Optimal Operation of Offshore Wind Farms With Line ... - IEEE Xplore

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 2, JUNE 2010

Optimal Operation of Offshore Wind Farms With Line-Commutated HVDC Link Connection Miguel Eduardo Montilla-DJesus, Member, IEEE, David Santos-Martin, Santiago Arnaltes, Member, IEEE, and Edgardo D. Castronuovo, Senior Member, IEEE

Abstract—This paper aims to calculate the optimal operation of offshore wind farms (OWF) working with wind turbines based on doubly fed induction generator (DFIG) technologies, and with a high-voltage dc transmission connection. The objective of the optimization problem is to maximize the active power output of the arrangement, while considering certain factors, i.e., the restrictions imposed by the available wind power, the capability curve of the DFIG, the dc-link model, and the operative conditions. The accomplishment of this aim implies setting, adjusting, and operating the system under study in order to produce a reliable and cost-efficient electric energy supply. A realistic simulation test case is performed to evaluate the proposed method, and the optimal operation analysis takes into account different wind speeds and high-voltage dc-link lengths. The results show the effectiveness of the proposed method, demonstrating the advantages of using the reactive control performed by the DFIG to manage the operational requirements of the dc link. Index Terms—Converters, doubly fed induction generator (DFIG), high-voltage dc transmission, nonlinear programming, optimization methods, wind power generation.

I. INTRODUCTION HE EUROPEAN UNION (EU) aims to generate 20% of its total primary energy from renewable sources by 2020 [1]. Offshore wind farms (OWFs) will decisively contribute to the achievement of this goal, as it is estimated that their power production will continue to rise and will overtake by up to 30% of the production generated from onshore wind farms [2], [3]. The problems involved in onshore wind turbines, such as land use restrictions and environmental issues (visual intrusion and noise), favor the development of OWF installations. On the other hand, investment costs for OWFs are higher than those required for onshore installations. The transmission system to the shore entails approximately 30% of the OWFs overall investment [3], and its design is, therefore, highly significant. High-voltage ac (HVac) and highvoltage dc (HVdc) connection methods are currently available for connecting OWF to the grid. Nowadays, most operating OWF have adopted the ac alternative, due to the comparatively small size of the farms (up to a few hundred megawatt), and to their short transmission distances to the connection grid node

T

Manuscript received March 3, 2009; revised June 20, 2009 and August 21, 2009; accepted September 21, 2009. Date of publication January 12, 2010; date of current version May 21, 2010. This work was supported by Spanish Ministry of Education and Innovation under Project ENE2008-06588C04-02. Paper no. TEC-00093-2009. The authors are with the Department of Electrical Engineering, University Carlos III de Madrid, Madrid 28911, Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TEC.2009.2033575

(below 50 km) [4], [5]. Nevertheless, the HVdc method involves three factors that may considerably increase its use in the future: a remarkable decrease in HVdc links investment costs, an increase in wind turbine capacity, and the possibility of exploiting isolated locations with high wind-power potential [6]. Larger OWF must inject their power to the grid in stronger connection buses, at high voltage levels that require larger transmission lines. As distances increase (mainly undersea) so do the ac cable costs, which become prohibitive beyond certain distances [7]. Long ac cables produce large amount of capacitive reactive power, and thus, reduce the transmission capacity. In these applications, HVdc transmission links may offer some advantages when compared with ac connections as they have lower costs, clear grid architecture, a simple operation manner, and less impact on the receiving power grid [8]. HVDC transmission can be based on two alternative technologies: line-commutated converter (LCC) using thyristors and voltage source converter using insulated-gate bipolar transistors (IGBTs). The HVdc voltage source converter technology limits the power transmission to 330 MW, mainly due to the maximum power rating of the semiconductors [9], [10]. The HVdc LCC is generally used in high-power applications. This transmission system approach has been operating with high reliability and little maintenance for more than 30 years [11]. Today, the HVdc LCC is considered the best solution (with regards transmission losses) for wind farms generating more than 500 MW and with transmission distances of more than 100 km [7]. This could be a suitable solution for the connection of large OWF to the grid, as in the wind farms to be built offshore the northwest coast of Great Britain, with installed capacities of around 1000 MW and distances to the nearest grid connection point over 100 km [9]. It is expected that HVdc technology will, in the near future, be competitive for both the capacity and the transmission distances of the smaller OWF. Larger wind farms can introduce additional control requirements, as requested for the adequate operation of the grid system. The real production of the wind farm must, generally, differ from the production values predicted by forecasting tools. In order to minimize this unwanted issue, the system operators (SOs) have encouraged the development of operational guidelines and regulations to maintain system reliability and security. Therefore, new wind farms must be able to control the active power production following optimal approaches [12], [13]. In Spain, all energy production installations with nominal power more than 10 MW must be associated to a Control Center for Renewable Production, which restricts the operation of wind farms under contingencies [14]. Wind farm control systems that

