A Novel Discrete Resonant Control Method for Cascaded H-Bridge Rail Balancers

Synchronisation with the three-phase grid voltage must provide fast and accurate estimation of the voltage vector (i.e. voltage vector amplitude U_m, voltage vector angle ϑ_u12, and grid frequency ω). The DSOGI-PLL synchronisation method has been chosen for its good immunity to input signal noise and disturbances. This method is well known from Suul et al. (2012).

To calculate the value of the compensation and symmetrisation currents accurately, it is necessary to evaluate the phasor of the single-phase current flowing through the TSS to the traction catenary I_{cat_SDFT_I} and I_{cat_SDFT_R}. The evaluation of the current phasor is based on the following SDFT Eqs (1) and (2): (1) Icat_SDFT_Rk+1Icat_SDFT_Ik+1=RIcat_SDFT_RkIcat_SDFT_Ik+2Nicatk−icatk−N0 \left[ {\matrix{ {{I_{cat\_SDFT\_R\left( {k + 1} \right)}}} \cr {{I_{cat\_SDFT\_I\left( {k + 1} \right)}}} \cr } } \right] = R\left[ {\matrix{ {{I_{cat\_SDFT\_R\left( k \right)}}} \cr {{I_{cat\_SDFT\_I\left( k \right)}}} \cr } } \right] + {2 \over N}\left[ {\matrix{ {{i_{cat\left( k \right)}} - {i_{cat\left( {k - N} \right)}}} \cr 0 \cr } } \right] (2) R=cosωkΔt−sinωkΔtsinωkΔtcosωkΔt R = \left[ {\matrix{ {\cos \left( {{\omega _{\left( k \right)}}\Delta t} \right)} & { - \sin \left( {{\omega _{\left( k \right)}}\Delta t} \right)} \cr {\sin \left( {{\omega _{\left( k \right)}}\Delta t} \right)} & {\cos \left( {{\omega _{\left( k \right)}}\Delta t} \right)} \cr } } \right] where N is the length of the SDFT buffer, ω_(k) is the analysed frequency (∼50 Hz) and Δt is the sampling period. The current amplitude and vector angle are calculated using the Eq. (3): (3) Icat_m=Icat_SDFT_R(k+1)2+Icat_SDFT_I(k+1)2ϑi=tan−1Icat_SDFT_Ik+1Icat_SDFT_Rk+1 \matrix{ {{I_{cat\_m}} = \sqrt {{I_{cat\_SDFT\_R(k + 1)}}^2 + {I_{cat\_SDFT\_I(k + 1)}}^2} } \cr {{\vartheta _i} = {\rm{ta}}{{\rm{n}}^{ - 1}}\left( {{{{I_{cat\_SDFT\_I\left( {k + 1} \right)}}} \over {{I_{cat\_SDFT\_R\left( {k + 1} \right)}}}}} \right)} \cr } The SDFT is a very effective and robust algorithm for spectral calculation. On the contrary, the main drawback is numerical instability. Compared to the common SDFT, we used a dual SDFT containing two independent phasor calculations using the same data. The calculated phasors are periodically reset to zero after two SDFT window length periods. However, the reset of the phasors of each SDFT is time-shifted. After resetting the first SDFT, the results of the second SDFT are used. After one window-length period following the first SDFT reset, the first SDFT converges to the correct value, its result can then be used, and the second SDFT can be reset, and vice versa. A detailed description of this algorithm and implementation information can be found in Talla and Blahnik (2017).

