19th Mediterranean Conference on Control and Automation Aquis Corfu Holiday Palace, Corfu, Greece June 20-23, 2011
WeCT2.3
New Reactive Power Control Concept for Converter Based Renewable Energy Sources S. Huseinbegovic, GSM, IEEE and B. Perunicic, Senior Member, IEEE Abstract—The paper presents a new approach to control of the reactive power supply from renewable energy sources to the grid. The aims of control algorithm are: maximal power extraction from the wind or solar power and its injection into the power grid and the control of reactive power supplied to the grid. The value of reactive power is controlled by magnitude of internal voltage E and phase angle δ between internal E and external V voltages vectors. The needed magnitude E and phase angle δ are calculated from the measured values of the supplied active power Pg and the desired supply of the reactive power Qg_ref. These expressions are obtained by comparing mathematical models of the power converter and the synchronous generator. This concept was implemented on the abc stationary reference frame and its effectiveness was demonstrated on a simulation model of the power converter connected to an infinite bus. Keywords – renewable energy sources, distributed generator, synchronous generator, voltage source converter, grid-connected converter, reactive power control
I
I. INTRODUCTION
N the last ten years, the use of renewable energy sources (RES) showed a spectacular growth [1][2]. Typical representatives of RES are wind turbines and solar energy sources. The RES are geographically spread out and commonly connected to the distribution network (DN). Such sources are known as distributed generators (DGs). The available active power of the DGs depends on weather and climate conditions, and cannot be directly controlled as in conventional power sources. The power electronics has a crucial role in DGs. Its application is necessary to connect a variable-speed wind power or a changeable supply from a solar generator units to the power system [3]-[7]. The power electronic converters match the characteristics of the DC to the grid operation requirements regarding frequency, voltage, active and reactive power, power quality, protections, etc. The voltage source converters (VSCs) are commonly used to transfer power between a DC system and an AC system or back-to-back connection for AC systems with different frequencies such as variable-speed wind power and solar generation units [4][7]-[10]. Functions of VSC for DGs can be summarized as follows [11]-[14]: Manuscript received January 30, 2011. S. Huseinbegovic is with the Faculty of Electrical Engineering, Department of Automatic Control and Electronics, University of Sarajevo, 71000 Sarajevo, Bosnia-Herzegovina (e-mail:
[email protected]). B. Perunicic is with the Faculty of Electrical Engineering, Department of Automatic Control and Electronics, University of Sarajevo, 71000 Sarajevo, Bosnia-Herzegovina (e-mail:
[email protected]).
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
− Power conversion from variable AC or DC voltage to AC voltage with given voltage and frequency. − Output power quality assurance regarding total harmonic distortion (THD), voltage and frequency deviation and flicker. − Protections of DGs and electric power systems from abnormal voltage, current and frequency values, and temperature condition. − Control of DGs power extraction from the wind or solar energy and injection of power into the power grid. Supply of reactive power to power systems is needed to maintain voltage value. Ideally all DGs should contribute to the voltage control by supplying reactive power. The DGs having a VSC are capable to control its output reactive power [15]-[17]. The VSCs can be operated to mimic the behavior of a synchronous generator (SG) [18][19]. In this way, the conventional control algorithms and equipment, developed for SG, can be implemented in VSC. This paper proposes a new reactive power control concept for VSC. The reactive power is controlled by adjusting magnitude of internal voltage and phase angle between internal and external voltages, while all the available active power is supplied to the greed. The paper is organized as follows. Section II gives the characteristics of the synchronous generator in the transient and steady state operation. The resemblance of mathematic models of VSC and the synchronous generator is topic of the Section III. Section IV presents a new control concept of the VSC reactive power. Results of simulation performed on MATLAB tool SIMULINK are in the Section V. Finally, conclusions and discussion are given in Section VI. II. SYNCHRONOUS GENERATOR A. Model of the synchronous generator Three phase synchronous generators are normally used for the generation of electrical power in conventional power supplies as nuclear, thermal and hydro plants. The theory of the synchronous generator was developed during the first half of the twentieth century. The model of a synchronous generator can be found in many references, for example in [18]-[21]. The abc, αβ0 or dq0 reference frames are mostly used to model synchronous generator operation.