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MONTILLA-DJESUS et al.: OPTIMAL OPERATION OF OFFSHORE WIND FARMS WITH LINE-COMMUTATED HVDC LINK CONNECTION

comply with controllability SO rules are described in [15] and [16]. The control ability depends on the wind turbine’s technology. Different wind turbine manufacturers have incorporated into their designs power electronic components for variable speed operation that comply with the newest SO requirements. The designs are based on the doubly fed induction generator (DFIG), which offers several important advantages, such as speed control, reduced flicker, and four quadrants active and reactive power capabilities. This paper details the optimal operation setup for the combined operation of an OWF with DFIG-based wind turbines, linked to the grid through a transmission line based on HVdc LCC technology. The method describes the inclusion of the DFIG capability limits, the HVdc restrictions and the operational requirements, in one unique optimization problem, and aims to improve the overall operation of the system. The main objective is to maximize the active power output of the complete configuration DFIG + HVdc. This objective implies setting, adjusting, and operating the system under study to produce a reliable and cost-efficient electric energy supply. The algorithm is suitable for short-term (0.5–24 h ahead) operation planning. The proposed approach is validated by realistic simulation tests that take into account the behavioral changes of OWF under different wind speed conditions and for various HVdc transmission-line distances, with an aggregated wind farm model. II. DFIG-BASED WIND TURBINE CAPABILITY LIMITS The typical DFIG configuration consists of a wound rotor induction generator with the stator directly connected to the grid and with the rotor interfaced through a back-to-back partialscale frequency converter, typically rated at around 30% of the generator rating for a given rotor speed variation range at ±25% [17]. This converter is a bidirectional frequency converter and has the capacity to regulate the active and reactive powers, uncoupled, depending on the network’s requirements [18]. In steady state, the DFIG capability limits are obtained by considering the stator- and rotor-rated currents; and then, calculating the total wind turbine capacity limits. These currents are responsive to stator and rotor heating due to Joule’s losses. Additionally, another capability limit appears when considering the steady-state stability limit of the generator [19]. A. Stator Current Limit The stator current limit considers the heating due to the statorwinding Joule’s losses. When considering the stator-rated current and rated voltage in per unit (p.u.), the expression of the stator current limit can be expressed as PS2 + Q2S = (US IS )2 .

(1)

Equation (1) shows that the locus of maximum stator current in the PQ plane is a circle centered at the origin, with a radius equal to the stator-rated apparent power (see Fig. 1), similar to the synchronous generator limits [20], [21].

Fig. 1.

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DFIG capability limits.

B. Rotor Current Limit The rotor current limit considers the heating due to the rotorwinding Joule’s losses. The active and reactive powers in the stator at the stator-rated voltage, where all the quantities are expressed in p.u. can be formulated as PS =

XM US IR sin δ XS

QS =

XM U2 US IR cos δ − S . XS XS (2)

From (2), we have 2 2 XM US2 2 = US IR . PS + Qs + XS XS

(3)

In the PQ plane represented, (3) is a circle, centered at [−US2 /XS , 0], and with a radius equal to XM /XS US IR . The offset in the reactive power axis reveals the DFIG magnetizing needs by means of a constant reactive power consumption that varies with the grid voltage (see Fig. 1). C. Steady-State Stability Limit Equation (2) shows that, as for the conventional synchronous generator, at constant rotor current and stator voltage, the active power generation is proportional to the sine of the load δ. When 0◦ < δ < 90◦ , the active power increases with the load angle, resulting in stable operation points. In this situation, an increase of the turbine torque will produce an increase of the load angle and, therefore, an increase of the generator torque. When 90◦ < δ < 180◦ , the active power production decreases with the load angle, resulting in unstable operation points. In these cases, an increase of the turbine torque will produce an increase of the load angle and, therefore, a reduction of the generator torque. As expected, the steady-state stability limit restricts the point of maximum active power generation to δ = 90◦ , i.e., the point where voltage and internal electromotive force (EMF) are orthogonal. In the PQ plane, the steady-state stability limit is given by a vertical line at [US2 /XS , 0] (see Fig. 1). Note that US2 /XS is approximately the no-load reactive-power consumption, which means that the generator becomes unstable when the reactive power absorption is higher than the no-load reactive power. Fig. 1 shows the resulting DFIG capability limits. The generator will be able to operate at any point in the shaded area within

