Control Algorithm for Parallel Connected Offshore Wind Turbine Generators

Abstract

A control algorithm for Parallel Connected Offshore Wind Turbines with permanent magnet synchronous Generators (PCOWTG) is presented in this paper. The algorithm estimates the optimal collective speed of turbines based on the estimated mechanical power of wind turbines without direct measurement of wind speed. In the proposed topology of the wind farm, direct-drive Wind Turbine Generators (WTG) is connected to variable low-frequency AC Collection Grids (ACCG) without the use of individual power converters. The ACCG is connected to a variable low-frequency offshore AC transmission grid using a step-up transformer. In order to achieve optimum wind power extraction, the collective speed of the WTGs is controlled by a single onshore Back to Back converter (B2B). The voltage control system of the B2B converter adjusts voltage by keeping a constant Volt/Hz ratio, ensuring constant magnetic flux of electromagnetic devices regardless of changing system frequency. With the use of PI pitch compensators, wind power extraction for each wind turbine is limited within rated WTG power limits. Lack of load damping in offshore wind parks can result in oscillatory instability of PCOWTG. In this paper, damping torque is increased using P pitch controllers at each WTG that work in parallel with PI pitch compensators.

1. Introduction

The first offshore wind farm installed on the coast of Sweden, with a rated power of 220 kW, dates back to 1990. Since then, the number and size of offshore wind farms have constantly been increasing, and their installation has become increasingly remote from the mainland. The main reasons for installing offshore wind turbines are the reduction of visual and acoustic impact on the environment, less wind turbulence, and higher average wind speeds than onshore wind turbines. Possible grid topologies for offshore windfarms include wind parks with High Voltage AC (HVAC) offshore transmission lines, and wind parks with High Voltage DC (HVDC) offshore transmission lines [1,2,3]. Most offshore wind parks installed today use the HVAC offshore transmission grid. In offshore wind parks with HVAC transmission lines, Wind Turbine Generators (WTG) are connected to an AC collector grid using an AC/AC power converter. The offshore substation contains a step-up transformer by which the collector grid voltage is raised to the high voltage of the transmission AC grid. Onshore substations also have a transformer. Wind parks with HVDC transmission grids usually have an identical collector grid solution as wind parks with HVAC transmission lines. However, the offshore transmission line is DC, where the collector grid frequency is decoupled from the mainland AC frequency. In this case, offshore and onshore substations contain AC/DC and DC/AC power converters in addition to transformers.

Deployment of power electronics in harsh offshore environments has been identified as a major challenge associated with any offshore wind system. Recent trends in offshore wind industry have been directed toward minimising the complexity of offshore assets, aiming to provide an adequate level of reliability in operation of wind farms. In Parallel (directly) Connected Offshore Wind Turbine Generators (PCOWTG) wind park configuration, proposed in this paper, directly-driven Permanent Magnet Synchronous Generators (PMSG) are connected to variable low-frequency AC Collection Grid (ACCG) without using a power converter. By using a step-up transformer, the ACCG is connected to a Variable Low-Frequency offshore AC transmission grid (VLFAC). The Back to Back (B2B) power converter acts as the point of connection for the VLFAC grid to the mainland grid, regulating the Wind Turbine Generator’s (WTG) collective speed and voltage at the B2B Point Of Interconnection (POI) to the VLFAC grid. The control system of the mainland grid side part of the B2B converter is responsible for controlling the DC link voltage of the B2B converter and reactive power injected (absorbed) to (from) mainland grid. DC link voltage of the B2B converter, without decreasing generality, is regarded as constant, and consequently, the control system of the mainland grid side part of the B2B converter is outside the interest of this paper.

A similar wind park (farm) configuration is proposed and analysed in [4,5] with the constant speed operation of Wind Turbine Generators. Another similar configuration of offshore wind parks under the name “slim wind turbine concept” is proposed and analysed in [6] at the level of feasible study, without designing the system’s controller. This paper extends this concept by developing a detailed simulation model using the Matlab/Simulink and Simscape Specialized Power System toolbox, designing the closed-loop control system and solving the problem of undamped oscillations of rotor angles. Each component of the system is carefully designed to make it as close to real-world components as possible. Due to the absence of gearboxes and offshore power converters, the proposed wind park configuration concept decreases both capital expenditure (CAPEX) and operational expenditure (OPEX), reduces subsystem complexity, and increases overall system reliability [7,8]. According to [9,10], the main reasons for offshore wind park outages are gearboxes and power electronics failures.

Underwater offshore AC cables have very high shunt capacitance, so the use of HVAC transmission cables is limited to 50–80 km distances due to large charging currents [11]. Consequently, the cables’ capability to transfer active power decreases either due to thermal limits or voltage stability. The simplified formula for charging cable currents calculation is:

$$I_c=2\pi fClV$$

(1)

where f is frequency; C is capacitance of the cable per unit length; l is the length of the cable; V is voltage. From (1), it can be seen that the reduction of cable charging currents is achieved by decreasing system frequency. Reactive power generated by the transmission cable with prevailing capacitive reactance by simplification can be calculated as:

$$Q_c=2\pi fCl{V}^2$$

(2)

From (2), it can be seen that Q_c depends on the product of frequency and cable length. Accordingly, the transmission cable length can be increased by decreasing system frequency for the same reactive power generated by the cable.

