State-feedback and observer-based control design of a hybrid system consisting of a steam power plant in cooperation with a doubly fed wind turbine

Abstract

This article presents a control strategy for a variable speed pitch-controlled wind turbine which supplies an autonomous system in cooperation with a small steam power unit. The wind turbine consists of a doubly fed induction generator and operates in variable speed mode in order to maximize the power absorbed from the wind and minimize the amount of steam consumed by the steam power unit. Many reliable control methods concerning doubly fed induction generator–based wind generation systems connected to the grid can be found in the literature. The proposed control schemes concern both of the generating units of the hybrid system and so there are two main controllers. The objectives of the control design are to deliver the total power from the wind turbine to the isolated load under constant voltage and frequency and suitably adjust the power from the steam unit at the same time. The pitch controller of the wind turbine drives the wind turbine at such value of mechanical velocity that the proper amount of power from the wind is absorbed according to the existing load so that the requirements of the voltage and the frequency are realized. The second controller is responsible for the regulation of the corresponding amount of flow of steam mass into the steam turbine. First, the whole system has been linearized according to the two-machine theory. In this model, there is also the electrical power of the load which is regarded as a disturbance in the linear model. In the first case, the controllers have been designed according to the state-feedback theory without taking the disturbance into consideration for the control design, and simple proportional–integral controllers have been developed. In the second case, an observer that estimates both the states of the disturbance and the system has been designed, in order for a better response of the power frequency to be achieved. The system has been tested with the real nonlinear model using SIMULINK software under two severe inputs: the first is a step (and almost unreal) change of the wind speed which brings about a huge alteration of the wind turbine output under constant load. The second is a step decrease of 40% of the load under constant wind speed. Under both tests, the controllers seem to cooperate very well and the variations of the voltage and the frequency are within acceptable limits.

Keywords

Wind turbine synchronous generators doubly fed induction machine steam power units

Introduction

Different control topologies about doubly fed induction generator (DFIG)-based wind power generation systems can be found in the literature for an excellent behavior of the wind turbine (WT) when it is connected to the grid (Abedinzadeh and Tohidi, 2016; Mohammadpour et al., 2015; Phan et al., 2016) or in the case it is a power generation unit in a hybrid system (Delgado and Dominguez-Navarro, 2015; Parida and Chatterjee, 2016; Schinas and Vovos, 2008).

The supply of a hybrid isolated load from renewable energy sources in cooperation with other power generation units presents many problems. In this article, a specific WT generation scheme is proposed which is able to supply an isolated load in cooperation with a steam power unit with a synchronous generator. There is a medium voltage network between the two generation units. Two control systems are designed in order for the voltage and the frequency of the load to be preserved in acceptable values under severe changes of the wind speed and the load. The system under study is shown in Figure 1.

Figure 1.

Hybrid system under study.

Description and modeling of the system

Mathematical model of steam turbine

The conventional power generation unit uses a synchronous generator with a steam turbine. The transfer function of a steam vessel is $Q_{o u t} = (1 / 1 + s T_{v}) Q_{i n}$ , where Q_out and Q_in are the output and input steam mass flow rates ( $kg / s$ ), respectively (Wang et al., 2008). The term T_v is the time constant of the steam volume effect. This is in fact the model of one-stage steam turbine. Today, modern steam turbine units include many stages of steam vessels like the one described above. Besides, there may be an intermediate reheater so that the overall thermal efficiency is increased. In Figure 2, a steam turbine of three stages is depicted (Wang et al., 2008).

Figure 2.

Configuration of three-stage steam turbine.

There are three time constants T_CH, T_RH, and T_CO, respectively, for every stage. Their values are 0.3, 5, and 0.5 s, respectively. Generally, the mechanical torque is proportional to the steam flow rate, and so in our case, it is $T_{m} = F_{H P} * Q_{1} + F_{I P} * Q_{2} + F_{L P} * Q_{3}$ , with values 0.3, 0.3, and 0.4, respectively. Finally, the transfer function of the whole system between the output, T_m, and the input, Q₀, is given by

G_{1} = T_{m} = \frac{1.5 s^{2} + 3.6 s + 2}{s^{3} + 7.2 s^{2} + 11.4 s + 2}

Mathematical model of the WT and the description of the proposed pitch controller

