that do not affect the accuracy of the analysis performed. A new general multimachine power system model is also developed which describes the topology and the complexity of Wind- Diesel systems in a compact form easy to implement in the simulation software.

b v w o r d s : Autonomous power systems, Diesel - wind turbines systems, transient stability analysis, multimachine power system modelling.

1. I"

In recent years there has been a growing interest in intermittent generation such as Wind Energy Conversion Systems interconnected with utility systems due to environmental, political and institutional reasons [l-31. Such systems are usually interfaced with the existing power grid for "fuel displacement" purposes as well as for earning some "capacity credit". Intermittent generation is quite different from conventional power plants. In the case of autonomous operation of WECS, some form of energy storage is required (pumped storage, hydrogen production, battery storage, etc.), thus reducing the economic attractiveness of the overall system.

The introduction of a relatively small amount of wind derived electrical power into the utility grid does not normally present any interfacing or operational problems. The situation is completely different when substantial amount of power is penetrating a conventional utility system. Penetration related problems are particularly acute when considering the installation and parallel operation of WECS with a small or medium autonomous power system.

95 WM 092-7 EC A paper recommended and approved by t h e IEEE Energy Development and Power Generation Committee of the IEEE Power Engineering Society f o r presenta t ion a t the 1995 IEEE/PES Winter Meeting, January 29, t o February 2, 1995, New York, Np. Manuscript submitted December 21, 1992; made a v a i l - ab l e f o r p r i n t i n g November 14, 1994.

In an autonomous grid supplied by Diesel - wind turbines system the voltage and frequency fluctuations will be considerably greater than in the main public grid. The assessment of the effects of the wind power penetration to an existing isolated power system necessitates the calculation of the disturbances (voltage and frequency deviations as well as instabilities) that may be provoked by the wind turbines (WT) under several abnormal operating conditions e.g. start-up or sudden disconnection of the wind farm or part of the wind farm, wind gusts, short circuits on the network e.t.c. The study of the previously mentioned abnormal system conditions enables power system stability studies.

that the energy delivered by a wind turbine is at the range of frequency 10-O"Hz to

Hz. This is the frequency range of the common transient stability power system studies.

In transient stability power system studies the concern is the stability of the power system following a disturbance which generally causes large excursions in many system variables [7-91. Hence the nonlinearities of the system equations must be included in the transient stability representation. Thus, advanced component modelling must be used. Such a system model because of the detailed component considerations consists a very realistic dynamic simulation model for isolated Diesel - wind turbines systems. Power system dynamic stability is out of the scope of the present paper.

Models of the Diesel engine, Diesel engine governor, synchronous generator, automatic voltage regulation (AVR), wind turbine (WT), induction generator, compensating capacitor banks (CB), transmission network and the load are presented and

Very detailed component models as Park equations [4] for induction and synchronous generators have been avoided, because these models do not add significant precision in the transient analysis power system studies.

It is a common practice for transient power system analysis to use simplified simulation models in which the stator electric transients of synchronous and induction machines are neglected [4, 6, 241. Moreover, the time constants associated with the transmission network transients are very small and can be considered to be zero without significant loss of accuracy [2, 6, 8, 19, 251.

A new general multimachine power system model is developed which incorporates in a compact form the transmission network and the machines stator equations by taking into account in a simple way the machine topology in the network. This approach offers a fast numerical solution of the numerous Wind-Diesel power system algebraic equations.

The paper finally presents the guidelines for incorporating the various components models in an integrated software able to perform transient stability studies for isolated Diesel - wind turbines interaction assessment.

The results presented in [ l ] have shown

analyzed.

0885-8969/95/$04.00 0 1995 IEEE

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2 . 2 G OF THE ES SUS=

In order to develop a simulation algorithm able to perform transient stability analysis of an autonomous Wind/Jliesel power system, appropriate models for all the power system and the WECS components are given in this Section. The various components models presented here are general and able to give accurate results for a wide range of the Windpiesel system perturbations.