The typical choice for nominal frequency in Low-Frequency AC systems (LFAC) is 50/3 Hz. This frequency in some countries is used for railway power supply (fractional frequency transmission systems). LFAC transmission systems are proposed and analysed in many papers [12,13,14]. In [15], a comparison of energy losses, total CAPEX and OPEX of LFAC and HVDC system was performed. It has been concluded that LFAC, with the extended length of the transmission, is comparable to HVDC (CAPEX costs are 6% lower than HVDC, and OPEX costs are 3% lower than HVDC). In a general case, the wind speed on each wind turbine may be different; this means that, in PCOWTG wind park configuration, the MPPT of the individual WTG’s cannot be achieved. The decreased amount of extracted wind power of individual WTs is compensated due to the absence of thermal losses of individual WTG power converters because they are not present in this configuration. A simpler system with less equipment also reduces the number of potential failures and reduces WTG’s downtime. Consequently, it can be concluded that the proposed configuration not only reduces the CAPEX and OPEX but could also potentially improve the Levelised Cost of Energy (LCOE).

A medium-frequency windfarm and modular multi-level converter-based high-voltage direct current transmission (MMC-HVDC) integration system are proposed in [16]. One of the features of the proposed scheme is the adoption of the rectifier station and inverter station due to high scalability and low switching frequencies with satisfactory output harmonics characteristics. One of the most important features of the proposed scheme is the increase of the operating frequency of the offshore AC system to the medium-frequency range (100–400 Hz) instead of the conventional 50/60 Hz.

The feasibility of employing the LFAC system for subsea transmission and distribution of 58 MW power is investigated in [17]. The authors developed simulation model of the LFAC-based subsea transmission and distribution system, which includes a hexverter as a frequency converter. A novel control strategy was developed to optimise its zero-sequence circulating current. The authors designed a dedicated control system to reduce the impact of real-time disturbances, such as wind speed fluctuations and harmonics due to heavy inductive load operating at 16 Hz.

In [18], the main characteristics of the control algorithm of PCOWTG are presented. The control algorithm is applied to an offshore wind park with a transmission grid 20 km in length and used for the limited wind speed region between cut-in wind speed and nominal wind speed. The wind speed is estimated from the Wind Turbine (WT) speed and active power of the Wind Generator (WG). Due to lack of load damping, a significant amplitude of oscillations of WG rotor angles was observed.

The rest of the paper is organised as follows. Section 2 provides an overview methodology of the paper. A detailed PCOWTG system description is given in Section 3, including a problem statement and control system architecture. In Section 4, the proposed PCOWTG control system is validated based on a realistic wind park simulator with a detailed discussion of the simulation results. Finally, Section 5 summarises concluding remarks and provides indications for further research.

2. Methodology

In this paper, the control algorithm is implemented on PCOWTG with a 150 km length of transmission grid. Additionally, a control algorithm is applied in case of wind speed changes from cut-in wind speed to above nominal wind speed. Instead of using the WGs active power, which was the case of the PCOWTG control system presented in [18], the calculated mechanical power of the WTG is used for estimating the wind speed of each WT. By using a P pitch controller with adequately adjusted parameters, the damping torque of the WTG is increased, and significant deviations of rotor angles are prevented, ensuring wind generators’ angle stability. In order to compensate for the reactive power generated by the transmission cable, shunt reactors are added on both sides of the transmission grid cable.

In order to achieve optimal wind power extraction (Maximum Power Point Tracking—MPPT), the collective rotational speed of WTGs is controlled by the B2B converter depending on wind speed. In order to keep magnetic flux in electromagnetic devices (transformers etc.) constant regardless of changes in system frequency, constant Volt/Hz ratio operation is adopted. This control strategy is well known in scalar control of asynchronous motors. It is also proposed in some papers that consider the variable frequency operation of wind parks [16,17], where the authors implemented the constant Volt/Hz operation on the AC collection grid for offshore wind with an HVDC transmission grid. Additionally, the wind speed is measured, which was not estimated in the approach proposed in [19,20]. In this paper, constant Volt/Hz operation is implemented in both collection and transmission offshore grids. In order to find the optimal collective rotational speed of WTG, the wind speed at each wind turbine is estimated by applying the original method, which is presented in this paper for the first time.