850
Where the voltage 𝒆𝒆 is equal to: 𝒆𝒆 = 𝐿𝐿𝑓𝑓𝑓𝑓
𝑑𝑑𝑖𝑖𝑓𝑓 𝑑𝑑𝑑𝑑
cos(ωt) sin(ωt) 2π� � cos�ωt − sin�ωt − 2π�3�� (5) � 3 � − 𝜔𝜔𝐿𝐿𝑓𝑓𝑓𝑓 𝑖𝑖𝑓𝑓 � cos�ωt − 4π�3� sin�ωt − 4π�3�
The above equation can be written as:
Fig. 1. Schematic diagram of a three phase synchronous generator
where:
This section reviews the equations related to electrical part of synchronous generators. The conventional model in abc reference frames is used for further calculations. Fig. 1 shows schematically the cross section of a three phase synchronous generator. The coils 11', 22' and 33' represent the distributed stator windings producing sinusoidal magneto-motive force (MMF) and flux density waves rotating in the air gap. The reference directions for the currents ik (k=1,2,3) are shown by dots and crosses. The frequency of the currents in the stator under steady state condition is a product of number of pole pairs and the radian speed of rotor denoted as ω. The distributed rotor field winding ff' generates sinusoidal MMF and flux density waves aligned with machine magnetic axis and rotating with the rotor. The electrical circuit equations for the three stator phase windings in time domain can be written using the Kirchhoff’s voltage law as:
where:
𝑖𝑖1 𝒊𝒊 = �𝑖𝑖2 � 𝑖𝑖3
𝒗𝒗 = 𝑅𝑅𝑠𝑠 𝒊𝒊 +
𝑑𝑑𝚿𝚿 𝑑𝑑𝑑𝑑
𝑣𝑣1 𝒗𝒗 = �𝑣𝑣2 � 𝑣𝑣3
𝑅𝑅𝑠𝑠 = 𝑅𝑅1 = 𝑅𝑅2 = 𝑅𝑅3
(1) 𝜓𝜓1 𝚿𝚿 = �𝜓𝜓2 � 𝜓𝜓3
For a balanced three phase generator, the total flux linkage of phases is: 𝜓𝜓1 = 𝐿𝐿𝑠𝑠 𝑖𝑖1 + 𝐿𝐿𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑓𝑓 cos(ωt) 𝜓𝜓2 = 𝐿𝐿𝑠𝑠 𝑖𝑖2 + 𝐿𝐿𝑓𝑓𝑓𝑓 𝑖𝑖𝑓𝑓 cos�ωt − 2π�3� 𝜓𝜓3 = 𝐿𝐿𝑠𝑠 𝑖𝑖3 + 𝐿𝐿𝑓𝑓𝑓𝑓 𝑖𝑖𝑓𝑓 cos�ωt − 4π�3�
sin(ωt + δ) sin�ωt + δ − 2π�3�� 𝒆𝒆 = 𝐸𝐸𝑚𝑚 � sin�ωt + δ − 4π�3� 𝑑𝑑𝑖𝑖
2
𝐸𝐸𝑚𝑚 = ��𝐿𝐿𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑𝑓𝑓 � + �𝜔𝜔𝐿𝐿𝑓𝑓𝑓𝑓 𝑖𝑖𝑓𝑓 �
(6)
2
𝜔𝜔𝑖𝑖
𝛿𝛿 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 �− 𝑑𝑑𝑖𝑖𝑓𝑓 𝑓𝑓 � � 𝑑𝑑𝑑𝑑
(7) (8)
In these expressions, Em is the magnitude of internal voltage (the induced emf) vector, and δ is the angle (phase difference) between internal e and external (terminal) v voltages. B. Generator's reactive power Conventional reactive power concept is associated with the oscillation of energy stored in capacitive and inductive components in a power system. Reactive power is produced in capacitive components and consumed in inductive components. A synchronous generator can either produce or consume reactive power. It may be done by controlling either the magnitude of the rotor field current (7) and (8),or the speed of the rotor (7) and (8). The complex power at the terminals of the synchronous generator can be determined using the equivalent one-phase circuit of synchronous generator (Fig. 2) [7][20][21]. The corresponding vector diagram of the synchronous generator is shown in Fig. 3. The generator terminals are used as the phase reference such that: 𝑬𝑬𝟏𝟏 = 𝐸𝐸1 𝑒𝑒 𝑗𝑗𝑗𝑗
𝑽𝑽𝟏𝟏 = 𝑉𝑉1 𝑒𝑒 𝑗𝑗 0
𝒁𝒁𝟏𝟏 = 𝑍𝑍1 𝑒𝑒 𝑗𝑗𝑗𝑗
where: δ – the phase angle of the generating voltage E1, φ – the phase angle of the synchronous impedance Z1.