2.1

The ability of a wind turbine to extract power from varying winds is a function of three main factors : the wind power available, the power curve of the machine, and the ability of the machine to respond to wind fluctuations. The mechanical power that is produced by a wind turbine is given by [ 111 :

Ph = 0.5 pCdU.! (watts) (1)

where the various notations are provided in a Nomenclature at the end of this paper.

In [12, 131 it is demonstrated that the power coefficient of any wind turbine can be given analytically either by a third degree polynomial or by other more complex expressions l i e the one presented below :

- 0.57 (2) 2 c -16 ' x

P-- 27 + 1.32 + [(X-8)/20J2 L(x+J-)

D 2B B where k is the tip speed ratio defied as: k = m.

U W In the above expression, B is the number of blades of the wind

turbine and LID is the lift to drag ratio. B and LID are known from the technical characteristics of the WT. Equation (2) is valid for 41h120, for B=1, 2 or 3. and LID225 with a high degree of accuracy (e.g. Cp within k 0.005) [13].

The case of a variable-speed WT with a pitch mechanism, that alters the effective rotor aerodynamic efficiency, can be easily considered if an appropriate expression for C,. as a function of the pitch angle, is applied. Such expressions, as well as the equations governing the control action, can be found in [ l l ] .

2.2 The mechanical Dart of the wind turbine mode]

The shaft dynamics of the wind turbine are characterized in terms of the wind turbine blade speed (aB), the hub speed (aH). the gear box speed (ac) and the generator mechanical speed (a,,,). The equations describing the motion of the several shaft sections are the following [ l l ] :

(3)

The mechanical (a,,,) and the electrical (aA) angular velocities of the generator are related by aA=(p/2)w,,,, while the angular velocities at the two sides of the gear-box by a,,,=R,oc, where P is the number of poles and R , is the transmission ratio. Angular

velocities are related to shaft angles by

phase voltampere rating.

= Am : i =B, H , G. dt

Equations (3) can be normalized, based on the generator three

The above model is convenient for the case in which the

machine behavior will be studied during high turbulent wind, wind gusts, etc. It is a common practice however to neglect the dynamics of the mechanical part as considerably slow compared to the fast electrical ones, especially for machines with great inertia or when the necessary parameters are not available. Then the system of equations (3) can be reduced to one differential equation by neglecting the hub dynamics and considering the shaft as inflexible (@ = 0, =2 OA) :

P

(4)

In equation (4). called the swing equation, angular velocities are in rads=. while the other quantities are given in the per-unit (P.u.) system where the machine kVA rating is chosen as basis. The well-known per-unit transformation is extensively developed in [4. 19, 231.

2.3. bsvnchronous generator model

It is considered that the WTs are equipped with asynchronous generators. The asynchronous machine equations are derived from Park equations after some simplifications [4, 8, 91. The most important is that the stator electrical transients are neglected as much faster compared to the rotor ones. These equations are given here in the per-unit (P.u.) system with respect to the synchronous reference frame.

Wind generators are considered as asynchronous motors in generating operation (negative loads). The operation of the asynchronous machine will be determined from the sign of the electromagnetic torque and the slip, that is negative torque and slip correspond to generator operation while positive torque and slip to motor operation [41.

Thus, the algebraic stator equations are :

v d = EA + rJd - xi,, v, = E; + r,Iq + x ' I d (5 )

The differential equations describing the dynamics of the rotor windings are :

a; - 1 [-E& - (X-X')Iq + sa;] dt T i

The electromagnetic torque equation is :

TAc= Ed Id + E; Iq (7) Output active and reactive power, voltage and current are given

by the following relations :

In the case of a wind park consisting of many units, the number of differential equations becomes very high. Then, in order to reduce the total number of equations it can be assumed that a number of wind generators operate under the same wind conditions. The wind park can be partitioned into groups of n- machines which operate identically and each group can then be replaced by an equivalent machine. In the same group should belong identical machines. The parameters of the equivalent machine replacing n-machines identically operating are then given by the following relations :

HAn=nHA, T L = T i , r,='., & = x , n xi=L n (9) n The case of WTs operating in the variable-speed mode by

employing power electronics to control the electrical link to the grid, can be easily considered given the modularity of the

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proposed simulation algorithm. The necessary additional modelling considerations can be found in [3].