The WT consists of a doubly fed induction machine (DFIM). As it is known, when the wind speed (V_w) is smaller than the rated one (V_wN) for the DFIM, maximum power from the wind must be captured. Then, the angular velocity (ω_t) of the WT has to operate in a specific value for every V_w so that the tip speed ratio, λ, of the WT will be equal to its optimum value. Thus, the WT operates in variable speed mode. In this mode of operation, the pitch angle is zero. This is true when the machine is connected to a grid and so all the energy from the wind can be inserted into the grid. In the case of a hybrid isolated system, this is not true. The pitch angle can have a positive value for every wind speed, even when V_w < V_wN because the energy from the wind has to be modulated according to the needs of the load. In order for this to be achieved, the pitch controller has to track the frequency of the grid voltage (or consequently, the angular frequency of the synchronous generator ω_sg) as an index of the load of the system. According to the difference between the set value (ω_sg, _ref) and the actual value ω_sg, the pitch controller decides about the necessary output of the WT and so the pitch angle is regulated. For wind speeds above the rated one, the pitch controller also takes on to stabilize the produced energy into its maximum value or another (smaller) one. The whole scheme is depicted in Figure 3.

Figure 3.

Description of the WT control scheme.

The mechanical power extracted from the wind is described by the following familiar equation

P_{m} = \frac{1}{2} p V_{w}^{3} A C_{P}

(1)

where P_m is the wind power (W), ρ is the air density ( ${kg / m}^{3}$ ), V_w is the wind speed (m/s), A is the swept area ( $m^{2}$ ), and $C_{P}$ is the power coefficient. The drive train system of the WT consists of the rotating masses, the gearbox, and so on. A two-mass model is used for the simulation of the drive system and is depicted in Figure 4.

Figure 4.

Two-mass system for the WT drive system (Vittal and Ayyanar, 2013).

The following are the equations of motion (Vittal and Ayyanar, 2013)

\begin{array}{l} T_{g} - D_{t g} (ω_{g} - ω_{t}) - K_{t g} (Θ_{g} - Θ_{t}) = 2 J_{g} {\dot{ω}}_{g} \\ D_{t g} (ω_{g} - ω_{t}) + K_{t g} (Θ_{g} - Θ_{t}) - T_{t} = 2 J_{t} {\dot{ω}}_{t} \end{array}

where ω_g and ω_t (p.u.) are the generator and the turbine speed, respectively; θ_g − θ_t is the shaft twist angle; J_g and J_t are the generator and the turbine moment of inertia, respectively; K_tg (per unit torque/rad) and D_tg (per unit torque/per unit speed) are the shaft stiffness and damping coefficient, respectively; and finally, T_t and T_g are the turbine torque and the generator torque, respectively.

System linearization

In order for the controllers to be designed, a general model of the hybrid system with both the machines included has to be developed, and this is shown in Figure 5.

Figure 5.

Two-machine control.

From the first equation of the mass mechanical model, we have

\begin{array}{l} T_{g} - D_{t g} (ω_{g} - ω_{t}) - K_{t g} (\frac{ω_{g}}{s} - \frac{ω_{t}}{s}) = 2 J_{g} {\dot{ω}}_{g} \Rightarrow T_{g} - D_{t g} ω_{g} + D_{t g} ω_{t} - \frac{K_{t g} ω_{g}}{s} + \frac{K_{t g} ω_{t}}{s} = 2 J_{g} s ω_{g} \\ ω_{g} T_{g} - D_{t g} ω_{g}^{2} + D_{t g} ω_{t} ω_{g} - \frac{K_{t g} ω_{g}^{2}}{s} + \frac{K_{t g} ω_{t} ω_{g}}{s} = 2 J_{g} s ω_{g}^{2} \end{array}

By linearization around the nominal point of operation (ω_to, ω_go), we get

Δ P_{e l} + (\frac{1}{s} K_{t g} ω_{g o} + D_{t g} ω_{g o}) Δ ω_{t} = (4 J_{g} s ω_{g o} + 2 D_{t g} ω_{g o} - D_{t g} ω_{t o} + \frac{1}{s} K_{t g} 2 ω_{g o} - \frac{1}{s} K_{t g} ω_{t o}) Δ ω_{g}

where ΔP_el is the electrical power of the load and is given by ΔP_el = T_gω_g.