3. PCOWTG System Description

3.1. Overview

A schematic diagram of PCOWTG is presented in Figure 1. In Figure 1, n is the number of WTGs (or the number of clusters of WTGs); V_r is the space vector of regulated voltage (voltage at POI); I is the space vector of converter current; θ_f1, θ_f2, …, θ_fn are PMSGs rotor field angles; ω_t1, ω_t2, …, ω_tn are rotational speeds of WTGs; β₁, β₂, …, β_n are pitch angles of WTs; ω_opt is the optimal collective speed of WTGs in wind parks; ω_t is the collective (average) speed of WTGs in wind parks; I₁, I₂, …, I_n are the space vectors of currents of wind generators; V₁, V₂, …, V_n are the space vectors of voltages of wind generators; P_mc1, P_mc2, …, P_mcn are estimated mechanical powers of WTs; v_w1, v_w2, …, v_wn are wind speeds of each WT; R_c and L_c represent resistance and inductance for the converter coupling inductor, respectively. The rotor field angles and turbine speeds are measured using encoders.

3.2. Problem Statement

Generally speaking, the PCOWTG is a complex nonlinear multiple input multiple output systems with spatially distributed elements and a random source of mechanical (wind) energy. Due to the variable speed operation, the reactance of electrical elements of the system is changing with the change of system frequency, causing the analysis of the system and control system design to be more complex. Although the exclusion of power converters at WTGs reduces overall system complexity, at the same time, it reduces the controllability of the system. Consequently, the design of a control system of PCOWTG is a demanding task. The main controlled variables of the system are the collective speed of WTGs, the voltage at POI of the B2B converter, and pitch angles of WTs. The PCOWTG control system should simultaneously ensure the system’s angle stability of WGs and voltage stability, ensuring that voltages and currents are within rated limits of all equipment. Reactive power flow should be minimised in order to reduce thermal losses. Additionally, by adjusting the speed of the turbines to the wind speed, maximum extraction of wind power should be ensured. In order to accomplish these tasks, the novel control system architecture combined with reactive power compensation and constant Volt/Hz operation is proposed.

3.3. Control System Architecture

A block diagram of the offshore B2B converter control system of the Voltage Source Converter (VSC) on the wind park side is presented in Figure 2. Inputs into the control system are measured using WTG speeds and PMSG rotor field angles, space vectors of currents from wind generators (that is, phase currents of PMSGs), space vectors of regulated voltage V_r (that is, phase voltages of regulated voltage), and space vectors of the current I of the B2B converter (that is, phase currents of B2B converter). The control system is realized as a multi-loop control system. The inner control loops represent a decoupled current controller of direct (d) and quadrature (q) components of the current converter. The outer loops realize the WTG (collective) Speed controller and the B2B converter Voltage controller. The WTG Speed controller calculates reference I_d current command (I_dcmd) for the Current controller while the Voltage controller calculates I_q current command (I_qcmd) for the Current controller (see Figure 3).

The WTG Speed controller, the Voltage controller, and the Current controller are realized in a rotating dq frame, where d axis is aligned with the space vector of voltage V_r. The Phase-Locked Loop (PLL) is used to calculate system frequency and calculate angle θ_r (angle of space vector V_r) of the abc to dq transformation. In this way, the q component of voltage V_r equals zero. Consequently, active and reactive powers of the B2B converter are (per unit—pu)

$$P=V_{rd}{I}_d$$

(3)

$$Q=-V_{rd}{I}_q$$

(4)

where V_rd is the d component of regulated voltage V_r; I_d and I_q are the d and q components of the B2B converter currents, respectively. The reference value of the regulated voltage V_r is found according to a constant Volt/Hz ratio operation using the formula:

$$V_{ref}=\frac{V_{nom}}{{\omega }_{tn}}{\omega }_t$$

(5)

where V_nom is the nominal converter voltage (pu); ω_tn is the nominal WTG’s rotational speed (pu), and ω_t is the WTG’s (collective) rotational speed (pu). The PI Voltage controller calculates reactive power order Q_ord while I_{q cmd} signal is found using Formula (4). The reference value of WTG collective speed ω_opt is found by the Calculator block shown in Figure 2. The Calculator block calculates WT mechanical power, wind speed at each WT, and optimal collective speed of wind turbines ω_opt. WTG Speed controller calculates active power order P_ord, by means of the I_{d cmd} signal found using Formula (3).

The block diagram of a decoupled Current controller of the B2B converter is shown in Figure 4. In Figure 4, R_c and L_c represent resistance and inductance of the converter coupling inductor, respectively; ω is the system rotational frequency. The “P, Q priority” block in Figure 4 limits the I_q converter current component (P priority) or I_d converter current component (Q priority) in order to limit the converter current according to the P, Q priority algorithm described in Appendix A, Figure A1. In the same Figure, V_{d cmd} and V_{q cmd} represent command signals for the d and q components of voltage V_c for the B2B converter, as shown in Figure 1. These components are inputs into the PWM modulator of the B2B converter. I_d and I_q represent d and q components of the space vector for current I of the B2B converter.

The converter coupling circuit is driven by the following set of equations:

$$\begin{gathered} \frac{d{I}_d}{dt}=\left(V_{cd}-V_{rd}-R_c{I}_d+\omega L_c{I}_q\right)/L_c \\ \frac{d{I}_q}{dt}=\left(V_{cq}-V_{rq}-R_c{I}_q-\omega L_c{I}_d\right)/L_c \end{gathered}$$

(6)

From this set of equations, receiving a block diagram of the decoupled Current controller of the B2B converter is straightforward, as shown in Figure 4. The block diagram of a decoupled Current controller of B2B, as shown in Figure 4, is a “standard“ block, which can be found in many textbooks discussing power converters in wind energy conversion systems and is not be further explained.