2.4 Diesel enpine modelling

The Diesel engine model gives a description of the fuel consumption rate as a function of speed and mechanical power at the output of the engine.

The Diesel engine is usually modeled by a simple first order model relating the fuel consumption (fuel rack position) to the engine mechanical power. Notice that the transfer function of a reciprocating engine involves a small but significant time delay associated with the mean time between firing [ 141.

The efficiency of the combustion E is the ratio of the effective horsepower developed by the engine and available on its crankshaft to the heat consumed during the same time, i.e., [I61 :

Incomplete combustion is the main reason for which the indicated efficiency is lower than the ideal efficiency. The mean effective pressure p i of the engine is defined as [ 161 :

(1 1)

By solving (IO) with respect to W i and substituting into (11)

p i = w, v h

we get :

where C, is the appropriate proportionality constant. Note that for normal or stable power system operation v is almost constant and its value is imposed in order to keep the system frequency constant at 50 Hz.

The mean pressure of mechanical losses is taken in a first approximation proportional to the mean piston speed U,, i.e., p f I U , [I61 , where U,=2Sfm since the piston travels a distance of twice the stroke per revolution. Thus, p f can be generally written as p f =C,o (w=rCPf,) valid for any engine with appropriate constant C,.

The real mean effective pressure p k of the engine must be :

p k = p i - p f (13)

The real mechanical power PDm of the Diesel engine is given by the equation :

PDni = zvhvpk = VHVpk = VH cr)n PI. (14)

The mechanical torque Tomof the engine is then given by the nK

following relation in the p.u. system :

The combustion efficiency E of the engine depends on the combustion quality as it has been mentioned before. A fuel- combustion model of the Diesel engine involving characterizing wave and flame propagation is needed for deeper study. Being out of the scope of the present paper, combustion efficiency is represented as a function of the air-fuel ratio m&, as following :

E = f (2) Analytic expressions off(.) for a specific engine can be found

in [16]. At normal operation of the engine we consider E to be a constant in the simulations.

The block diagram of the Diesel engine derived using the above considerations is presented in Fig. 1. Typical values for the

F

ENGINE

Fig 1: Diesel engine - Governor model.

parameters of the above model can be found in [14].

2.5 Diesel engine speed repulator (Povernor)

A governor can be defined as a mechanical or electromechanical device for automatically controlling the speed of an engine by relating the intake of fuel. Several types of governors exist as mechanical-hydraulic, direct mechanical type, electrohydraulic, electronic and microprocessor based governors.

For a constant setting of the speed changer the static increase in Diesel engine power output is directly proportional to the static frequency droop [7]. This result points out the physical significance of the regulator feedback parameter R. It is reminded that the physical unit for R is H a W . The value of R is considered always positive and if frequency and power variations are in p.u. then R is also in p.u.

After the primary control function, which brings the system to an equilibrium state with a permanent frequency error, a secondary control is needed which establishes eventually nominal rotational speed by eliminating the static frequency error. The device which performs this action is the frequency error signal integrator [5, 151.

The IEEE General Model for Speed-Governing Systems, defined in [IO], has been used in this study. In the case of governors for Diesel units, this model is simplified by setting all the time constants that correspond to steam or hydro units [19] equal to zero and by neglecting the gate servo rate limits [6, 151. Then, a first order model with a time constant ‘52 representing the governor delay, is obtained.

The complete Diesel engine and speed regulator block diagram is shown in Fig. 1.

The load is generally frequency dependent [9, 191. This fact can be taken into account by considering the p.u. load damping

coefficient D, defined as : D = - (PJ. kW 1 P.U. Hz). The load

damping coefficient D can be neglected in practical cases, due to the positive effect that it always has on the system stability. The omission of D during the system transient simulations gives more pessimistic results than the real situations.