Defining symmetrisation and reactive power compensation of a single-phase load connected to the TSS catenary is based on Steinmetz’s equivalent circuit and explained by a phasor diagram in Figure 3. The balancing current calculation depends on the catenary current amplitude and its phase shift relative to the grid voltage [Eqs (4) and (5)]. Eqs (6)–(8) describe ARB reactive power compensating and balancing currents of individual CHB branches based on the catenary current. The I_{cat_m} is the catenary current magnitude, ϑ_u12 is the voltage vector angle (line to line U_g12 voltage angle) and ϑ_i is the current vector angle of the catenary load. The calculated currents i_CHB12, i_CHB23 and i_CHB31 are used as a fundamental harmonic current reference for the ARB inner current loop. (4) Icat_I=Icat_msinϑu12−ϑi {I_{cat\_I}} = {I_{cat\_m}}\sin \left( {{\vartheta _{u12}} - {\vartheta _i}} \right) (5) Icat_R=Icat_mcosϑu12−ϑi {I_{cat\_R}} = {I_{cat\_m}}\cos \left( {{\vartheta _{u12}} - {\vartheta _i}} \right) (6) iCHB12_com=Icat_Icosϑu12+π2 {i_{CHB12\_com}} = {I_{cat\_I}}\cos \left( {{\vartheta _{u12}} + {\pi \over 2}} \right) (7) iCHB23_bal=Icat_R13cosϑu12−5π6+cosϑu12=Icat_R3cosϑu12−π6 {i_{CHB23\_bal}} = {I_{cat\_R}}\left( {{1 \over {\sqrt 3 }}\cos \left( {{\vartheta _{u12}} - {{5\pi } \over 6}} \right) + \cos \left( {{\vartheta _{u12}}} \right)} \right) = {{{I_{cat\_R}}} \over {\sqrt 3 }}\cos \left( {{\vartheta _{u12}} - {\pi \over 6}} \right) (8) iCHB31_bal=Icat_R−13cosϑu12−π6+cosϑu12=Icat_R3cosϑu12+π6 {i_{CHB31\_bal}} = {I_{cat\_R}}\left( { - {1 \over {\sqrt 3 }}\cos \left( {{\vartheta _{u12}} - {\pi \over 6}} \right) + \cos \left( {{\vartheta _{u12}}} \right)} \right) = {{{I_{cat\_R}}} \over {\sqrt 3 }}\cos \left( {{\vartheta _{u12}} + {\pi \over 6}} \right)

Figure 3.

Steinmetz’s symmetrising circuit and corresponding phasor diagram illustrating the function of the electronic balancer.

The feedforward calculation is based on a simplified electrical steady-state model of the device: a series connection of the grid voltage, filter inductance L_CHB with the effect of filter resistance neglected and converter voltage. The resulting feedforward converter voltages u_ff12, u_ff23, u_ff31, depending on the grid voltage U_m, ϑ_u12, frequency ω and catenary current I_{cat_I}, I_{cat_R} are presented in Eqs (9)–(11). The filter inductance voltage drop is shifted by −π2 {\raise0.7ex\hbox{${ - \pi }$} \!\mathord{\left/ {\vphantom {{ - \pi } 2}}\right.}\!\lower0.7ex\hbox{$2$}} to CHB currents as presented by Eqs (6)–(8). (9) uff12=Umcosϑu12+ωLfIcat_Icosϑu12 {u_{ff12}} = {U_m}\cos\left( {{\vartheta _{u12}}} \right) + \omega {L_f}{I_{cat\_I}}\cos\left( {{\vartheta _{u12}}} \right) (10) uff23=Umcosϑu12−2π3+ωLfIcat_R3cosϑu12−2π3 {u_{ff23}} = {U_m}\cos\left( {{\vartheta _{u12}} - {{2\pi } \over 3}} \right) + {{\omega {L_f}{I_{cat\_R}}} \over {\sqrt 3 }}\cos\left( {{\vartheta _{u12}} - {{2\pi } \over 3}} \right) (11) uff31=Umcosϑu12+2π3+ωLfIcat_R3cosϑu12−π3 {u_{ff31}} = {U_m}\cos\left( {{\vartheta _{u12}} + {{2\pi } \over 3}} \right) + {{\omega {L_f}{I_{cat\_R}}} \over {\sqrt 3 }}\cos\left( {{\vartheta _{u12}} - {\pi \over 3}} \right)

The DC-link voltages and CHB currents of each ARB branch are controlled individually. The DC-link voltage control provides an approximately constant sum of DC-link voltages of floating capacitors of all cascaded-connected H-bridges per CHB phase (u_{DC_w} is the total reference voltage of the whole CHB branch). The DC-link capacitor’s energy covers the power losses of the converter and the fluctuation of reactive power. Since the power losses are low, fast dynamics of the DC-link voltage controller are not required. Therefore, a proportional integral (PI) voltage controller with a dominant integral component was chosen. The output signal of the voltage controller is the required current I_{12_Δu}, which is in phase with the grid voltage (active current). This current is summed with the required symmetrising current from Steinmetz’s Eqs (6)–(8), and with the required higher-order harmonics filtration current i_{CHB12_fil}. This creates the total reference current of the CHB branch i_{CHB12_w}.