(9)
(3)
where 𝐿𝐿𝑠𝑠 is known as the synchronous inductance. Substituting the above expression of flux linkage into the circuit equations (1), leeds to the following equation: 𝑑𝑑𝐢𝐢
𝐿𝐿𝑠𝑠 𝑑𝑑𝑑𝑑 = −𝑅𝑅𝑠𝑠 𝒊𝒊 + 𝒆𝒆 − 𝒗𝒗
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
(4)
Fig. 2. Steady-state equivalent circuit of a synchronous generator
851
Fig. 3. Vector diagram of a synchronous generator
From Fig. 3, the three phase complex power supplied by the synchronous generator terminal is: 𝑺𝑺𝒈𝒈 = 3𝑽𝑽𝟏𝟏 𝑰𝑰∗𝟏𝟏
Three phase active power and reactive power are: 𝑃𝑃𝑔𝑔 = 3
𝐸𝐸1 𝑉𝑉1
𝑄𝑄𝑔𝑔 = 3
𝑍𝑍1
𝑉𝑉 2
𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑 − 𝛿𝛿) − 3 𝑍𝑍1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑍𝑍1 𝐸𝐸1 𝑉𝑉1
𝑠𝑠𝑠𝑠𝑠𝑠(𝜑𝜑 − 𝛿𝛿) − 3
1
𝐸𝐸12 𝑍𝑍1
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
(11) (12)
If the effective armature resistance is neglected (𝑅𝑅1 ≈ 0), the magnitude of the synchronous impedance is equal to the magnitude of the synchronous reactance (Z1=x1) with φ = 90o and the per-phase expressions for P1 and Q1 reduce to: 𝐸𝐸 𝑉𝑉 𝑃𝑃𝑔𝑔 = 3 1 1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 (12) 𝑥𝑥 𝑄𝑄𝑔𝑔 = 3
𝐸𝐸1 𝑉𝑉1 𝑥𝑥 1
1
𝐸𝐸 2
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 3 𝑥𝑥1
(13)
𝑉𝑉1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
(15)
1
Using expressions (10), (13) and (14), the power factor PF of the generator is: 𝑃𝑃𝑃𝑃 =
𝑃𝑃𝑔𝑔 𝑆𝑆𝑔𝑔
=
�𝑉𝑉12 −2𝑉𝑉1 𝐸𝐸1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 +𝐸𝐸12
= 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
where φ is the angle between external voltage V1 and current I1 vectors (Fig. 3). The ratio of the reactive Qg and active Pg power is defined as: 𝑡𝑡𝑡𝑡𝑡𝑡 =
𝑄𝑄𝑔𝑔 𝑃𝑃𝑔𝑔
= 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −
Fig. 4. VSC connected to the AC grid
(10)
𝐸𝐸1 𝑉𝑉1
1
∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
(16)
From (13), (14), (15) and (16), follows that the output active power Pg and power factor PF are functions of the magnitude of the internal voltage vector E1 and the angle δ between internal E1 and external V1 voltage vectors. III. VOLTAGE SOURCE CONVERTER MODEL A typical configuration of VSC structure is shown in Fig. 4 [4][6][7]. The voltage Edc is the input DC voltage. The R and L represent the resistance and inductance between the converter internal voltage ek (k=1,2,3) and the external grid voltage vk (k=1,2,3), and ik (k=1,2,3) is the current injected into the grid. The switches Skn (k=1,2,3; n=1,2) in the VSC are controllable semiconductor devices, such as IGBT, MOSFET or FET.