The differential equations describing the Diesel engine and the speed regulation system of Fig. 1, are :

apL

af

An expression is given in [I41 to compute ‘5, with respect to

The electrical angle of the rotor 6 is related to the electrical the Diesel engine speed variations.

angular velocity o by the following relation :

580

&=,-a=& dt

The mechanical motion equation (swing equation), is in p.u. :

In some practical cases the necessary parameters for the Diesel engine model, which is proposed in this paper, may not be available. Then the Diesel engine system can be simplified to a first order transfer function model with a time constant To [15]. in which the mechanical losses and the delay are neglected (C,=O, r,=O).

2.6 -nous e-

The equations of synchronous generator are obtained from Park equations after some simplifications [4, 171. The most important is that stator transients are neglected as much faster compared to the rotor ones. In the model presented here synchronous machine rotor consists of three windings. A field and a damping winding are considered on the direct axis in order to take into account the transient and subtransient behavior respectively in this axis. On the quadrature axis a damping winding is considered having a subtransient (or transient) time constant. With this model both subtransient and transient phenomena can be examined [17, 251.

The algebraic stator equations are in p.u. :

v d = E: -rsld +x;Iq, vq = E; - r,I, -xi& (22) The differential equations corresponding to the rotor winding

dynamics are in p.u. :

The electromagnetic torque equation is in p a . :

Toe= E& + EiIq - ( X i - Xq-)ldq (26) The machine current, output voltage and power given by (8) for

the asynchronous generator are also valid here.

2.7 Automat ic voltage reg ulator (AVR)

The purpose of an AVR is to provide to the synchronous machine the proper field voltage, so as to maintain the desired voltage and reactive power that the synchronous machine generates. An important characteristic of an AVR system is that it responds rapidly to voltage deviations during both normal and emergency system operation.

The most commonly used AVR general models are those defied by the IEEE and especially Type 1 model [18. 191. The differential equations of the IEEE Type 1 AVR model can be written in a matrix form convenient for the system simulation as following :

where S, = flVJ is the exciter saturation function. Expressions for

S, can be found in [18, 241.

excitation voltage V, by : The synchronous generator field voltage Efd is related to the

where ~f = Ldd is characteristic parameter of the generator (&$

mutual inductance between stator and field windings, R,d: resistance of the field winding). All the variables must be in the per unit-system. The one per-unit (1 P.u.) generator voltage is defied as rated voltage. The one per-unit (1 P.u.) exciter output voltage is that voltage required to produce rated generator voltage on the generator air gap l i e . Hence in the per-unit system Efd

Other AVR schemes may easily be tested since the interface variables between the AVR system and the synchronous generator remain the same : the output of all four IEEE Types is the p.u. excitation voltage V,, while the input is the p.u. synchronous generator terminal voltage Vr (in Type 3 the p.u. synchronous generator terminal and field currents should be added as input) [le].

2.8 Load m o a

6 Rfd

equals v,.

For a realistic model of the load it is necessary to estimate qualitatively and quantitatively its composition that is the nature of the load components and their percentage in the total load. An analysis for the estimation of the load composition is presented

In the simplest but common case the whole load or a part of this can be modeled as an impedance 2, = RL + j d,. When the load active (P,), the reactive (aL) power and the bus-bar voltage (V,) are known at steady-state then the values of resistance (R,) and inductance (LJ can be calculated by the relation :

in [20-221.

PL +~QC = IVr P Y i (29)

where YL = Yi' and * denotes the conjugate of Y,. The load components which represent motors can be

aggregated into groups depending on the machines size. Each group of motors is represented by an equivalent motor which is modeled by the induction machine equations (5) to (7) with the appropriate parameters. The motor acceleration dynamics can be described also by (4).