Figure 5 shows the block diagram representing the Pitch control system. It consists of a PI Pitch compensator and a P pitch controller, which work in parallel. If the extracted mechanical power of WT is larger than rated WT power P_rated, then the PI pitch compensator increases pitch angle β and limits extracted mechanical power to P_rated. The algorithm for estimating WT mechanical power P_mci (P_mcipu is an abbreviation for P_mci pu) is explained in Section 3.4. The P Pitch controller increases the damping torque of WTG, preventing oscillatory instability of wind generators. The reference signal for this controller is an averaged rotational speed of WTG ω_t. In Figure 5, P_mci and ω_ti represent the estimated mechanical power and rotational speed of ith WT, respectively. In order to mitigate blade passing effects and prevent excessive pitch activity in the practical realization of the proposed control system, adaptive notch filters for both error signals in Figure 5 should be used, i.e., adaptive notch filter blocks should be added before the P and PI blocks. The notch frequencies of these filters should be selected as multiples of the WTG’s rotational speed and as the resonant frequencies of the WTG’s drive system.

3.4. Estimation of WTG Mechanical Power, Wind Speed and Optimal WTG Speed

The extracted mechanical wind power P_mi of ith WT is provided as:

$$P_{mi}=\frac{1}{2}\rho A{v}_{wi}^3{C}_p\left({\lambda }_i,{\beta }_i\right), i=1, 2, 3,\dots , n$$

(7)

where: ρ is air density (kg/m³); A is the area swept by blades of WT (m²); v_wi is wind speed at ith WT (m/s); C_p is the power coefficient (coefficient of performance); λ_i is the tip speed ratio (ratio of speed at the tip of blades and wind speed); β_i is the pitch angle (deg); n is the number of WTs. In Equation (7), it is assumed that uniform air density over the area of the offshore wind park and all WTGs are identical. According to [21], the power coefficient is provided as:

$$c_p\left({\lambda }_i,{\beta }_i\right)=c_1\left(c_2\frac{1}{K_i}-c_3{\beta }_i-c_4{\beta }_i^{c_5}-c_6\right)e^{-c_7\frac{1}{K_i}}$$

(8)

$$\frac{1}{K_i}=\frac{1}{{\lambda }_i+c_8{\beta }_i}-\frac{c_9}{1+{\beta }_i^3}$$

(9)

where c₁, c₂, …, c₉ are constants that depend on the type of WT used. Different approximations of Formula (8) can be found in the literature. For convenience, polynomial approximations provided in [22] are used in this paper:

$$C_{pi}={\displaystyle \sum }_{j=0}^4{\displaystyle \sum }_{k=0}^4{\alpha }_{jk}{\lambda }_i^k{\beta }_i^j={\displaystyle \sum }_{j=0}^4{\displaystyle \sum }_{k=0}^4{\alpha }_{jk}{\left(\frac{R{\omega }_{ti}}{v_{wi}}\right)}^k{\beta }_i^j , i=1, 2, 3,\dots , n$$

(10)

where R is the radius of the blade of WT (m); ω_ti is ith WTG speed (rad/s); and coefficients α_jk are provided in [22]. These coefficients depend on the type of WT used. If coefficients α_jk are chosen according to [22], the maximum of power coefficient C_pmax = 0.517 at tip speed ratio λ_opt = 8.805 and β_i = 0 (deg) is achieved.

A common control strategy of variable speed WTG is shown in Figure 6, which is also applied in the PCOWTG control system [23]. In the region of wind speed [0, v_wcut-in] (region I), the wind turbine is turned off by moving the blades toward the feather position. When the wind speed is between v_wcut-in and the base wind speed v_wb (region II), the WTG control system adjusts WTG speed in a way that achieves optimum tip speed ratio:

$${\lambda }_{opt}=\frac{R·{\omega }_t}{v_w}=\frac{R·{\omega }_{tn}}{v_{wb}}$$

(11)

where at v_w = v_wb nominal WTG speed ω_tn is achieved. In this region, the PCOWTG control system keeps pitch angle β_i = 0 (deg). In this way, the optimum mechanical power (P_m) extraction from wind power is achieved. At v_w = v_wb, the extracted mechanical power of WTG is P_mb.

When the wind speed is between v_wb and nominal wind speed v_wn (region III), the PCOWTG control system keeps WT speed at nominal level ω_tn and pitch angle at 0 deg. Consequently, the tip speed ratio is not at the optimal level; that is, C_p is slightly below C_pmax. This region was introduced to achieve a better transition from region II to region IV. In region IV, the PCOWTG control system limits extracted mechanical power to nominal WTG mechanical power P_mn while keeping WTG speed at nominal level ω_tn; this is achieved by increasing the pitch angle. In region V, WTG is turned off by moving the blades toward the feather position.