From the second equation of the mass mechanical model, we have

\begin{array}{l} D_{t g} (ω_{g} - ω_{t}) + K_{t g} (\frac{ω_{g}}{s} - \frac{ω_{t}}{s}) - T_{t} = 2 J_{t} {\dot{ω}}_{t} \Rightarrow \\ D_{t g} ω_{g} ω_{t} - D_{t g} ω_{t}^{2} + \frac{1}{s} K_{t g} ω_{g} ω_{t} - \frac{1}{s} K_{t g} ω_{t}^{2} - ω_{t} T_{t} = 2 J_{t} s ω_{t}^{2} \end{array}

By linearization, we get

(D_{t g} ω_{t o} + \frac{1}{s} K_{t g} ω_{t o}) Δ ω_{g} - Δ P_{m} = (- D_{t g} ω_{g o} + 2 D_{t g} ω_{t o} - \frac{1}{s} K_{t g} ω_{g o} + \frac{1}{s} K_{t g} 2 ω_{t o} + 4 J_{t} s ω_{t o}) Δ ω_{t}

where ΔP_m is the mechanical power from the wind and is given by ΔP_m = T_tω_t.

The power coefficient, C_P, is related to the pitch angle, β, by the following equation (Schinas and Vovos, 2008)

C_{P} = (0.44 - 0.0167 β) s i n \frac{π (λ - 3)}{15 - 0.3 β} - 0.00184 (λ - 3) β

The linearization of equation (1) that gives the mechanical (wind) power results in

Δ P_{m} = 286290 Δ ω_{t} - 92851 Δ β

Working on the previous equations and replacing with the values from Appendix 1, we finally conclude to the following linear equation

\begin{array}{l} Δ ω_{g} = 5.56 \frac{s (s + 96.15)}{s^{4} + 43.72 s^{3} - 326.11 s^{2} + 2665.8 s - 29592.37} Δ β \\ + 1.77 * 10^{- 5} \frac{s (s^{2} + 214 s + 215.55)}{s^{4} + 43.72 s^{3} - 326.11 s^{2} + 2665.8 s - 29592.37} Δ P_{e l} = G_{3} Δ β + G_{4} Δ P_{e l} \end{array}

From the circuit given in Figure 5, we have

\begin{array}{l} Δ ω_{s g} = G_{2} (Δ P_{m} - Δ P_{e l} - T_{12} (Δ δ_{1} - Δ δ_{2})) \overset{}{\Rightarrow} \\ Δ ω_{s g} = G_{2} Δ P_{m} - G_{2} Δ P_{e l} - G_{2} T_{12} (\frac{1}{s} Δ ω_{s g} - \frac{1}{s} Δ ω_{g}) \overset{}{\Rightarrow} \\ (1 + \frac{1}{s} G_{2} T_{12}) Δ ω_{s g} = G_{1} G_{2} Δ μ - G_{2} Δ P_{e l} + \frac{1}{s} G_{2} T_{12} Δ ω_{g} \overset{}{\Rightarrow} \\ Δ ω_{s g} = G_{1} G_{2} {(1 + \frac{G_{2} T_{12}}{s})}^{- 1} Δ μ + \frac{G_{2} T_{12}}{s} {(1 + \frac{G_{2} T_{12}}{s})}^{- 1} G_{3} Δ β + (G_{4} - G_{2}) {(1 + \frac{G_{2} T_{12}}{s})}^{- 1} Δ P_{e l} \end{array}

The transfer function G₂ is the transfer function of the synchronous generator and is of the form

G_{2} (s) = \frac{K_{p}}{1 + T_{p} s}

From Appendix 1, we have

G_{2} (s) = \frac{100}{1 + 2.4 s}

It is obvious that G₄ << G₂, so by approximation, we finally have

Δ ω_{s g} = G_{1} G_{2} {(1 + \frac{G_{2} T_{12}}{s})}^{- 1} Δ μ + \frac{G_{2} T_{12}}{s} {(1 + \frac{G_{2} T_{12}}{s})}^{- 1} G_{3} Δ β - G_{2} {(1 + \frac{G_{2} T_{12}}{s})}^{- 1} Δ P_{e l}

(2)

We see that Δω_sg is generally of the form $Δ ω_{s g} = h_{1} Δ μ + h_{2} Δ β + h_{3} Δ P_{e l}$ .