The mechanical load torque is represented by the following polynomial function of the motor speed [22] :

T,= T , (A& + B% + c) (30)

The values of A, B and C are determined from the steady-state loading of the motor and hence the steady-state value of speed U,. A simpler relation that can also be used is T,,, = T,& where k= 1 for fan-type loads or k=2 for centrifugal pumps [24]. The value of T , is estimated in the same way as the parameters of (30).

Another important category of load is lighting load which is modeled by static models, that is models depending only on frequency and voltage and not on time, like the motor (dynamic) loads. For details on lighting load models the reader is referred to [20]. In [22] more complex aggregated models for static load can be found that take into account the effect of frequency and voltage variations on this load type [18].

2.9 -ti-tor

Capacitor banks (CBs) play an important role when connected to Diesewind power systems especially when large penetration levels are examined. Wind generators absorb reactive power for their excitation and hence reactive load in the power system is increased. Capacitor banks contribute in maintaining the power

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factor of synchronous generators to acceptable levels. A positive effect also in voltage regulation is provided.

CBs are connected in steps to the power system. In the simulations they are expressed as injected currents in the network. When a CB is applied in a particular node the injected current in the bus-bar, after the elimination of the transients, is :

(31)

Equation (31) is valid when there are not large deviations from the nominal frequency. The reactance X c can be introduced in the network admittance matrix. In order to take into account large frequency variations, parameter X, must be expressed as oC. Transients from the CBs interconnection are neglected as very fast compared to the transient stability phenomena time scale examined here.

- 9 v, xc

Ic=J-

3. ,4 COMPACT MULTIM ACHINE PO WER SYSTEM MODEL

In this Section the more general case of a power system, where the transmission network is not neglected and an arbitrary number of synchronous generators and/or asynchronous machines is connected at each bus-bar, is considered. This general case as well as the differences between the machines reference frames, imply normally a large amount of consecutive intermediate calculations in order to calculate the machine stator currents. In the following analysis, a unique relation which gives analytically by a simple matrix inversion the internal machine currents is derived by eliminating all the intermediate calculations.

Transmission lines are modeled depending on their size. They are characterized as short when their length is less than 80 km. as medium when it is between 80 and 240 km and as long lines when their length is greater than 240 km [23]. A small line is modeled by the series resistance RTL and inductance LTL which are considered as concentrated for the total length of the line. The capacitance C T L of small transmission lines can be neglected while in the case of medium lines it has to be included (equivalent 1c circuit). The modelling of long lines is out of interest in this paper since small or medium isolated power systems are examined.

Although in similar studies [12, 151 it is usual to neglect transmission network, especially for small power systems, th is practice will not be followed here in order to develop a general algorithm able to cover cases with rather extensive networks which can not be neglected.

Since transmission line characteristics (RT? LTL. C T L ) are given per length, they are multiplied by the line length and the resulting values are considered as concentrated over the transmission line. Then the network admittance matrix Y can be easily formulated as explained in 119, 231. Other power system components, which can be represented by an equivalent admittance (capacitor banks, transformers, loads, etc.), can be also introduced in the network admittance matrix.

The admittance matrix Y can be reduced [19] so as to obtain matrix YN with dimension 2Nx2N, where N is the number of power system bus-bars where current is injected.

The bus-bar voltages V N and the corresponding injected currents IN are related by the following equation which is written in the synchronous reference frame D-Q :

I , = Y , VN (32) The vector of the currents injected into the N bus-bars is

Bk

IC = MI II (33) i =1

and Bc (k=l,.. N) is the number of machines connected at the k-th bus-bar.

The matrix M: is introduced in order to consider the appropriate sign of the current 1; of the i-th machine connected to the k-th bus-bar, that is :

J Mf = -J for an asynchronous machine.

for a synchronous generator (J is the 2x2 identity matrix),

N

1 The total number G of machines in the power system is the sum

of machines connected at each bus-bar : G = C Bk. Equation (33)

expresses that the total injected current lk at the k-th bus-bar is the algebraic sum of currents injected (absorbed) by the B , generators (or motors) which are connected to this bus-bar.