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tance XS are constant for all the feasible operating area shown in Fig. 2. Therefore, when considering Vr = US and Pg = PS , (3) transforms into 2 (6) Pg2 + Qg + BVr2 = AVr2 where A = ((XM /XS ) IR )2 , and B = 1/XS . For the lower and upper limits of the reactive power, the following equations can be written:

Fig. 2.

Total DFIG-based wind turbine capability limits.

the specified limits. It is important to stress that the three operational limits of the DFIG activity (stator current, rotor current, and stability) depend on the stator voltage; therefore, operating the generator at a variable stator voltage will affect the capability limits of the generator. The generator-rated power factor is obtained from the intersection of the limits given by the stator and rotor currents, as it is at that point where the generator works with the rated stator and rotor currents. D. Total Capability Limits The DFIG-based wind turbine’s total capability limits are obtained by adding the rotor active-power generation to the stator active-power generation. The rotor active-power generation, neglecting stator and rotor resistance, can be expressed [18] PR = −sPS

(4)

where s is the slip. Therefore, the total active power generation of the DFIG can be summarized as PT = (1 − s)PS .

(5)

Equation (5) clearly shows that the total power capacity increases with rotational speed, i.e., maximum power capacity is obtained at maximum speed. Fig. 2 shows the generator total capability limits, increasing the stator capability limits with the proportion of power fed through the rotor. Reactive capability is not modified from (2), because the grid-side inverter of the DFIG is usually operating with a unity power factor. The shaded area in Fig. 2 represents the feasible area of operation for the DFIG generator, including the restrictions related to the maximum active power available at wind turbine Pgm ax . E. Maximum and Minimum Reactive Power Limits Maximum and minimum reactive power limits take into account the DFIG-based wind turbine’s total capability limits and steady-state stability limits, as shown in Fig. 2. The value of these limits is determined by (3). The stator voltage US is equal to the rectifier voltage Vr in the HVdc link. Rotor current IR is the rated current by the rotor windings and remains constant. The DFIG parameters, magnetizing reactance XM , and stator reac-

Qg

m ax

= + AVr2 − Pg2 − BVr2

Qg

m in

=−

1 2 V . XS r

(7) (8)

Equations (7) and (8) represent the DFIG capability limits, as shown in Fig. 2. Note that the maximum reactive power depends on both the rectifier voltage and the generated active power, whereas the minimum reactive power depends solely on the rectifier voltage. Several authors have used the control abilities of a DFIG wind farm to reach different objectives [15], [16], [22]–[24]. Most of them do not consider the DFIG’s complete capability limits. This paper considers the control abilities of an advanced DFIG model to improve the operation of an OWF, also taking into account the control capacities of the HVdc transmission link.

III. HVDC EQUATIONS SYSTEM For a steady-state analysis, the HVdc converters are simplified using the following considerations [25]–[27]. 1) The ac source is, at the converter terminal, a sinusoidal voltage waveform with constant amplitude and frequency. 2) All harmonic voltage and currents produced by the converter are filtered out and are not considered by the ac system. 3) The converter transformers have no resistance and no magnetizing reactance. 4) The converter has no active power losses, considering ideal valves those without voltage drop. 5) The dc current and voltage are approximately constant and ripple-free. Using LCC technology for the HVdc link, the converters perform ac/dc and dc/ac conversions that consist of valve bridges and transformers with tap changers. The valve bridge is an array of thyristors that sequentially connect each phase of the ac three-phase system to the appropriate point of the voltage cycle, producing dc (when operating as a rectifier) and supplying ac power (when operating as an inverter). Monopole and bipolar configurations are the most frequent arrangements of HVdc links. The monopole HVdc link uses one conductor to deliver the power and either the ground or a metallic conductor as a return path [26]. The bipolar HVdc configuration uses two conductors, one positive and one negative, and may require a return path. The HVdc rectifier and inverter substations are usually integrated by cascaded groups of several converters that each has a transformer bank and a group of valves. The converters are connected in series to the dc side (valve) to give the desired level of voltage


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by means of the commutation resistance. When considering an overlap angle µ less than 60◦ , the ignition and extinction angles, and the angles ϕr and ϕi are expressed as follows [25]:

Fig. 3.