The last equation gives the relationship between the output ω_sg (the synchronous generator angular speed) and the two controlling inputs of the system: the first input is the steam flow in the first stage of the steam governor (µ_ο) and the second input is the pitch angle (β) of the WT. By controlling these two inputs properly, the frequency of the voltage of the hybrid isolated system will be kept constant. ∆ $P_{e l}$ is regarded as a disturbance.

As it has already been stated, there is a medium voltage line between the two power stations, because generally, the wind generators may not be located in the neighborhood of the steam power plant.

The term T₁₂ is equal to

T_{12} = \frac{V_{1} V_{2}}{X}

From Appendix 1, we get T₁₂ = 16.8 p.u. Substituting G₁, G₂, and T₁₂ in equation (2), we finally get the expression of h₁ as

h_{1} (s) = \frac{360 s^{4} + 1014 s^{3} + 840 s^{2} + 200 s}{5.76 s^{6} + 46.27 s^{5} + 4133 s^{4} + 30780 s^{3} + 58080 s^{2} + 27218 s + 3360}

(3)

Similarly and approximately we have

h_{2} (s) = \frac{0.03 s}{s^{3} - 7.75 s^{2} + 55.95 s - 625.25}

(4)

State-feedback control design

From the above analysis, we have two loops for the control of the real power coming out from each generation unit by not taking into account the term ΔP_el which is regarded as a disturbance. The two control loops are depicted in Figure 6.

Figure 6.

Control loop for the (a) steam turbine and (b) WT.

Let us start from the design of the controller, G_c₂(s), for the pitch angle of the WT. The real nonlinear system is depicted in Figure 7, in which all the constraints of a pitch actuator have been taken into consideration. For instance, there are limits for the value of the angle and the rate of change of the angle.

Figure 7.

Actual nonlinear system of the pitch actuator.

Our intention is the design of as much simple controllers as possible so that their software implementation is easy. We use the state-feedback technique in order for the parameters of a simple proportional–integral (PI) controller to be found. We ignore the third order of “s” in the denominator and so the system is regarded as a second-order system. By transforming the single input–single output (SISO) system with the transfer function h₂ given by equation (4) into state-space model ( $\dot{x} = A x + B u$ ), we have

A = [\begin{matrix} 7.2194 & - 80.6742 \\ 1 & 0 \end{matrix}], B = [\begin{matrix} 1 \\ 0 \end{matrix}], C = [- 0.00390], D = 0

The reachability matrix of the system is $W = [B A * B] = [\begin{matrix} 1 & 7.2194 \\ 0 & 1 \end{matrix}]$ (Astrom and Wittenmark, 2013). We arbitrarily set the desired characteristic polynomial to be P(s) = s² + 2s + 2. Ackermann’s formula now gives

L = [0 1] * W^{- 1} * P (A) = [11.21 - 78.67]

The elements of L are the values the states must be multiplied with. By means of MATLAB software, we can find an equivalent series controller as it is depicted in Figure 7. This is given as follows

G_{c 2} = \frac{- 2898 s + 20320}{s}

We can see that this is actually a PI controller with K_P = −2898 and K_I = 20,320.

We repeat the same procedure with the design of the steam turbine controller. The SISO system with the transfer function h₁ is given by equation (3). In this case, we neglect the sixth and fifth orders of “s” in the denominator and so the system is of fourth order.