The stator equations of both synchronous and asynchronous machines, given by (5) and (22). can be expressed by the following global matrix relation :

A=1

= E: - Z,"MfI,^ (34)

where the index i denotes the machine connected to the bus-bar k. In (34) VI and I; are the vectors of each connected machine

terminal voltage and current coordinates, given as VI = [Vd. V q r

and I f = [ I d , Iq]' respectively. E: is each connected machine voltage behind transient (or subtransient) reactance defined as

E! =[E; , E ~ T for an asynchronous machine or as EI = [G, E;Y for a synchronous. Finally the impedance Zf is given by

Z:=[ i , :] for an asynchronous and by Z I = rJ . -" for a

synchronous machine respectively. In order to write the stator equations of a general multimachine

system in a compact matrix form, the following definitions are necessary :

[ X , .I v, 9 [v l',..., vi',, v: ,..., vi2,v,N ...., V&P

E,: [E! ,.... EJl, E? ,..., E & E f ...., EB"]T

Each element of the above vectors and matrices refers to a specific machine. Elements with the same indices are related through (34). Hence the following general equation can be written for the multimachine system :

Vdq=E,+q-ZnMml* (36) Equation (36) contains quantities expressed in the d-q reference

frame of each machine, thus the elements of system variables vectors in (36) are not expressed in a common reference frame.

In order to be able to express all the elements of vectors in (36) in a common reference frame, that is in the synchronous reference frame. the following transformations are used :

V W = T V + , E W = T E , , IW=TI+ (37) where T is the rotation matrix defined as :

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k where 6; is the angle of the i-th machine connected to the k-th

cos(6) -sin(@ sin (@ cos (@

bus-bar and rot (S) =

The transformation from the synchronous to the machine reference frame (for the vector of voltage) is demonstrated in Fig. 2.

We introduce the matrix L, in order to take into account the machines topology in the network. The dimension of L, is 2Nx2G. The matrix L, is composed by submatrices 2x2 each one characterizing the topology of only one machine. Each submatrix is either the 2x2 identity matrix J for the Bk machines that are connected to the k-th bus-bar or the 2x2 null matrix for the other machines that are not connected to this bus-bar. The algorithm to create matrix L,,, is given below :

It can be easily shown that :

IN = Lm Mm IDQ (39)

From (32), (39) and (40) the following relation is obtained in the synchronous reference frame :

rDQ = E 24 L MJ'VDQ (41)

where & = Yil. Finally from (36). (37) and (41) the relation which gives the

internal machine currents in the machine reference frame is obtained :

I , =[T-'Z;T (42)

where: &=c&L,, ,M,, , and &=Z.M,. Relation (42) provides the current coordinates of each machine

(synchronous or asynchronous generator or motor) connected to the network, in the individual machine reference frame. For a system of G machines it provides 2xG algebraic equations which are solved together with the differential equations of the various components (see Section 2).

Equation (42). which has the same simple form as (32), incorporates both the network and all the machines stator equations. Its solution requires minimum calculation effort since a unique matrix inversion must be performed. It is reminded that for the transformation matrix T -' = T'. The matrices & and & are calculated only once at the initialization of the integration process.

4. INTEGRATION OF THE VARIOUS COMPONENTS ELS IN A SIMULATION SOFTW-

In this Section guidelines to integrate the various components models presented above in order to develop a simulation software able to perform accurate transient stability analysis of isolated WmdDiesel systems are given.

In order to construct the network admittance matrix it is convenient to number the various bus-bars following a specific order which depends on the type of components that are connected at each bus-bar. It is recommended to number f i s t bus- bars where synchronous generators are connected, while buses with asynchronous generators, motors and static load are then numbered. If this order is followed it is an easy task to formulate the described above multimachine power system matrices necessary for the numerical computations.

Fig. 2 Transformation fran the synchronous to the machine ref. frsme.