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
There are other configurations of VSC structure such as multilevel converter or matrix converter [3][4][6][7][11][22]. Without a loss of generality, only the VSC circuit given in the fig. 4 is examined in this paper, since other configurations of VSC can be analyzed in a similar way. Using Kirchhoff’s law, dynamics of a VSC converter in the three-phase reference frame in time domain is described as: 𝑑𝑑𝒊𝒊 𝐿𝐿 ∙ = −𝑅𝑅 ∙ 𝒊𝒊 + 𝒆𝒆 − 𝒗𝒗 (17) 𝑑𝑑𝑑𝑑 𝒆𝒆 =
where: 𝑖𝑖1 𝒊𝒊 = �𝑖𝑖2 � 𝑖𝑖3
𝐸𝐸 𝑑𝑑𝑑𝑑 3
2 −1 −1 ∙ �−1 2 −1� ∙ 𝒖𝒖 −1 −1 2
𝑒𝑒1 𝒆𝒆 = �𝑒𝑒2 � 𝑒𝑒3
𝑅𝑅 = 𝑅𝑅1 = 𝑅𝑅2 = 𝑅𝑅3
𝑣𝑣1 𝒗𝒗 = �𝑣𝑣2 � 𝑣𝑣3
(18)
𝑢𝑢1 𝒖𝒖 = �𝑢𝑢2 � 𝑢𝑢3
𝐿𝐿 = 𝐿𝐿1 = 𝐿𝐿2 = 𝐿𝐿3
The elements of the vector u = (u1 u2 u3 )T belong to the set {−1, +1} and they define the states of the switches as follows: 𝑢𝑢𝑘𝑘 = �
−1 , +1 ,
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑆𝑆𝑘𝑘1 𝑖𝑖𝑖𝑖 𝑂𝑂𝑂𝑂 𝑘𝑘 = 1,2,3 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑆𝑆𝑘𝑘2 𝑖𝑖𝑖𝑖 𝑂𝑂𝑂𝑂
Comparison with SG (4), shows that a VSC (17) can generate or absorb reactive power in the same way as a synchronous generator [18][19]. From this analysis it follows that the controlled reactive power generation could be generated by DGs converters. In conventional power sources the value of voltage/active power or active/reactive power of a synchronous generator are set The output values of active and reactive power are kept relatively unchanged for the sake of the power system proper operation. The active power available for conversion to electrical power in distributed generators (wind and solar) changes unpredictably. For example, in wind generators it depends on the wind speed (𝑃𝑃~𝑣𝑣 3 ). The task of the control algorithm is to forward all available active power into the power grid, since this decreases overall costs of active power 852
generation, and the since the changes of supplied active power are relatively small the normal grid operation is not in jeopardy. Reactive power in the grid is needed to keep the external (grid) voltage at the desired value. The correction of deviation of the external voltage from the nominal require the supply of the reactive power to the grid. The needed amount depends on the current value of the active power supply. This is usually the task of synchronous generators, and their burden can be eased by DGs supply of reactive power. The aim of this paper is to define an algorithm to calculate the desired magnitude of internal voltage vector E1, and the desired phase angle δ between internal E1 and external V1 voltages vectors, from the measurement of the current active power Pg; the given external voltage of the power grid V1, and the desired reactive power Qg_ref . The formulas are derived from ones given in Section II. The equations defining the control algorithm conclude this section.