The required catenary filtration current i_{cat_fil} is calculated as the difference between the actual catenary current i_cat and the fundamental harmonic of the catenary current i_cat(1st) estimated by SDFT. The required filtration current is equally divided into two parallel branches: (1) compensation branch CHB12 i_{CHB12_fil} = −1/2 i_{cat_fil} and (2) the entire symmetrisation branch, consisting of a series connection of CHB23 and CHB31 branches i_{CHB23_fil} = i_{CHB31_fil} = 1/2 _{icat_fil}. However, the harmonic current distribution in parallel branches can be freely selected, for example, with respect to branch power losses.

The total current reference i_{CHB12_w} is used as a reference for a proportional controller and several advanced PR controllers regulating the fundamental harmonic component (50 Hz) and higher-order harmonics (150 Hz, 250 Hz, 350 Hz and 450 Hz). The PR controller generates a signal u_PR12, which is summed in the following step with a feedforward value, u_ff12 Eq. (9), and a harmonics filtration component, also denoted as u_hf12. The result is the converter modulation signal for the CHB 1 kHz phase shift pulse width modulation (PS-PWM) modulator, including the DC-link balancing algorithm. Low switching and sampling frequency, large control latency, and the requirement to eliminate higher-order current harmonics up to 450 Hz are the biggest challenges in ARB control, which led to the development of the advanced R controller algorithm presented in Section 3.

In the given application, a multilevel voltage source inverter (VSI) is used in combination with PS-PWM. To better understand the delays introduced into the system, see Figure 4. The base sampling frequency of the system is 8 kHz (sampling is at the top and bottom of all phase-shifted PWM timers). However, the compare register of each H-bridge is only changed at its top or bottom of the PWM timer (two times per PWM frequency, i.e. 2 kHz). It creates the variable control delay of individual inverters.

Figure 4.

Timeline visualising control and modulation delay of the system.

This means that, at a given instant, it takes 1 sample time for the measured values to be processed and set as the active output voltage reference for the first converter, 2 sample times for second converter up to 4 sample times for the fourth converter. This reference remains active for a duration of four samples before being replaced by a new value.

The latency in the system that needs to be compensated corresponds to one sample delay due to computation, plus half of the time during which the reference value is active. In this case, the total compensated delay is equivalent to three sampling steps. This value is primarily influenced by the number of levels in the multilevel VSI used.

To model the real PS-PWM latency and demonstrate its effect, discrete state-space equations have been derived for the PS-PWM: (12) uCHB_X.1kuCHB_X.2kuCHB_X.3kuCHB_X.4k=0000100001000010uCHB_X.1k−1uCHB_X.2k−1uCHB_X.3k−1uCHB_X.4k−1+1000uCHB_Xk \left[ {\matrix{ {{u_{CHB\_X.1\left( k \right)}}} \cr {{u_{CHB\_X.2\left( k \right)}}} \cr {{u_{CHB\_X.3\left( k \right)}}} \cr {{u_{CHB\_X.4\left( k \right)}}} \cr } } \right] = \left[ {\matrix{ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \cr } } \right]\left[ {\matrix{ {{u_{CHB\_X.1\left( {k - 1} \right)}}} \cr {{u_{CHB\_X.2\left( {k - 1} \right)}}} \cr {{u_{CHB\_X.3\left( {k - 1} \right)}}} \cr {{u_{CHB\_X.4\left( {k - 1} \right)}}} \cr } } \right] + \left[ {\matrix{ 1 \hfill \cr 0 \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right]{u_{CHB\_X\left( k \right)}}

A method for balancing individual cells of CHB based on energy formula at the modulation signal level was used (Blahnik et al., 2018). The PS-PWM is modulator implemented in an Altera/Intel Cyclone III, FPGA.