According to previous considerations, it follows:

$$P_{mbpu}=\frac{\frac{1}{2}\rho A{v}_{wb }^3{C}_p\left({\lambda }_{opt}=\frac{R·{\omega }_{tn}}{v_{wb}}, \beta =0^0 \right)}{S_b}=\frac{\frac{1}{2}\rho A{v}_{wb }^3{C}_{pmax}}{S_b}$$

(12)

where S_b is base system power (W). The subscript “pu” in (12) is added to differentiate per-unit quantities. In direct-drive, WTG generator speed ω_g equals turbine speed ω_t. If at base WTG speed is chosen ω_b = ω_tn, then for any wind speed and ith WTG we have:

$$P_{mipu}=\frac{1}{2}\rho A{v}_{wi }^3{C}_p\left({\lambda }_i={\lambda }_{opt}\frac{\frac{{\omega }_t}{{\omega }_b}}{\frac{v_{wi}}{v_{wb}}}, {\beta }_i \right)/S_b=\frac{1}{2}\rho A{v}_{wi }^3{C}_p\left({\lambda }_{opt}\frac{{\omega }_{tpu}}{v_{wipu}}, {\beta }_i \right)/S_b$$

(13)

where ω_tpu is the collective rotational speed of turbine per-unit of the base rotational speed ω_b; v_wipu is wind speed of ith WT per-unit of base wind speed v_wb. By combining (12) and (13) we have:

$$P_{mipu}=v_{wipu }^3·\frac{1}{C_{pmax}}·C_p\left({\lambda }_{opt}\frac{{\omega }_{tpu}}{v_{wipu}}, {\beta }_i \right)·P_{mbpu}$$

(14)

The overall extracted mechanical power of the wind park, which consists of n parallel connected WTG, is:

$$P_{mpu}={\displaystyle \sum }_{i=1}^n{P}_{mipu}=\frac{P_{mb}}{C_{pmax}}·\left(v_{w1pu}^3{c}_{p1}+\cdots +v_{wipu}^3{c}_{pi}+\cdots v_{wnpu}^3{c}_{pn}\right)$$

(15)

where

$$c_{pi}=c_p\left({\lambda }_{opt}·\frac{{\omega }_{tpu}}{v_{wipu}},{\beta }_i\right)={\displaystyle \sum }_{j=0}^4{\displaystyle \sum }_{k=0}^4{\alpha }_{jk·}{\left({\lambda }_{opt}·\frac{{\omega }_{tpu}}{v_{wipu}}\right)}^k·{\beta }_i^j,i=1, 2,\dots ,n$$

(16)

From (14) it follows:

$$\frac{P_{mipu}}{P_{mbpu}}·v_{wipu}={\displaystyle \sum }_{j=0}^4{\displaystyle \sum }_{k=0}^4{\alpha }_{jk·}{\left({\lambda }_{opt}·{\omega }_{tpu}\right)}^k·v_{wipu}^{4-k}·{\beta }_i^j$$

(17)

Formula (17) can be written in the form:

$$\begin{gathered} c_{i0}{v}_{wipu}^4+c_{i1}·{\lambda }_{opt}·{\omega }_{tpu}·v_{wipu}^3+c_{i2}·{\left({\lambda }_{opt}·{\omega }_{tpu}\right)}^2·v_{wipu}^2+ \\ \left[c_{i3}·{\left({\lambda }_{opt}·{\omega }_{tpu}\right)}^3-\frac{P_{mipu}}{P_{mbpu}}\right]·v_{wipu}+c_{i4}·{\left({\lambda }_{opt}·{\omega }_{tpu}\right)}^4=0 \end{gathered}$$

(18)

where $c_{ik}={\alpha }_{ok}+{\alpha }_{1k}{\beta }_i+{\alpha }_{2k}{\beta }_i{}^2+{\alpha }_{3k}{\beta }_i{}^3+{\alpha }_{4k}{\beta }_i{}^4, k=0, 1,\dots , 4, i=1, 2,\dots ,n$.

The left-hand side of Equation (18) is a polynomial of the 4th order in v_wipu. Positive zero of this polynomial, between v_cut-in and v_cut-out, represents a wind speed of ith WT. The issue with Equation (18) is that mechanical power P_mipu of WT is challenging to measure. In this paper, the estimation of the mechanical power of WT by using the WTG swing equation is proposed as follows:

$$2H\frac{d{\omega }_{tipu}}{dt}+D{\omega }_{tipu}=T_{mipu}-T_{eipu}$$

(19)

where T_eipu is the electromagnetic torque (pu) of ith WG; T_mipu is the mechanical torque (pu) of ith WT; H is WTG inertia constant in seconds (MWs/MVA), and D is the WTG damping coefficient. Electromagnetic torque of a non-salient pole PMSG can be found using formula:

$$T_{eipu}={\lambda }_{fpu}{i}_{qipu}$$

(20)

where λ_fpu is rotor flux linkage (pu) produced by permanent magnets; i_qipu (pu) is q component of ith PMSG stator current in the rotor field synchronous dq frame, whose angle θ_fi is measured using an encoder.