In state-space, it is

\begin{array}{l} A = [\begin{matrix} - 7.44 & - 14.052 & - 6.58 & - 0.813 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}] \\ B = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}] \\ C = [- 0.4034 - 1.0208 - 0.5252 - 0.0708] \\ D = 0 \end{array}

(5)

The state variables matrix x for the system given by the transfer function h₁ is

x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]

(6)

The reachability matrix of the system is

W = [B A * B A^{2} * B A^{3} * B] = [\begin{matrix} 1 & - 7.44 & 41.41 & - 210.33 \\ 0 & 1 & - 7.44 & 41.41 \\ 0 & 0 & 1 & - 7.44 \\ 0 & 0 & 0 & 1 \end{matrix}]

The characteristic equation is set to be P(s) = s⁴ + 3.4s³ + 6.41s² + 5.848s + 2.9684. Then, we get

L = [0 0 0 1] * W^{- 1} * P (A) = [- 1.075 - 4.662.232.15]

(7)

The equivalent series controller is obtained if we neglect high orders of “s” and we finally reach in

G_{c 1} = \frac{40.72 s + 4.456}{s}

We can see that that this is another PI controller with K_P = 40.72 and K_I = 4.456.

So, by means of calculating the state-feedback multiplication factor matrix, we can find equivalent series controllers and very simple ones such as PI controllers. The next paragraph shows that the performance of the system under these two controllers is very good for step changes in the wind speed and the power of the load.

Observer-based design control

We have seen that the speed of the synchronous generator is actually given by

Δ ω_{s g} = h_{1} Δ μ + h_{2} Δ β + h_{3} Δ Ρ_{e l}

Now, the disturbance, ΔP_el, has been taken into consideration for the design of the controller, G_c₁, which regulates Δω_sg in the steam turbine. In this way, the output from the WT will be inserted more easily into the system. The whole scheme is depicted in Figure 8.

Figure 8.

Synchronous speed as a function of µ and ΔP_el.

We have defined in the previous paragraph the state variable matrix of h₁ as x given by equation (6). We define the augmented matrix Z with all the state variables of the system as

Z = [\begin{matrix} x \\ w \end{matrix}] = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ w_{1} \\ w_{2} \end{matrix}]

where $w = [\begin{matrix} w_{1} \\ w_{2} \end{matrix}]$ is the state variable matrix of the disturbance system given by h₃(s).

So, converting transfer function model h₃ into state-space, the following are derived

\frac{d w}{d t} = A_{w} * w + B_{w} * u_{w}, y_{w} = C_{w} * w, A_{w} = [\begin{matrix} - 0.4167 & - 700 \\ 1 & 0 \end{matrix}], B_{w} = [\begin{matrix} 1 \\ 0 \end{matrix}], C_{w} = [0 700]

(8)

So, the whole system in state-space will be

\dot{z} = [\begin{matrix} A & 0 \\ 0 & A_{w} \end{matrix}] * z + [\begin{matrix} B & 0 \\ 0 & B_{w} \end{matrix}] [\begin{matrix} u_{1} \\ u_{2} \end{matrix}], y = [C C_{w}] * z

with u₁ = Δµ and u₂ = ΔP_el, A, B, and C are given by equation (5) and A_w, B_w, and C_w are given by equation (8).

By not knowing the states of the disturbance input, we have to use an observer which would be able to estimate the state x of the system and the state w of the disturbance. In order for a closed loop to be configured, we design an observer which estimates the w ( $\hat{w}$ ) and the difference $\hat{e}$ between the reference value of x (x_ref) and its estimated value $\hat{x}$ , that is, $\hat{e} = x_{r e f} - \hat{x}$ . We define $ε = \hat{y} - y$ , in which y is the output of the system (Δω_sg). Then, from the above data, we get

ε = - y + C * \hat{x} + C_{w} * \hat{w} = - y + C * \hat{x} + C_{w} * \hat{w} + C * x_{r e f} - C * x_{r e f} = - y + y_{r e f} - C * \hat{e} + C_{w} * \hat{w}

There is a general control law u = u_f + u₁. The control law, u_f, will be according to the state feedback, $u_{f} = - L * \hat{e} - L_{w} * \hat{w},$ in which L has already been defined by equation (7), and the new multiplication matrix L_w will be calculated. Then, the first equation of the observer will be

\begin{array}{l} \hat{e} . = A * \hat{e} + B * u + k * ε \Rightarrow \\ \hat{\dot{e}} = A * \hat{x} + B * (- L * \hat{e} - L_{w} * \hat{w}) + K * (y_{r e f} - y - C * \hat{e} + C_{w} * \hat{w}) \Rightarrow \\ \hat{\dot{e}} = (A - B * L - K * C) * \hat{e} + (K * C_{w} - B * L_{w}) * \hat{w} + K * (y_{r e f} - y) \end{array}