The integration of (18) and (19) and all the necessary intermediate calculations (see Paragraph 2.4) for the Diesel engine model implementation can be made in the physical units while only the mechanical torque is transformed in the per-unit system in order to be used in (21). The same methodology can be followed for the solution of the differential equations of the wind turbine mechanical part.

The swing equation (21). the synchronous generator and also the speed regulator equations are given in a common per-unit system which is based on the synchronous generator kVA rating. The interconnection of the voltage regulator (AVR) and synchronous generator models is described in Paragraph 2.7. In order to integrate (27). the steady-state values of Vnf and V, are calculated in p a . by solving (27) when derivatives are equal to zero. The values of V, and EP needed in order to determine Vn, and V , are known from the initial conditions of the synchronous generator.

When equations written in the machine individual per-unit system must be interconnected, a common per-unit system has to be chosen. The methodology for choosing the base values for the common per-unit system is presented in [7. 8, 231. All the parameters that their definitions are based in per-unit quantities related to each machine rating, have to be transformed in the chosen common per-unit system. Such parameters are the machine inertia time constant H. the Diesel engine speed regulator parameters R and K,, and the load self regulation parameter D.

The integration of the entire system of differential equations provides the time response of the power system and WECS state variables. Details of the integration procedure as well as the calculation of the necessary initial conditions are given in the Part II of this two-part paper.

5. CONCLUSIONS In the fxst part of this two-part paper the modelling of the

various power system components was examined. Models were given for the various components of an autonomous power system that is for conventional units (Diesel engine, synchronous generator and speed and voltage regulators). for wind generators (wind turbine, asynchronous generator), for capacitor banks, for the transmission network and finally for various types of load.

The common practice to neglect the a.c. machines stator electrical transients and also the transmission network transients as very fast with respect to the time scale of the transient phenomena examined here was analyzed. From this approximation it was possible to derive a simple but general multimachine model in a compact form for the power network. This model permits to describe in a simple way isolated power systems of arbitrary topology with any number of interconnected synchronous generators and asynchronous machines. In addition the derived equation provides a fast solution of the necessary power system algebraic equations.

The integration of the models presented here in a simulation software will provide a general and robust tool for the accurate assessment of isolated Diesel-wind turbines systems interaction.

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REFERENCES A. D. Garrad and U. Hassan. "What is Power Quality ?", Proceedings of the IO-th British Wind Energy Association Conference, D . J. Milborrow. London. 22-24 March 1988. G. J. Vachtsevanos, K. C. Kalaitzakis, "Penetration of wind electric conversion systems into the utility gr id , IEEE Trans. on Power Apparatus and System, vol. PAS-104, no. 7. pp. 1677-1683.1983. M. Papadopoulos. P. Malatestas, "Operational Advantages from the Use of AC/DC/AC Interface for the Connection of Wind Turbines in Diesel Power Systems", Proceedings of the 1990 European Community Wind Energy Conference. Madrid, Spain, 10-14 September, 1990. Paul C. Krause, Analysis of Electric Machinery, McGraw Hill, 1987. J. D. Glover, M. Sarma. Power System A ~ l y ~ i s and Design, Boston :

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P. M. Anderson, A. A. Fouad, Power System Control & Stability, Iowa : The Iowa State University Press, 1977. F. Iliceto, A. Ceyhan, G. Ruckstuhl, "Behaviour of loads during voltage dips encountered in stability studies. Field and Laboratory tests", IEEE Winter Meeting, Winter 1972. C. Concordia, S. Ihara, "Load Representation in Power System Stability Studies". IEEE Trans. on Power Apparatus and System. vol. PAS 101, no. 4, April 1982. W. W. Price et al. "Load Modelling for Power Flow and Transient Stability Computer Studies", IEEE Trans. on Power System, vol. 3, no. 1. February 1983. W. D. Stevenson, Elements of Power System Analysis, Singapore : Mc Graw Hill, 1982. J. Anillaga, C. P. Amold, Computer Modclling of Electrical Power System, John Wiley & Sons Ltd., 1983. C. D. Voumas, N. D. Hatziargyriou, B. C. Papadias, "Programme interactif de simulation de dseaux. Application au &eau grec". CIGRE Session 1990.26 AoCt - lkre Septembre.