Fig. 5. The family of curves EQ (blue line: tgφ=-0.8; green line: tgφ=-0.4; red line: tgφ=0; cyan line: tgφ=+0.4; magenta line: tgφ=+0.8) (Pg=2kW, V1=230V, x=6.9Ω)
From the last condition, with some calculation, we obtain the necessary condition for existence of the solutions:
𝐸𝐸1 =
𝑥𝑥𝑃𝑃𝑔𝑔 3𝑉𝑉1
1
1
𝑔𝑔
(19)
∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝐾𝐾 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝑄𝑄𝑔𝑔 = 3 ∙
𝑥𝑥
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
∙ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 3 ∙
The function error is equal to:
𝐾𝐾 2 𝑥𝑥
1
∙ 𝑠𝑠𝑠𝑠𝑠𝑠 2
𝛿𝛿
𝐸𝐸𝑄𝑄 (𝑦𝑦, 𝑡𝑡𝑡𝑡𝑡𝑡) = 𝑄𝑄𝑔𝑔 − 𝑄𝑄𝑔𝑔_𝑟𝑟𝑟𝑟𝑟𝑟 = 0
(21)
𝑥𝑥𝑃𝑃𝑔𝑔2
𝑦𝑦 ≝ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐, Π ≝ 3𝑉𝑉 2 , 𝑄𝑄𝑔𝑔_𝑟𝑟𝑟𝑟𝑟𝑟 ≝ (𝑡𝑡𝑡𝑡𝑡𝑡)𝑟𝑟𝑟𝑟𝑟𝑟 ∙ 𝑃𝑃𝑔𝑔 and substituting
Fig. 5 shows a family of curves EQ for various values of the parameter (𝑡𝑡𝑡𝑡𝑡𝑡)𝑟𝑟𝑟𝑟𝑟𝑟 . These curves suggest that the value of reactive power 𝑄𝑄𝑔𝑔_𝑟𝑟𝑟𝑟𝑟𝑟 can be adjusted by controlling the angle δ. The solution of the above equation follows:
where 𝑦𝑦 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.
2∙Π
(23)
The equation (22) has the solution (23) only under the following condition: 𝑃𝑃𝑔𝑔2 − 4 ∙ Π ∙ �𝛱𝛱 − (𝑡𝑡𝑡𝑡𝑡𝑡)𝑟𝑟𝑟𝑟𝑟𝑟 ∙ 𝑃𝑃𝑔𝑔 � ≥ 0
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𝑔𝑔
Π
𝑃𝑃𝑔𝑔
(26)
𝑃𝑃𝑔𝑔
(27)
4∙𝛱𝛱
The value of the supplied reactive power 𝑄𝑄𝑔𝑔_𝑟𝑟𝑟𝑟𝑟𝑟 has a lower bound (inequality (27)) depending on the system parameters and current active power. The reactive power control algorithm is derived in two steps. The first step is the calculation of variable y, and the second step is the calculation of controllable magnitude of internal voltage E1 from two expressions:
1
𝐸𝐸𝑄𝑄 (𝑦𝑦, 𝑡𝑡𝑡𝑡𝑡𝑡) = 𝛱𝛱 ∙ 𝑦𝑦 2 − 𝑃𝑃𝑔𝑔 ∙ 𝑦𝑦 + 𝛱𝛱 − (𝑡𝑡𝑡𝑡𝑡𝑡)𝑟𝑟𝑟𝑟𝑟𝑟 ∙ 𝑃𝑃𝑔𝑔 = 0 (22)
Π
≥ 𝑃𝑃 − 𝑔𝑔
𝑄𝑄𝑔𝑔 from (20) into (21), the function error becomes:
𝑦𝑦1,2 =
𝑃𝑃𝑔𝑔
𝑄𝑄𝑔𝑔_𝑟𝑟𝑟𝑟𝑟𝑟 ≥ � − � 𝑃𝑃𝑔𝑔 𝑃𝑃 4∙𝛱𝛱
(20)
where 𝑄𝑄𝑔𝑔_𝑟𝑟𝑟𝑟𝑟𝑟 is desired value of reactive power. Defining
𝑃𝑃𝑔𝑔 ±�𝑃𝑃𝑔𝑔2 −4∙Π∙�𝛱𝛱−(𝑡𝑡𝑡𝑡𝑡𝑡 )𝑟𝑟𝑟𝑟𝑟𝑟 ∙𝑃𝑃𝑔𝑔 �
𝑄𝑄𝑔𝑔 _𝑟𝑟𝑟𝑟𝑟𝑟
or:
This value of E1 inserted into the equation (14) gives:
(25)
Using equation (16), the above condition can be written as:
where K is a positive constant.