(9)

The second equation will be

\begin{array}{l} \hat{w} . = A_{w} * \hat{w} + K_{w} * ε \underset{}{\Rightarrow} \hat{w} . = A_{w} * \hat{w} + K_{w} * (y_{r e f} - y + C_{w} * \hat{w} - C * \hat{e}) \Rightarrow \\ \hat{w} . = (A_{w} + K_{w} * C_{w}) * \hat{w} + K_{w} * (y_{r e f} - y - C * \hat{e}) \end{array}

(10)

In order to minimize the disturbance impact on the states of the system, the term $- B * L_{w} + K * C_{w}$ in equation (9) must be equal to zero, and the matrix L_w can be calculated by

- B * L_{w} + K * C_{w} = 0

(11)

Equation (8) becomes

\hat{\dot{e}} = (A - B * L - K * C) * \hat{e} + K * (y_{r e f} - y)

(12)

Equations (9) and (12) actually describe the observer which estimates the states e and w. In these equations, L has already been calculated and is given by equation (7). There is also the gain K of the observer. This is calculated as follows: at first, the matrix W_o is calculated as (Astrom and Wittenmark, 2013)

W_{o} = [\begin{matrix} C \\ C * A \\ C * A^{2} \\ C * A^{3} \end{matrix}]

and then

K = P (A) * W_{o}^{- 1} * [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}] = [- 3.2 - 2.6 - 4 - 7.5]

The second part, u₁, will be equal to y_ref − y. So, all the parameters of equations (9) and (12) are known and the observer can be simulated. The system is depicted in Figure 9.

Figure 9.

Observer-based control scheme.

Nonlinear system simulation results

The system under study has been simulated using SIMULINK software with the nonlinear models for the synchronous machine, the DFIM, as well as the turbines and the governors, from the software library. The rated values for the machines are listed in Appendix 1. It has to be noted that the rated power of the WT is 1.5 MVA while the load is 3 MVA with a power factor 0.9 inductive, that is, the rated power of the WT is 50% of the load.

In the first case, we consider the power of the load to be kept steady and the wind speed to be altered at t = 35 s (Figure 10). The change of the power from the WT follows the change of the wind speed. The output power from the steam generator acts in such a way that its angular speed and the frequency vary in acceptable limits despite the vast variation in the wind speed. The response of the system is shown in Figures 11 and 12.

Figure 10.

Wind speed for the simulation.

Figure 11.

System response for a step change in wind speed with the two PI controllers.

Figure 12.

System response for a step change in wind speed with the observer control.

In the second case, there is a step load decrease at t = 55 s. The response of the system is shown in Figures 13 and 14.

Figure 13.

Main components’ response under a step load decrease with the two PI controllers.

Figure 14.

System response for a step load decrease with the observer control.

Generally, the response of the system is satisfactory in all situations with both the control schemes. When there is enough load demand for real power, the pitch controller of the WT permits the available power from the wind to be inserted into the grid. When the power of the load decreases, both the steam governor controller and the WT controller cooperate and suitably adjust their power outputs so that the load frequency varies in acceptable limits. The only apparent difference between PI control and observer-based control is that in the second case, there is not only the low-frequency oscillation of the synchronous generator speed which is present at the frequency but also the rms voltage of the load.

Conclusion

In this article, a hybrid isolated system with two generating units has been studied. The system consists of a steam turbine with a synchronous machine and a WT with doubly fed induction generator. Each machine is supplied with a basic controller for the regulation of the produced real power according to the continuous variations of the wind speed and the alterations of the load. Both the controller of the WT and the controller of the steam turbine were designed through the state-feedback theory and observer-based theory for load disturbances taken into account. The control design was based on the linearized system of the original one. The results of the nonlinear system are very good under vast changes of the wind speed, and the load and the cooperation of the two controllers ensure that the voltage and the frequency are within acceptable limits.

Footnotes

Appendix 1

Table 3.

Parameters of synchronous generator.

Nominal power, S_N	4.7 MVA
Nominal stator voltage, V_sN	1 kV
Nominal frequency, f	60 Hz
Inertia coefficient, H	4.7 s
Friction factor, F	0.01 p.u.
Pole pairs	1

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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