PWS-KENT Publishing Company, 1989.

1460-1464,1968.

POMENCLATURl$

r wind turbine blade radius (m). wind turbine rotor swept area (m'). indices corresponding to blades (B), hub (H) and generator (G) subsystems of a wind turbine. axis angle of the wind generator subsystem i. moment of inertia of the wind turbine subsystem i (kg ms. damping coefficient of the wind turbine subsystem i (when the index is ij the quantity corresponds to the axis connecting the subsystems i and]) (J sechad).

Kij;ij=BCI.G spring constant of the axis connecting the subsystems i and j ("/rad).

TArn mechanical torque developed by a wind turbine @.u.). H A total wind turbine and induction generator inertia time constant. 0 0 synchronous angular velocity [=2 x 50 Hz] (radsec). 0 A wind generator angular velocity (rad/sec). 0, angular velocity of the synchronous reference frame.

Q

A B, H , G

6; = B ~ , G J i = BJ,,C Di=BH,G

S slip of induction machine defined as the ratio s = q-0, . induction or synchronous generator stator resistance @A). induction generator open circuit reactance @.u.). induction generator open circuit transient reactance @A.). induction generator rotor open circuit transient time constant (sec). a.c. machine terminal (bus-bar) voltage @.u.). voltage coordinates : V, = v d + jVq @.u.). induction machine voltages behind transient reactance X' @.u.) induction machine current magnitude @.U,). current coordinates : I , = Id + j I , @.u.). Diesel engine controller reference speed (rad/sec).

> X'

2, Vq Ed', E,' [ A Id , I ,

r o

Diesel engine speed (radlsec). Diesel engine efficiency of the combustion. Number of poles in the synchronous generator. base toque for the per-unit transformaion [4, pp.1801. Diesel engine stroke cycles per second [=o,,,/(~cK)]. where : K=2 for two-stroke engine or K=4 for a four-stroke engine. Diesel engine number of cylinders (operating during a combustion cycle). Diesel engine one stroke volume [=zD2S/4. where D : cylinder diameter, S : stroke] (m'). Diesel engine total stroke volume [= zV,] (m') Diesel engine heat value of the fuel (kJ / kg). Diesel engine time delay (sec). Fuel actuator gain constant Diesel engine mean effective work (developed by one piston during a combustion cycle) (kWh). Diesel engine consumption rate (kg/sec). Diesel engine combusted fuel rate (kg/sec). Diesel engine mechanical losses expressed as equivalent pressure droop (bar). govemor summing loop amplification factor @.u.kW/p.u.Hz) Diesel engine permanent speed droop (pm Hz / p.u. kW). synchronous generator inertia time constant (sec). synchronous generator reactance of direct ( i=d) or quadrature ( i=q) axis (P.u.). synchronous generator transient reactance of direct ( i=d) or quadrature ( i=q) a x i s (PA.). synchronous generator subtransient reactance of direct ( i=d) or quadrature ( i=q) (PA.). synchronous generator voltage behind the transient reactance

synchronous- generalor voltages behind the subtransient

synchronous generator open circuit subtransient time constant of quadrature axis (sec). synchronous generator open circuit transient and subtransient respectively time constants of direct a x i s (sec). AVR subsystems indices : Regulator input fdter (R), Amplifier (A), Stabilizer (S). Exciter (E).

x d (p.U.1.

reactances x d and x (P.U.).

gain constant of the i subsystem of AVR. time constant of the i subsystem of AVR (sec). AVR reference voltage @.U,)

P atmospheric density (kg m-'). U, instantaneous wind speed averaged over the wind turbine rotor Acknowledgements as well as the authors

swept area (m sec-'). biographies are found in the Part I1 of the paper.