Equivalent representation of high-voltage dc transmission.

from pole to ground. Independent to the kind of connection and the number of bridges, the HVdc links can be represented through an equivalent HVdc model [25], as shown in Fig. 3. In the present paper, a p.u. representation is adopted for both ac and dc grids. The relationship between ac and dc quantities, when considering multiple bridges, is included in Appendix A. Using the p.u. system, the dc magnitudes of voltages Udr and Udi at the terminals of the rectifier and inverter, respectively, are represented as Udr = ar Vr cos αr − rcr Id

(9)

Udi = ai Vi cos αi − rci Id

(10)

where αr is the ignition angle for the rectifier operation; αi is the extinction angle for the inverter operation; rcr and rci are the commutation resistances in the rectifier and inverter, respectively; ar and ai are the converter transformer taps setting in the rectifier and inverter, respectively; Vr and Vi are the effective voltage magnitudes of the phasor at the ac terminals of the rectifier and inverter, respectively; and Id is the dc by the HVdc link. The real power flowing into Pdr or out Pdi of the dc network at the terminals of the rectifier and inverter respectively, can be expressed as follows: Pdr = Udr Id

(11)

Pdi = Udi Id .

(12)

The reactive power flowing into the ac rectifier terminals Qdr and inverter Qdi , can be calculated as follows: Qdr = Pdr tan ϕr

(13)

Qdi = Pdi tan ϕr

(14)

where ϕr and ϕi are the angles by which the angle of fundamental line current ζ lags the angle of the line-to-neutral source voltage δ. During the commutation stage, the phase currents cannot change instantly. Therefore, the transfer of current from one phase to another requires a finite time, referred to as commutation time or overlap time. The corresponding overlap or commutation angle is denoted by µ [26]. A short-circuit of small duration between the two commuting thyristors could happen during this period. This short-circuit produces a temporary reduction of dc voltage. This effect is not explicitly considered in the aforementioned set of equations, but is taken into account

tan ϕr =

2µr + sin 2αr − sin 2(αr + µr ) cos 2αr − cos 2(αr + µr )

(15)

tan ϕi =

2µi + sin 2αi − sin 2(αi + µi ) . cos 2αi − cos 2(αi + µi )

(16)

The overlap angle µ may be calculated, on both the tifier and inverter sides, using the following equations Appendix C): 2rcr Id −1 µr = cos cos αr − − αr ar Vr 2rci Id µi = cos−1 cos αi − − αi . ai Vi

rec(see

(17) (18)

Additionally, the current in the dc link between the two terminals is expressed by (19). As shown in Fig. 3, in the equivalent model, the HVdc link is modeled as a cable with a resistance Rcc Id =

Udr − Udi . Rcc

(19)

In [27]–[29], an HVdc model similar to the one considered here is included for optimal power flow, aiming to represent the ac–dc power systems. The present paper considers the utilization of the available controllability of the HVdc link to improve the OWF operation for different wind speeds. IV. OPTIMIZATION PROBLEM The aim of this paper is to maximize the active power output of OWF, taking into account the restrictions imposed by the available wind power, the DFIG, and the HVdc-link models. In Spain (as in other power systems), all wind-farm production is accepted by a pool-type market at a zero bid price. Wind farms are remunerated in relation to the quantity of active power injected to the external grid. Maximizing the active power production is, therefore, an important economic objective in the operation of an OWF. The ac system consists of an aggregated model of a DFIG wind farm connected to the HVdc rectifier through a transformer. The HVdc inverter is also connected to the grid with a transformer (see Fig. 4). The aggregated wind farm model represents a large number of grid-connected wind turbines within a large OWF [3]. Both transformers used for the ac sides have tap settings. Passive filters are also connected to both the HVdc rectifier and inverter terminals, represented by constant shunt admittance YS . In Fig. 4, Pg r + jQg r and Pdr + jQdr are the powers of the aggregated model of a DFIG offshore wind farm and the rectifier converters, respectively, and Pg i + jQg i and Pdi + jQdi are the powers of the grid and the inverter converter, respectively. PL r + jQL r and PL i + jQL i are load powers at the rectifier and inverter terminals, respectively. The optimal operation for the OWF + HVdc link, represented in Fig. 4, can be obtained from the solution of the optimization

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Fig. 4.