The digital implementation of the R controller can be derived by transforming the transfer function (13) into a Laplace equation: (14) ys=1sKRωus−ω21sys {y_{\left( s \right)}} = {1 \over s}\left( {{K_R}\omega {u_{\left( s \right)}} - {\omega ^2}{1 \over s}{y_{\left( s \right)}}} \right)

Eq. (14) can be reformulated into a state-space equation using the general formula: (15) sx=Ax+Buy=Cx+Du \matrix{ {sx = Ax + Bu} \hfill \cr {y = Cx + Du} \hfill \cr }

From Eqs (14) and (15), we can obtain the state-space equations of the R controller: (16) sxαxβ=0−ωω0xαxβ+KRω0uy=10xαxβ \matrix{ {s\left[ {\matrix{ {{x_\alpha }} \cr {{x_\beta }} \cr } } \right] = \left[ {\matrix{ 0 & { - \omega } \cr \omega & 0 \cr } } \right]\left[ {\matrix{ {{x_\alpha }} \cr {{x_\beta }} \cr } } \right] + \left[ {\matrix{ {{K_R}\omega } \cr 0 \cr } } \right]u} \hfill \cr {y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_\alpha }} \cr {{x_\beta }} \cr } } \right]} \hfill \cr } where x_α = y is the R controller output and xβ=1sωxα=1sωy {x_\beta } = {1 \over s}\omega {x_\alpha } = {1 \over s}\omega y creates orthogonal (imaginary complement) part of the R controller output.

The matrix A can be expressed as follows: (17) A=0−ωω0=jω0j−j0 A = \left[ {\matrix{ 0 & { - \omega } \cr \omega & 0 \cr } } \right] = j\omega \left[ {\matrix{ 0 & j \cr { - j} & 0 \cr } } \right] where j is imaginary operator.

By using the exact discretisation formula for matrix A and sampling time Δt, (18) Ad=eAΔt=ejωΔt0j−j0 {A_d} = {e^{A\Delta t}} = {e^{j\omega \Delta t\left[ {\matrix{ 0 & j \cr { - j} & 0 \cr } } \right]}}

We get the following analytical solution: (19) Ad=cosωΔt−sinωΔtsinωΔtcosωΔt {A_d} = \left[ {\matrix{ {\cos \left( {\omega \Delta t} \right)} & { - \sin \left( {\omega \Delta t} \right)} \cr {\sin \left( {\omega \Delta t} \right)} & {\cos \left( {\omega \Delta t} \right)} \cr } } \right]

The result is an orthonormal (rotational) matrix (A_d⁻¹ = A_d^T).

We can use the exact discretisation formula for the B matrix: (20) Bd=A−1Ad−IB {B_d} = {A^{ - 1}}\left( {{A_d} - I} \right)B

So: (21) Bd=1ω01−10cosωΔt−1−sinωΔtsinωΔtcosωΔt−1KRω0=KRsinωΔt1−cosωΔt {B_d} = {1 \over \omega }\left[ {\matrix{ 0 & 1 \cr { - 1} & 0 \cr } } \right]\left[ {\matrix{ {\cos \left( {\omega \Delta t} \right) - 1} & { - \sin \left( {\omega \Delta t} \right)} \cr {\sin \left( {\omega \Delta t} \right)} & {\cos \left( {\omega \Delta t} \right) - 1} \cr } } \right]\left[ {\matrix{ {{K_R}\omega } \cr 0 \cr } } \right] = {K_R}\left[ {\matrix{ {\sin \left( {\omega \Delta t} \right)} \cr {1 - \cos \left( {\omega \Delta t} \right)} \cr } } \right]

Resulted R controller equations are: (22) xαkxβk=cosωΔt−sinωΔtsinωΔtcosωΔtxαk−1xβk−1+KRsinωΔt1−cosωΔtukyk=10xαkxβk=xαk \matrix{ {\left[ {\matrix{ {{x_{\alpha \left( k \right)}}} \cr {{x_{\beta \left( k \right)}}} \cr } } \right] = \left[ {\matrix{ {\cos \left( {\omega \Delta t} \right)} & { - \sin \left( {\omega \Delta t} \right)} \cr {\sin \left( {\omega \Delta t} \right)} & {\cos \left( {\omega \Delta t} \right)} \cr } } \right]\left[ {\matrix{ {{x_{\alpha \left( {k - 1} \right)}}} \cr {{x_{\beta \left( {k - 1} \right)}}} \cr } } \right] + {K_R}\left[ {\matrix{ {\sin \left( {\omega \Delta t} \right)} \cr {1 - \cos \left( {\omega \Delta t} \right)} \cr } } \right]{u_{\left( k \right)}}} \hfill \cr {{y_{\left( k \right)}} = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_{\alpha \left( k \right)}}} \cr {{x_{\beta \left( k \right)}}} \cr } } \right] = {x_{\alpha \left( k \right)}}} \hfill \cr } where u_(k) = e_(k) is the difference between required and actual/measured value and y_(k) is the R controller output. If ωΔt ≅ 0 then: (23) Ad≅1−ωΔtωΔt1, Bd≅KRωΔt0 {A_d} \cong \left[ {\matrix{ 1 & { - \omega \Delta t} \cr {\omega \Delta t} & 1 \cr } } \right],\;\;\;{B_d} \cong {K_R}\left[ {\matrix{ {\omega \Delta t} \cr 0 \cr } } \right] leads to the forward Euler discretisation of the R controller Eq. (23).