From (19), it follows that the mechanical power of ith WT can be found from:

$$P_{mipu}=\left(2H\frac{d{\omega }_{tipu}}{dt}+D{\omega }_{tipu}+T_{eipu}\right){\omega }_{tpu}$$

(21)

Since the filtered derivation (approximation of ideal derivation) is used in simulations, the estimated P_mcipu is an approximation of P_mipu.

After estimating wind speed at each WT, the optimal collective speed of WT can be found from (18) as follows:

$$\begin{array}{c}{P}_{mipu}=P_{mbpu}·[c_{i0}{v}_{wipu}^3+\left(c_{i1}·{\lambda }_{opt}·v_{wipu}^2\right)·{\omega }_{tpu}+\left(c_{i2}·{\lambda }_{opt}^2·v_{wipu}\right)·{\omega }_{tpu}^2\\ +\left(c_{i3}·{\lambda }_{opt}^3\right){\omega }_{tpu}^3+\left(c_{i4}·{\lambda }_{opt}^4·\frac{1}{v_{wipu}}\right){\omega }_{tpu}^4]\end{array}$$

(22)

The overall extracted mechanical power of the wind park is:

$$\begin{array}{c}{P}_{mpu}={\displaystyle {\displaystyle \sum }_{i=1}^n}{P}_{mi}=\\ =P_{mbpu}·[c_{10}{v}_{w1pu}^2+\cdots +c_{n0}{v}_{wnpu}^2)+(c_{11}{v}_{w1pu}^2+\cdots c_{n1}{v}_{wnpu}^2)·{\lambda }_{opt}·{\omega }_{tpu}+(c_{12}{v}_{w1pu}+\cdots c_{n2}{v}_{w2pu})\\ ·{\lambda }_{opt}^2·{\omega }_{tpu}^2+ \left(c_{13}+\cdots +c_{n3}\right)·{\lambda }_{opt}^3·{\omega }_{tpu}^3+\left(c_{14}·\frac{1}{v_{w1pu}}+\cdots +c_{n4}·\frac{1}{v_{wnpu}}\right)·{\lambda }_{opt}^4·{\omega }_{tpu}^4]\end{array}$$

(23)

The right-hand side of (23) is the fourth-order polynomial in ω_tpu. The maximum of P_m is achieved for ω_t = ω_opt, which can be determined by using derivate or non-derivate optimisation algorithm to find the maximum value of the right-hand side of (23). The estimated ω_opt is used as a reference value of the WTG Speed controller. A schematic diagram of the Calculator block is shown in Figure 7. In Figure 7, variable s represents the Laplace operator and T_f is the time constant of a low-pass filter.

4. Case Study

4.1. System Parameters

The simulations are based on realistic models of all wind park components using Matlab/Simulink and Simscape Specialized Power System toolbox. The wind park consists of 3 WTGs to only simulate the main characteristics of the system. The model of the WTGs is based on the NOWITECH 10 MW direct-drive WTG model [24]. In this model, the optimum (maximum) tip speed ratio is 7.8, the number of poles of the WG is 198 and rated WTG speed is 13.54 rpm. In order to comply with the new maximum (optimum) tip speed ratio λ_opt = 8.805 used in this paper, some parameters of WTG were changed, which is explained in the following text. These changes do not influence the main results achieved in this paper in any way. Since it is supposed that system nominal frequency is f_n = 16.67 Hz, it follows that the nominal WTG speed is:

$${\omega }_{tn}=\frac{2·\pi ·16.67}{number of pair poles}\frac{rad}{s}$$

(24)

For convenience, the number of poles equal to 130 is proposed in this paper. Accordingly, it follows ω_tn = 1.61 rad/s = 15.39 rpm. It is also supposed that the base wind speed is v_wb = 12 m/s.

From (11) the radius of WT blades is found to be R = 65.75 m, while WT mechanical power, achieved at base wind speed, is P_mbpu = 0.755 pu found using (12). From (18), by setting P_mipu = 1 pu, nominal wind speed is equal to v_wn = 13.24 m/s. Since we changed the nominal WTG speed compared with the original NOWITECH WTG, the inertial constant of WTG was also changed to H = 3.48 MWs/MVA. It is also assumed that the air density is ρ = 1.25 kg/m³. The rest of the parameters of the WTG are shown in

Table 1. WTG parameters.

WTG Nominal/Rated Parameters	Value	Unit
Power output	10	MW
Line to line RMS voltage	4	kV
Stator frequency	16.67	Hz
Number of pole pairs	65	-
Phase resistance	36.6	mΩ
Phase inductance	5.29	mH
Rotor flux linkage	1	pu
Inertial constant	3.48	MWs/MVA
Damping coefficient	0.01	pu

. The base system power is S_b = 10 MVA.

The parameters of transformers T_r1, T_r2 and T_r3, are similar to parameters of transformers in the Matlab/Simulink “power_wind_dfig” example, which are given in

Table 2. Transformer parameters.