Equivalent representation of OWF + HVdc.

problem as follows: Pg i

(20)

Pg r − Pdr − PL r = 0

(21)

− Pg i + Pdi − PL i = 0

(22)

Maximize Subject to

2

Qg r − Qdr + |Vr | YS − QL r = 0 2

(23)

− Qg i + Qdi + |Vi | YS − QL i = 0 Udr − Udi Udr − ar Vr cos αr + rcr =0 Rcc Udr − Udi =0 Udi − ai Vi cos αi + rci Rcc Udr − Udi =0 Pdr − Udr Rcc Udr − Udi =0 − Pdr − Udi Rcc

(24)

Qdr − Pdr tan ϕr = 0

(29)

− Qdi − Pdi tan ϕi = 0

(30)

tan ϕr −

2µr + sin 2αr − sin 2(αr + µr ) =0 cos 2αr − cos 2(αr + µr )

2µi + sin 2αi − sin 2(αi + µi ) =0 cos 2αi − cos 2(αi + µi ) 2rcr Udr − Udi cos(µr + αr ) − cos αr + =0 ar Vr Rcc 2rci Udr − Udi =0 cos(µi + αi ) − cos αi + ai Vi Rcc 1 2 Vr ≤ Qg r ≤ + AVr2 − Pg2 − BVr2 − XS

tan ϕi −

(25) (26) (27) (28)

(31) (32) (33) (34) (35)

Vk

m in

≤ Vk ≤ Vk

m ax

k = r, i

(36)

ak

m in

≤ ak ≤ ak

m ax

k = r, i

(37)

Pg k

m in

≤ Pg k ≤ Pg k

m ax

k=i

(38)

Pdk

m in

≤ Pdk ≤ Pdk

m ax

k = r, i

(39)

Qdk

m in

≤ Qdk ≤ Qdk

Udk

m in

≤ Udk ≤ Udk

m ax

m ax

k = r, i k = r, i

(40) (41)

αk

m in

≤ αk ≤ αk

m ax

k = r, i

(42)

µk

m in

≤ µk ≤ µk

m ax

k = r, i

(43)

ϕk

m in

≤ ϕk ≤ ϕk

m ax

k = r, i

(44)

where additional subscripts g, r, i, and d are used to denote the quantities of the generator, rectifier, inverter, and dc terminals, respectively. (∗)m ax and (∗)m in represent the maximum and minimum limits of (∗), respectively. In the optimization problem, Vr , Vi , ϕr , ϕi , Udr , Udi , αr , αi , µr , µi , ar , ai , Pg i , Pdr , Pdi , Qg i , Qg r , Qdr , and Qdi are continuous variables and Pg r is a fixed parameter. The objective function defined in (20) aims to maximize the active power output of the OWF + HVdc system. Equations (21)–(24) represent the active and reactive power balance at the ac terminals of the rectifier and inverter, respectively. It must be stressed that the active power generation Pg r of the DFIG is a fixed quantity, imposed by the wind speed and with maximum value 1.01 p.u. Note how, in Fig. 4, the active and reactive powers at the inverter terminal have an opposite direction to those shown in Fig. 3. In (25)–(34), the HVdc model is explicitly represented in the optimization problem, as considered in Section III. Equations (25) and (26) calculate the Udr and Udi dc voltages of the HVdc link. In these equations, the current Id is replaced by (19). The active and reactive powers at the dc terminals are represented in (27)–(30). Equations (31)–(32) show the relationship between angles αr and αi and angles ϕr and ϕi of the rectifier and inverter, respectively. Equation (35) takes into consideration the DFIG-based wind turbine capability limits, this equation is represented as an inequality constraint that remains active during the optimization process. AC voltage (Vr and Vi ) limits for both ac terminals are considered in (36). Typically, these limits are imposed by the SO. Tap positions (ar and ai ) in (37) are adjusted between limits, in order to maintain the voltage range imposed by (36). Equation (38) represents the maximum and minimum active powers of the aggregated model’s generator, and (39) represents the maximum and minimum active power at the rectifier and inverter sides, respectively. Equation (40) represents the maximum and minimum reactive power at the rectifier and inverter sides, respectively. Equation (41) shows the maximum and minimum limits of dc voltage that can operate at the rectifier and inverter sides, respectively. Equation (42) indicates the operating ranges of the ignition and extinction angles. According to [25], the equations that model the HVdc, in Section III, are adequate when µ is less than 60◦ . Thus, this variable is restricted between [0◦ ; 60◦ ] through (43). Finally, the limits of ϕr and ϕi are defined in (44). The optimization problem defined in (20)–(44) is characterized as a nonlinear optimization problem, solved by using the predictor–corrector primal–dual interior point method, as described in [30]. V. CASE STUDY To illustrate the proposed method of optimization, a wind farm of 600 MW and 230 kV has been modeled. This wind farm is connected to the HVdc link, as shown in Fig. 5, using a