The control algorithm’s latency can reduce the efficiency of the R or even make it unstable. However, the R controller in the state-space form can be easily extended with feedforward latency compensation included in the C matrix. The original output matrix contains only the selection operation of the first state (x_α(k)). If we define the control latency as Δt_lat, we can calculate feedforward latency compensation by multiplying the original C matrix by orthogonal rotational matrix as: (24) yk=10cosωΔtlat−sinωΔtlatsinωΔtlatcosωΔtlatxαkxβk=cosωΔtlat−sinωΔtlat xαkxβk {y_{\left( k \right)}} = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {\cos \left( {\omega \Delta {t_{lat}}} \right)} & { - \sin \left( {\omega \Delta {t_{lat}}} \right)} \cr {\sin \left( {\omega \Delta {t_{lat}}} \right)} & {\cos \left( {\omega \Delta {t_{lat}}} \right)} \cr } } \right]\left[ {\matrix{ {{x_{\alpha \left( k \right)}}} \cr {{x_{\beta \left( k \right)}}} \cr } } \right] = \left[ {\matrix{ {\cos\left( {\omega \Delta {t_{lat}}} \right)} & { - \sin\left( {\omega \Delta {t_{lat}}} \right)} \cr } } \right]\;\left[ {\matrix{ {{x_{\alpha \left( k \right)}}} \cr {{x_{\beta \left( k \right)}}} \cr } } \right] The matrix rotates the R controller outputs by an angle equivalent to the time latency multiplied by the angular velocity of the R controller. The resulting C matrix is shown in the Eq. (24).

To completely eliminate latency effect on R controller, the direct link between control error and the controller output (D matrix in state space model) behaving like a proportional controller can be calculated according to Eq. (25). This component can usually be neglected when PR controller is used due to the low impact to the complete proportional gain. (25) yk=cosωΔtlat−sinωΔtlat xαkxβk+KRsinωΔtlatuk {y_{\left( k \right)}} = \left[ {\matrix{ {\cos\left( {\omega \Delta {t_{lat}}} \right)} & { - \sin\left( {\omega \Delta {t_{lat}}} \right)} \cr } } \right]\;\left[ {\matrix{ {{x_{\alpha \left( k \right)}}} \cr {{x_{\beta \left( k \right)}}} \cr } } \right] + {K_R}\sin\left( {\omega \Delta {t_{lat}}} \right){u_{\left( k \right)}}

The complete equations with latency compensation of advanced PR controller are: (26) xαkxβk=cosωΔt−sinωΔtsinωΔtcosωΔtxαk−1xβk−1+KRsinωΔt1−cosωΔtukyk=cosωΔtlat−sinωΔtlatxαkxβk+KP+KRsinωΔtlatuk, \matrix{ {\left[ {\matrix{ {{x_{\alpha \left( k \right)}}} \cr {{x_{\beta \left( k \right)}}} \cr } } \right] = \left[ {\matrix{ {\cos \left( {\omega \Delta t} \right)} & { - \sin \left( {\omega \Delta t} \right)} \cr {\sin \left( {\omega \Delta t} \right)} & {\cos \left( {\omega \Delta t} \right)} \cr } } \right]\left[ {\matrix{ {{x_{\alpha \left( {k - 1} \right)}}} \cr {{x_{\beta \left( {k - 1} \right)}}} \cr } } \right] + {K_R}\left[ {\matrix{ {\sin \left( {\omega \Delta t} \right)} \cr {1 - \cos \left( {\omega \Delta t} \right)} \cr } } \right]{u_{\left( k \right)}}} \hfill \cr {{y_{\left( k \right)}} = \left[ {\matrix{ {\cos\left( {\omega \Delta {t_{lat}}} \right)} & { - \sin\left( {\omega \Delta {t_{lat}}} \right)} \cr } } \right]\left[ {\matrix{ {{x_{\alpha \left( k \right)}}} \cr {{x_{\beta \left( k \right)}}} \cr } } \right] + \left( {{K_P} + {K_R}\sin\left( {\omega \Delta {t_{lat}}} \right)} \right){u_{\left( k \right)}},} \hfill \cr } where u_(k) = e_(k) is the difference between required and actual/measured value and y_(k) is the PR controller output.