Transformer Rated Parameters	Tr1	Tr2	Tr3
Power	12 MVA	47 MVA	47 MVA
Frequency	16.67 Hz	16.67 Hz	16.67 Hz
Primary winding inductance	0.025 pu	0.08 pu	0.08 pu
Primary winding resistance	0.025/30 pu	0.08/30 pu	0.08/30 pu
Secondary winding inductance	0.025 pu	0.08 pu	0.08 pu
Secondary winding resistance	0.025/30 pu	0.08/30 pu	0.08/30 pu
Magnetization inductance	500 pu	500 pu	500 pu
Magnetization resistance	500 pu	500 pu	500 pu

. Resistance and inductance of the converter coupling inductor are R_c = 0.0015 pu and L_c = 0.15 pu. These values are also taken from the same example.

The transmission and collection cables used are three core cross-linked polyethylene (XLPE) cables, modelled using a three-phase π section line in Matlab/Simulink. The length of the transmission network used in this paper is 150 km, and the lengths of collection network lines are 1 km, 3 km, and 5 km. The parameters of the XLPE cables are given in

Table 3. ABB XLPE cables parameters.

Cable Parameters	Collection Line			Transmission Line	Unit
	LINE 1	LINE 2	LINE 3
Length	1	3	5	150	km
Nominal voltage	30	30	30	132	kV
Cross section area	70	70	70	185	mm2
Resistance	0.2956	0.2956	0.2956	0.119	Ω/km
Inductance	46	46	46	47	mH/km
Capacitance	0.16	0.16	0.16	0.13	μF/km

[25]. In order to compensate for the reactive power generated by the cables, shunt reactors L_s1 = 0.5 pu and L_s2 = 1.25 pu were applied on both ends of the transmission grid cable. All simulations were performed based on per-unit system base quantities. Simulation time was set to 360 s.

4.2. Simulation Results

4.2.1. Steady-State Analysis

Figure 8 shows voltages of nodes from the system at a steady-state for different wind speeds. In Figure 8, E_int1 represents the internal voltage of PMSG1. Approximate reactive power absorbed by reactors L_s1 and L_s2 can be represented as:

$$Q_L=\frac{V^2}{2\pi f\left(L_{s1}+L_{s2}\right)}$$

(25)

where f is system frequency and V is the voltage applied to shunt reactors. Since constant Volt/Hz ratio operation is employed and system frequency is proportional to wind speed, it follows:

$$V\sim f\sim v_w$$

(26)

From (2), (25) and (26) it follows:

$$Q_c\sim v_w^3, Q_L\sim v_w$$

(27)

When wind speed decreases, capacitive reactive power production decreases at a higher rate than the decreasing inductive reactive power consumption; this is the reason why, in Figure 8, by decreasing wind speed, the voltage at the inner nodes of the system decreases to a more significant extent than voltages at the outer (generators and converter) nodes of the system. In order to achieve even better voltage profiles than those presented in Figure 8, tapped reactors can be used [26] (p. 630).

4.2.2. Dynamic Responses

Figure 9 displays wind profiles applied to the wind park; as shown in Figure 9, the wind speed changes from 9.5 m/s to 14 m/s. In this way, wind speed varies from region II to region IV (see Figure 6), in which the control system is active. Regions I and region V were not used since WTGs are shut down in these regions. Additionally, the wind profiles are delayed relative to each other to consider the spatial distribution of WTGs and the limited travel speed of the wind disturbance front. The time between decreasing and increasing wind profiles is long enough to allow the system to settle in a steady-state.

Deviations of rotor angles around the average rotor angle of all WTGs in the wind park, and responses from the B2B converter current and WGs currents for using the control system of PCOWTG with a disabled P pitch controller are shown in Figure 10 and Figure 11, respectively.

Due to the lack of load damping, oscillations of rotor angles in Figure 11 increase, leading to system instability (unbounded oscillations). In this case, simulation time is shortened to 260 s since, due to oscillatory instability, the signals rapidly increase after 220 s.

Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 show simulation results with the enabled P pitch controller. Figure 12 shows the power coefficients of WTs. It can be seen in the steady-state and for the case v_w ≤ v_wn, that power coefficients are close to the optimal value C_pmax = 0.517. Pitch angles of WTs are shown in Figure 13. When wind speed is below v_wn, it can be seen that pitch angles are equal to zero for most of the time. During wind disturbance, P pitch controllers slightly increase their pitch angles in order to damp rotor angle oscillations. If wind speed is above v_wn, PI pitch compensators increase their pitch angles to limit the extracted mechanical power of WT to a rated power of WTGs.

The reference (optimal) speed (estimated by the Calculator block) and speeds of WTGs are shown in Figure 14. Since PMSGs rotate synchronously, only a one-time response from the speed of WTGs can be observed. It should be emphasised that the speeds of WTGs may be slightly different during disturbances of active power, that is, during wind disturbances. Otherwise, WTGs rotate with the same collective speed or ω_t.