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Fig. 5. Equivalent representation of OWF + HVdc bipolar configuration with 12-pulse bridge. TABLE I AC VARIABLES RANGE

Fig. 6. DC voltage in rectifier and inverter for three HVdc lengths (100, 150, and 200 km), and Y s = 0.1 p.u.

TABLE II DC-LINK VARIABLES RANGE

bipolar configuration with a 12-pulse bridge. The HVdc link has a rating of 600 MW and ±300 kV. Each converter has a phase commutation resistance equal to 5.65 Ω (0.0176 p.u.). The filters on both sides of the HVdc link are modeled as shunt admittance equal to 0.1 p.u. Active power PL r and PL i , and reactive power QL r and QL i are zero in all the simulations. The ac bases are: Vac base = 230 kV and Sbase = 600 MVA. The dc bases are defined by using the equations in Appendix A. For these parameters, the DFIG-based wind turbine total reactive capability limits are between −0.35 and 0.95 p.u. when the input active power available at wind turbine equals zero, according to Fig. 2. Other wind generator parameters are included in Appendix B. Tables I and II show the setting for the ac and dc variables, respectively. VI. SIMULATION RESULTS As previously stated, active power generation Pg r of the DFIG is considered as a fixed value in the optimization problem. This

value mainly depends on the wind speed and the settings of the DFIG. To simulate the model’s behavior for all operational wind speeds, optimization problem (20)–(44) was sequentially solved for Pg r between [0.01; 1.01] (in p.u.), with an increase step power of 0.01 p.u. Three HVdc with different lengths were considered in the simulations: 100, 150, and 200 km. When the resistance per kilometer in the HVdc cable is 0.0217 Ω/km, the equivalent resistance Rcc of the equivalent HVdc model adopts the values of R1 = 2.17 Ω (0.0034 p.u.), R2 = 3.26 Ω (0.0051 p.u.), and R3 = 4.34 Ω (0.0067 p.u.), respectively, for the three HVdc distances. Filter admittance Ys of the HVdc link is also an important parameter of the model, limiting the suitable range of the OWF + HVdc operation. The explicit constraints of the DFIG (as shown in Section II) restrict the reactive power that can be delivered by the DFIG. Three simulations of the model’s behavior when Ys equals 0.1, 0.2, and 0.3 p.u, respectively, are analyzed in this section. Fig. 6 shows the curves of dc-voltage variation on both sides of the HVdc link for the three HVdc distances, when the active power injected by the DFIG Pg r varies from 0.01 to 1.01 p.u., and the shunt admittance Ys is 0.1 p.u. As shown in Fig. 6, in order to maintain the active and reactive balances in the ac bus of the HVdc rectifier side [restricted by the capability limits of the DFIG; see (35)] and to minimize the active power losses during operation, the optimization problem reaches optimal solutions when maximizing Udr . This result is coherent with the theory. The active power losses during operation increase with the equivalent of the HVdc link (Rcc ) and the dc current Id . In each simulation, the power injected by the DFIG (Pdr ) is a fixed quantity. Therefore, reducing dc current Id implies increasing dc voltage Udr , as expressed in (11). Following (27), larger injected wind powers may decline the inverter voltage Udi . This reduction is more significant when increasing the equivalent resistance (and the length) of the dc line. For (11) and (27), the dc current in the HVdc link is solely dependent on the active power injected by the DFIG (as previously

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Fig. 7. DC-link current and active power losses at the HVdc link for three HVdc lengths (100, 150, and 200 km), and Y s = 0.1 p.u.


Fig. 9. Reactive power generations and consumption in the rectifier converter, filter, and DFIG generator with three values of the admittance filter, HVdc length = 150 km.