For comparison purposes, three implementation alternatives were selected: the Basic R controller Eqs (27) and (29) as an effective common R implementation, the R controller based on Tustin discretisation (Yepes et al., 2010) Eq. (30), and the first-order hold (FOH) discretisation (Yepes et al., 2010) Eq. (31).

The Basic R controller implementation is based on modification of forward Euler discretisation Eqs (27) and (28): (27) yk=yk−1+KRωuk−ωxβk−1Δt {y_{\left( k \right)}} = {y_{\left( {k - 1} \right)}} + \left( {{K_R}\omega {u_{\left( k \right)}} - \omega {x_{\beta \left( {k - 1} \right)}}} \right)\Delta t (28) xβk=xβk−1+ωyk−1Δt, {x_{\beta \left( k \right)}} = {x_{\beta \left( {k - 1} \right)}} + \omega {y_{\left( {k - 1} \right)}}\Delta t, where u_(k) = e_(k) is the difference between the required and the actual/measured value, y_(k) is the R controller output. Using pure forward Euler discretisation, the sampling frequency of the PR controller discretised by Euler method must be around 1,000 times higher, than its resonant frequency to function properly (Yepes et al., 2010). Furthermore, the forward Euler implementation shifts the R resonant frequency. A basic and straightforward improvement of the forward Euler algorithm applies the result of the output Eq. (27) directly to update the second Eq. (28) resulting in Eq. (29): (29) xβk=xβk−1+ωykΔt {x_{\beta \left( k \right)}} = {x_{\beta \left( {k - 1} \right)}} + \omega {y_{\left( k \right)}}\Delta t

A similar result can be achieved when the Eq. (29) is calculated first and its result x_β(k) is immediately used for the calculation of the R output y_(k) Eq. (27) instead of x_β(k−1). This Basic algorithm of the R controller is computationally very effective (especially with variable frequency), suitable for many applications and works with a lower sampling frequency compared to forward Euler. However, for a high-power converter with low switching frequency (i.e. low sampling frequency and large control delays) which contains R controllers for higher harmonics, this R controller implementation is still insufficient and even unstable. This fact is demonstrated in later section see Figure 14.

The Tustin and FOH R controller implementations are in Eqs (30) and (31), respectively. Both the Basic and the Tustin discretisation methods shift the resonant frequency, each in the opposite direction (Figure 5). In contrast, the proposed advanced R and the FOH R controllers yield identical results, which perfectly match the ideal R controller characteristics. (30) Rtustinz=2Δt1−z−2ω2Δt2+4+z−12ω2Δt2−8+z−2ω2Δt2+4 {R_{tustin\left( z \right)}} = 2\Delta t{{1 - {z^{ - 2}}} \over {\left( {{\omega ^2}\Delta {t^2} + 4} \right) + {z^{ - 1}}\left( {2{\omega ^2}\Delta {t^2} - 8} \right) + {z^{ - 2}}\left( {{\omega ^2}\Delta {t^2} + 4} \right)}} (31) RFOHz=1−cosωΔt1−z−2ω2Δt1−2z−1cosωΔt+z−2 {R_{FOH\left( z \right)}} = {{\left( {1 - \cos\left( {\omega \Delta t} \right)} \right)\left( {1 - {z^{ - 2}}} \right)} \over {{\omega ^2}\Delta t\left( {1 - 2{z^{ - 1}}\cos\left( {\omega \Delta t} \right) + {z^{ - 2}}} \right)}}

Figure 5.

Influence of different discretisation techniques on R controller Bode’s characteristics. FOH, first-order hold.

The influence of the control latency on the R control part of the system in the frequency domain is shown in Figure 6. The figure illustrates the effect of delay introduced by both the computation and the modulator (see Section 2.5). This delay causes a significant phase shift of uncompensated R controllers. The uncompensated R controller shown in the figure uses the FOH discretisation method. The similar results would be obtained with an uncompensated advanced R controller. The ideal frequency characteristic, used as a reference, corresponds to a continuous-time R controller without latency. The advanced R controller with delay compensation closely follows the ideal curve up to the resonant frequency, exhibiting only a small, negligible phase error at heavily damped frequencies.