𝐾𝐾∙𝑉𝑉1
𝑃𝑃
Π
(𝑡𝑡𝑡𝑡𝑡𝑡)𝑟𝑟𝑟𝑟𝑟𝑟 ≥ − 𝑔𝑔 𝑃𝑃 4∙𝛱𝛱
From the equation (13), the magnitude E1 is:
𝑦𝑦 =
𝑃𝑃𝑔𝑔 +�𝑃𝑃𝑔𝑔2 −4∙Π∙�𝛱𝛱−(𝑡𝑡𝑡𝑡𝑡𝑡 )𝑟𝑟𝑟𝑟𝑟𝑟 ∙𝑃𝑃𝑔𝑔 � 2∙Π
𝐸𝐸1 = 𝐾𝐾 ∙ �1
(28)
+ 𝑦𝑦 2
This can be accomplished only under following condition:
where:
(𝑡𝑡𝑡𝑡𝑡𝑡)𝑟𝑟𝑟𝑟𝑟𝑟 ≥ 𝑥𝑥𝑃𝑃𝑔𝑔2
Π = 3𝑉𝑉 2 1
4𝐾𝐾 2 −𝑉𝑉12 4𝐾𝐾𝑉𝑉1
𝐾𝐾 =
(29) 𝑥𝑥𝑃𝑃𝑔𝑔 3𝑉𝑉1
(30)
To conclude, the above reactive power control needs as an input the following parameters: Pg - instantaneous active power, V1 - external voltage of power grid, x - reactance between the VSC and the power grid.
(24) 853
𝑟𝑟𝑟𝑟𝑟𝑟
𝒊𝒊
50.5
Pout Pav
Pg [kW]
The control algorithms of VSC needs the value the loop currents and also reference current. For a balanced three phase system, the reference currents are deduced from the equations (12) and (13) as:
50.0
1
(𝐾𝐾𝐾𝐾 − (𝐾𝐾𝐾𝐾 − 𝑉𝑉1 )𝑐𝑐) 𝑥𝑥 ⎛1 ⎞ = ⎜2𝑥𝑥 ��−𝐾𝐾�1 + 𝑦𝑦√3� + √3𝑉𝑉1 �𝑠𝑠 − �𝐾𝐾�𝑦𝑦 − √3� − 𝑉𝑉1 �𝑐𝑐�⎟ (31) 1
⎝2𝑥𝑥
��𝐾𝐾�−1 + 𝑦𝑦√3� − √3𝑉𝑉1 �𝑠𝑠 + �𝐾𝐾�𝑦𝑦 + √3� − 𝑉𝑉1 �𝑐𝑐�
⎠
49.5 0.1
0.4 Time [sec]
0.5
0.7
0.6
12
Qout
11
Qref
Qg [kVA]
10 9 8 7 6 5 4 0.1
0.2
0.3
0.4 Time [sec]
0.5
0.4 Time [sec]
0.5
0.6
0.7
Fig. 8. The desired (red line) and output (blue line) reactive power Qg of the VSC at different power factor 650 640
E1 [V]
The above control concept has been implemented in the model in MATLAB-SIMULINK (Fig. 6). The simulation parameters were : − the active power from the VSC: Pg=50kW; − the voltage of 3-phase AC grid: V1rms=560V; − the input DC voltage in the VSC: Edc=1600V; − the inductance: L=22mH; − the resistor: R=0.5Ω;
0.3
Fig. 7. The available (red line) and output (blue line) active power P of the VSC at different power factor
where s=sin(ωt) and c=cos(ωt). These variables can be extracted of the grid voltages using PLL or some other structure. IV. SIMULATION RESULTS
0.2
In this example, the necessary existence condition to implement the reactive power control algorithm is:
630 620 610 600 0.1
(𝑡𝑡𝑡𝑡𝑡𝑡)𝑟𝑟𝑟𝑟𝑟𝑟 ≥ −0.316
0.2
0.3
0.7
0.6
Fig. 9. The computed magnitude of internal voltage E1 of the VSC at different power factor
Control results for various supply power factors are shown in Fig. 7-11. The output reactive power reference Qg_ref presents a step from 0.12·Pg → 0.20·Pg →0.16·Pg, while the available active power Pg is constant at Pg=50kW.