Fig. 8. Reactive power in rectifier converter side, filter, and generator for three HVdc lengths (100, 150, and 200 km), and Y s = 0.1 p.u.

Fig. 10. AC voltage in the rectifier converter for three HVdc lengths (100, 150, and 200 km), and Y s = 0.1 p.u.

said, for optimal solutions, the value of Udr is in all simulations the maximum value, 1.10). Therefore, in Fig. 7, the dc-current magnitude Id increases with the power transmitted by the HVdc link, regardless of the equivalent dc line resistance. The active power losses, as expected, depend on the HVdc line resistance. Fig. 8 shows the reactive power balance in the ac bus of the HVdc rectifier side, for the three HVdc lengths, Ys = 0.1 p.u. HVdc links require, for optimal operation, the injection of reactive power on the rectifier side. The passive filter generates a fixed quantity of the reactive power needed by the HVdc-link operation. The DFIG cooperates in the optimal HVdc operation, varying its reactive power injection when the active power generation is modified. Note that the total reactive capability limits of the DFIG are met for all the range of active power input, (35). The effect of the different values of the shunt admittance on the reactive power balance is discussed in Fig. 9. Fig. 9 shows the changes in reactive power generation and consumption in the filter, rectifier converter and generator for different reactance Ys values. Three admittance values of the

filter Ys are considered: Ys1 = 0.1 p.u., Ys2 = 0.2 p.u., and Ys3 = 0.3 p.u. As previously stated, the HVdc has a Rcc = 3.26 Ω (150 km). When the admittance of the filter falls, its reactive power injection decreases, as shown in Fig. 9. To maintain the balance of the reactive power, there is a higher demand of the DFIG, increasing the reactive power injection. In the present simulations, the DFIG can reach the reactive power requirements, and maintain both its operation within the capability curve (35) and the optimal HVdc-link reactive power consumption. This curve shows the important assistance the DFIG provides for the combined operation, allowing the optimal operation of the OWF + HVdc system. As shown in Fig. 10, the ac voltage on the rectifier side of the HVdc link adopts different optimal values in the simulations, depending on the wind power injection. The optimization problem calculates the optimal ac voltage needed to maintain the reactive power balance on this side (see Fig. 8). For most of the power values, the generator works with low ac voltages,


Fig. 11. AC voltage in rectifier converter for three values of the admittance filter, HVdc length = 150 km. (Y s 1 = 0.1 p.u., Y s 2 = 0.2 p.u., and Y s 3 = 0.3 p.u.)

reducing the magnetizing reactive power consumption. The ac voltage in the rectifier is not dependent on the HVdc length. Fig. 11 depicts the variation of this voltage with the admittance filter Ys value, for an HVdc length of 150 km. In Fig. 11, when Ys increases, the optimal solution requires high ac voltages on the rectifier side of the HVdc link to reach the reactive power balance. The discontinuities in the curves are due to the inequality restrictions reaching their limits, inducing changes in the best optimal operation. The optimal value of the dc voltage at the rectifier side is almost constant in all the simulations (see Fig. 6). Therefore, the ignition angle must vary, as expressed in (25), for suitable operation. Figs. 12 and 13 show the influence of the ignition angle on the reactive power consumption of the HVdc link (on the rectifier side), for the three HVdc lengths and admittance filters. The small ignition angle variation required for optimal operation, in all wind speed conditions, must be stressed. This result suggests the convenience to analyze the possibility of using a static rectifier, instead of the more expensive rectifier converter. The ignition angle for the rectifier operation is not variable in the optimal solutions of the simulations performed. If the ignition angle is set at zero, the solution does not change significantly. VII. CONCLUSION This paper presents an optimization method to calculate the optimal operation of an offshore wind farm connected to the grid through an LCC HVdc link. The problem has been formulated as a nonlinear optimization problem. Wind turbines have been represented as power injections constrained by the capability curve of the DFIGs. The operating limits of this kind of generator have been obtained by taking into account the maximum allowable stator and rotor currents and the steady-state stability limit of the generator. Results of the optimization problem for different wind speeds and parameter variation have shown the accuracy and robustness of the proposed algorithm. The obtained results

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Fig. 12. Reactive power of the rectifier converter versus ignition angle with three values of the admittance filter, HVdc length = 150 km. (Y s 1 = 0.1 p.u., Y s 2 = 0.2 p.u., and Y s 3 = 0.3 p.u.)