Figure 6.

Bode’s characteristics of discrete R controllers with ARB control system latency and ideal characteristic of continuous R controller without latency. ARB, advanced rail balancer; FOH, first-order hold.

The main advantage of the proposed controller over FOH lies in its simple implementation and effective control latency compensation. Furthermore, decomposition of the R controller output into two orthogonal (complex) components [x_α, x_β] allows determination of amplitude xα2+xβ2 \left( {\sqrt {{x_\alpha }^2 + {x_\beta }^2} } \right) and phase tan−1xαxβ {\rm{ta}}{{\rm{n}}^{ - 1}}\left( {{{{x_\alpha }} \over {{x_\beta }}}} \right) of the R controller output. This information can be used for anti-windup correction, feedforward computations and other control enhancements.

On the contrary, the computational effort required by the advanced R controller compared to the forward Euler or its Basic modification involves four additional operations (multiplications or additions) and three more operations for delay compensation with constant resonant frequency (constant R coefficients).

The AC RL load (P₍₁₎ = 8.66 kW, Q₍₁₎ = 3.4 kVAr) is used for testing. Figure 9 shows the operation of the TSS without ARB functionality. In this case, the catenary current I_cat equals the grid current I_g1, while I_g2 = −I_g1 and I_g3 = 0 A. This configuration results in the negative sequence component being 100% of the positive sequence component. The grid current is unbalanced and phase-shifted—I_g1 leads the grid phase voltage U_g1, by 8.5°, indicating a capacitive character relative to the grid voltage, in contrast to the expected inductive character with respect to the catenary voltage U_g12.

Figure 10 shows the same RL load, but with ARB balancing enabled, excluding harmonic current filtration. While the catenary current I_cat remains unchanged, the amplitude and phase shift of I_g1 relative to U_g1 are reduced. However, significant harmonic distortion remains in the grid current, primarily caused by the ARB itself, particularly due to dead times and voltage drops across the IGBTs.

Figure 11 displays the internal ARB symmetrising currents I_CHB without harmonics filtration, along with the 9-level output voltage U_CHB12 of the corresponding ARB branch during the symmetrisation process. The resulting grid currents are presented in Figure 12. The currents are successfully symmetrised, and the negative sequence component is reduced to 1.15%. Nevertheless, the grid currents still contain prominent odd-order harmonics (3rd, 5th, 7th and 9th).

Figure 13 illustrates the performance of a conventional R controller implemented using the basic discretisation method as shown in Eqs (27) and (29), and higher-order harmonics filtration. This controller exhibits divergent behaviour, especially at higher resonant frequencies, due to the ARB’s control delay, large sampling time and low switching frequency. It is caused by a large basic R numerical error and uncompensated control delay (see Section 3.4).

In contrast, the control employing the proposed advanced R controller with delay compensation demonstrates great performance even in harmonic current filtration, as shown in Figure 14.

The diode bridge rectifier with an RL load (P₍₁₎ = 6.7 kW, Q₍₁₎ = 4.3 kVAr) is used to simulate a non-linear and unbalanced load (similar to older types of locomotives). Figure 15 shows the operation of the TSS without ARB functionality. In this case, the catenary current I_cat equals the grid current I_g1, current I_g2 = −I_g1 and I_g3 = 0 A. The load introduces significant harmonic distortion, causes a phase shift and results in unbalanced current distribution.

Figure 16 presents the internal ARB (CHB) branch currents during operation with both balancing and harmonics filtration active. Approximately 50% of the harmonic content of the load current is filtered by the CHB12 branch (I_CHB12), and the remaining 50% is handled by the CHB23 and CHB31 branches (I_CHB23, I_CHB31).

The final grid currents are shown in Figure 17. The currents are balanced and exhibit no phase shift relative to the respective grid phase voltages, with no significant harmonic distortion. Minor current spikes are visible during diode rectifier commutation events, affecting only I_g1 and I_g2, which is typical for this type of load. The further current spike reduction is limited by the Gibbs phenomenon together with the maximum order of harmonic filtration (up to 9th), which is constrained by the switching frequency of the converter, sampling frequency, DC-link voltage and size of filtering inductance.