0.35 0.345
y
0.34 0.335 0.33 0.325 0.32 0.1
0.2
0.3
0.4 Time [sec]
0.5
0.7
0.6
Fig. 10. The computed variable y of the VSC at different power factor
Our second example is the operation VSC connected to the AC grid with stochastic available active power Pg. Fig. 11-14. show the response of simulated system. The reactive → 10kVA power reference Qg_ref is a step from 6kVA →8kVA. 51.0
Pout Pav
Fig. 6. MATLAB control scheme for VSC connected to the AC grid
Pg [kV]
50.5 50.0 49.5 49.0 48.5 0.1
0.2
0.3
0.4 Time [sec]
0.5
0.6
0.7
Fig. 11. The available (red line) and output (blue line) active power Pg of the VSC at stochastic active power
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
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12
REFERENCES
Qout
11
Qref
[1]
Qg [kVA]
10 9
[2]
8 7
[3]
6 5 4 0.1
0.2
0.3
0.4 Time [sec]
0.5
0.6
0.7
Fig. 12. The desired (red line) and output (blue line) reactive power Qg of the VSC at stochastic active power
[4]
640
[5]
635
E1 [V]
630 625
[6]
620 615 610 0.1
0.2
0.3
0.4 Time [sec]
0.5
0.6
0.7
Fig. 13. The computed magnitude of internal voltage E1 of the VSC at stochastic active power
[7] [8] [9]
3.0
y
2.95
[10]
2.9
2.85
[11] 2.8 0.1
0.2
0.3
0.4 Time [sec]
0.5
0.6
0.7
Fig. 14. The computed variable y of the VSC at stochastic active power
The previously plots are divided in the three time sections, starting with the reactive power reference of Qg_ref=6kVA; between time t=0.3sec and t=0.5sec, the reactive power reference is Qg_ref=10kVA and after time 0.5sec, the reactive power reference is Qg_ref=8kVA. Fig. 7 and 11 show available (red line) and output (blue line) active power Pg of the VSC for two examples. The maximal absolute error in the steady-state between available and output active power is about 80W for the first example and 300W for the second example. The desired (red line) and output (blue line) reactive power is illustrated in the fig. 8 and 12. The maximal absolute error in the steady-state between desired and output reactive power is about 23W for the both examples. The aforesaid errors are result of the simple control algorithm, the losses in the power electronic equipment in the VSC and the resistive losses. V. CONCLUSION Using the dynamical model of synchronous generator, the comparison between the performance of the synchronous generator and the typical configuration of the converter was derived showing that the converter can imitate the synchronous generator. Reactive power control algorithm is derived based on the relations for active and reactive power of the synchronous generator. The predicted performance of the algorithm is illustrated by simulations. Relations describing the algorithm are suitable for digital realization of the control systems. 978-1-4577-0123-8/11/$26.00 ©2011 IEEE
[12]
[13]
[14] [15] [16] [17] [18] [19] [20] [21] [22]
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