Fig. 13. Reactive power of the rectifier converter side versus ignition angle for three HVdc lengths (100, 150, and 200 km), and Y s = 0.1 p.u.

are coherent with those showed in previous works (see [3]–[7] and [15]–[22]). The results show that the DFIG’s control ability (mainly concerning its reactive power capability) is crucial in the optimal operation of OWF. The objective function is the maximization of the active power injected to the external grid. Other objective functions can be easily considered in further studies, to represent particular economic or security targets. Finally, it has to be remarked that the results suggest the convenience of considering the possibility of using a passive diode rectifier in the HVdc link, instead of the more expensive thyristor LCC rectifier. APPENDIX A PER UNIT SYSTEM A common base power Sbase is chosen for both the ac and dc systems. DC base quantities [29] Vdc base = Kb Vac base ,

Kb =

√ 3 2 nb π

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Idc base

√ 3 = Iac base Kb

Zdc base = Kb2 Zac base where nb is the number of series-connected bridges in a terminal. APPENDIX B WIND GENERATOR PARAMETERS

XM = 2.79 p.u.

|IR | = 1.34 p.u.

XS = 2.88 p.u. APPENDIX C

DERIVATION OF OVERLAP ANGLE µr AND µi The average direct voltage in rectifier Udr is given by Udr = ar Vr cos αr − rcr Id = ar Vr

cos αr + cos δ 2

(C1)

where δ = αr + µr . The power factor in [26] is approximately expressed as cos αr + cos δ . 2 Substituting (C.2) into (C.1) results in cos ϕr ≈

cos ϕr = cos αr −

(C2)

2rcr Id . ar Vr

(C3)

By equaling (C.3) and (C.2), followed by some algebraic manipulation, we obtain 2rcr Id δ = cos−1 cos αr − . (C4) ar Vr Hence µr = cos−1 Similarly, for µi µi = cos−1

cos αr − cos αi −

2rcr Id ar Vr

2rci Id ai Vi

− αr .

(C5)

− αi .

(C6)

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Miguel Eduardo Montilla-DJesus (M’08) received the B.S. degree in electrical engineering from the University of Los Andes, Mérida, Venezuela, in 1995, and the M.Sc. degree from the University Carlos III de Madrid, Madrid, Spain, in 2008. He is currently working toward the Ph.D. degree from the University Carlos III de Madrid, focusing on offshore wind farms and HVdc transmission. His current research interests include the control of doubly fed electrical machine and their use in renewable energy systems.

David Santos-Martin received the B.Sc. degree in electrical and electronic engineering from the Escuela Técnica Superior Industrial Engineering, Universidad Politécnica de Madrid, Madrid, Spain, in 1997, the M.Sc. degree in control engineering from ´ ´ The Ecole Supérieure d’Electricit´ e, Paris, France, and the Ph.D. degree from the University Carlos III de Madrid. He is currently an Assistant Lecturer with the Department of Electrical Engineering, University Carlos III de Madrid. Before joining University Carlos III de Madrid, he was with Iberdrola from 2001 to 2007, and with Ecotecnia-Alsthom from 2000 to 2001. His current research interests include focus on power electronics, application of power electronics to power systems, and advanced control techniques applied to the renewable energies.

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Santiago Arnaltes (M’02) received the Ph.D. degree in electrical engineering from the Polytechnic University of Madrid, Madrid, Spain, in 1993. Since 1997, he has been an Associate Professor with the Department of Electrical Engineering, University Carlos III de Madrid. His current research interests include grid integration of wind energy and control of electrical drives and flexible ac transmission system (FACTS), mainly for wind energy applications.

Edgardo D. Castronuovo (M’03–SM’07) received the B.S. degree in electrical engineering from the National University of La Plata, La Plata, Argentina, in 1995, and the M.Sc. and Ph.D. degrees from the Federal University of Santa Catarina, Santa Catarina, Brazil, in 1997 and 2001, respectively He was a Postdoctoral Fellow with Institute for Systems and Computer Engineering of Porto (INESC-Porto), Porto, Portugal. He was engaged with Centro de Pesquisas em Engenharia Eléctrica (CEPEL), Rio de Janeiro, Brazil and INESC-Porto, where he was involved in power system areas. He is currently a Professor with the Department of Electrical Engineering, University Carlos III de Madrid, Madrid, Spain. His research interests include optimization methods applied to power system problems and deregulation of the electrical energy systems.