Waveforms of regulated voltage V_r and PMSG voltages V₁, V₂, and V₃ are presented in Figure 15. It can be seen that, due to “electrical vicinity” and voltage control by the B2B converter, the voltages of all generators and the B2B converter are in close proximity to each other. Figure 14 and Figure 15 show that waveforms of the WTG’s speed, voltages of WTGs and the B2B converter are very similar, which is to be expected. The internal voltage of WG is inherently proportional to WTG speed, so the time constant of the internal voltage of WG equals the mechanical time constant of WTGs. Due to constant Volt/Hz ratio operation at steady-state, regulated voltage V_r is proportional to WTG speed. Since we have chosen the same parameters for the PI WTG Speed controller and the PI Voltage controller, the waveform of regulated voltage is very similar to the WTG speed waveform during wind disturbances (that is, during transients). Figure 15 shows that voltages achieve their maximum values during wind disturbances and if the wind speed is close to v_wn or above v_wn. The achieved maximum is less than 110% of V_nom, which is acceptable in most practical realisations.

In Figure 16, the active power of the B2B converter and PMSGs is shown. Due to the chosen direction of the B2B converter’s active power, shown in Figure 1, the active power of the B2B converter is negative. It is interesting to note that after the decrease in wind speed of the WTG1, the active power of PMSG2 and PMSG3 immediately start to increase. In this case, the PMSG1 temporarily runs slower than the other two generators, so the angular position of the PMSG1, relative to the faster PMSG2 and PMSG3, lags behind.

Similar behaviour can also be observed from Figure 17, where deviations of rotor angles of WGs around the average rotor angle of WGs are shown. As stated in [26] (p. 22), the resulting angular difference transfers part of the load from a slow machine to a faster machine. The situation is reversed if wind speed increases.

Waveforms of B2B converter currents and PMSG currents are shown in Figure 18. These waveforms are qualitatively similar to waveforms from the active power of the B2B converter and the active powers of PMSGs. The current of the B2B converter is negative due to the chosen direction of the current. Due to the use of P pitch controller, oscillations of rotor angles and currents are mitigated, which can be seen in Figure 17 and Figure 18, respectively.

Waveforms of reactive power from the B2B converter and PMSG are shown in Figure 19. It can be seen that, according to the chosen directions of reactive power in Figure 1, reactive powers are negative—that is, the B2B converter and generators absorb reactive power.

A unit variance is a band-limited white noise signal passed through a forming filter, an approximation of the Von Kármán velocity spectra (MIL—F-8785C) used as input into the system, as shown in Figure 20 [27]. The mean value of this random (turbulent) wind disturbance is 12 m/s while variance is 1.5 (m/s)², that is, turbulence intensity is 10.2%. Responses from the system are shown in Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28. In Figure 22, it can be noticed that more pitch action occurs than in non-turbulent cases, which can lead to bearing and actuator wear as well as mechanical fatigue. One of the measures to reduce these effects is introducing a dead zone in the P pitch controller. In this way, the P pitch controller will not react to small wind disturbances. Other techniques to increase system damping will be the subject of future investigation. In Figure 21, it can be seen that the power coefficient in the region below nominal wind speed, as a result of non-zero pitch angle, is below the optimal level for most of the time. This behaviour is quite different from wind parks with variable speed WTGs that use a power converter per WTG, leading to decreased amounts of extracted wind power for individual WTs than is optimal. This loss is partially compensated by the absence of thermal losses in individual WTG power converters in PCOWTG configurations.

5. Conclusions and Further Work

This paper presents the new control algorithm for PCOWTG with a 150 km length of transmission grid for cases where wind speed changes from cut-in wind speeds of above nominal wind speed. The problem of large charging currents in underwater cables is mitigated by decreasing system frequency. In order to achieve maximum extraction of wind power, WTG speed changes as the wind speed changes, while reference WTG speed is determined using an optimisation algorithm. Instead of measuring wind speed, the algorithm for wind speed calculation is used at each wind WT. Shunt reactors are added to both sides of the transmission grid cable to compensate for the reactive power generated by the transmission cable. The magnetic flux of electromagnetic devices is kept constant regardless of changing of system frequency by applying the constant Volt/Hz ratio controller. Damping torque is increased by using P pitch controllers at each WTG that work parallel with PI pitch compensators. The use of a well-tuned P pitch controller guarantees the stability of the PCOWTG.

The main contributions and novelty of this paper are summarised as follows:

• The first published control algorithm for control of PCOWTG;
• A novel algorithm for estimation of wind speed and optimal collective speed for PCOWTG.

In further research, more analytical studies of PCOWTG and the stability of PCOWTG will be investigated. Additionally, more advanced control algorithms such as adaptive control and fuzzy logic control of PCOWTG will be designed. Analysis of thermal losses in PCOWTG will be conducted, and a comparison with other wind topologies will be performed. Due to the low and variable voltages and frequencies associated with PCOWTG, we will consider the use of nonstandard equipment and power conversion topologies. Further increase in system damping using alternative methods combined with pitch angle control (to decrease pitch action and turbine elements fatigue) will also be the subject of future research, together with an investigation of alternative methods to reduce bearing wear and mechanical fatigue.