Finite Control Set Model Predictive Control with Increased Prediction Horizon for a 5 Level Cascaded H-Bridge StatCom N.D. Marks1 , T.J. Summers2 , R.E. Betz3 School of Electrical Engineering and Computer Science University of Newcastle, Australia, 2308 email:1
[email protected] 2
[email protected] 3
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Keywords , ,
Abstract Model Predictive Control (MPC) can provide performance benefits over traditional schemes in power electronic converter control. A key variable in predictive control schemes is the prediction horizon. This paper examines the impact of increasing the prediction horizon of a five level StatCom via simulation and experimental results.
Introduction MPC has recently found use in the field of power converter control due to the advantages it offers for simultaneously controlling multiple control objectives [1, 2]. MPC is particularly useful for the control of multilevel converters where significant switching redundancy exists [3, 4], where capacitor voltage balancing is a key control objective [5], and for emerging photovoltaic applications where multiple maximum power point tracking algorithms can be incorporated [6, 7]. The focus of this paper is to examine the impacts of increasing the prediction horizon on the control performance with particular focus on the number of switching transitions.
Model Predictive Control for Power Converters MPC for power converters is termed Finite Control Set Model Predictive Control (FCS-MPC) as the available control actions are limited to the set of switching states or vectors that the converter can produce [8]. The objective of FCS-MPC is to determine which switching state is the most desirable to apply by using a model of the system to predict the trajectories of system variables [1, 2]. The most desirable state minimises a cost function which defines the control objectives. The cost function penalises the control objectives based on their relative importance [8].
Fig. 1: Per phase grid connected 5 level cascaded h-bridge The general quadratic cost function form is shown in (1), and (2) shows a cost function where control objectives are weighted according to their relative importance, in this case current tracking and the number of switching transitions. g = |x∗ − x|2
(1)
g = |i∗ − iˆ|2 + λSW nSW
(2)
where, i∗ is the desired current, iˆ is the predicted current, λSW is the weighting coefficient for the number of switching transitions and nSW is the number of switching transitions between the switching combination under test and the last one applied. The weighting factors can be adjusted to tradeoff the performance of the control objectives [8]. The value of the weighting factors to achieve optimal performance continues to be the subject of current research, but in general they are found via empirical methods [2, 5]. The discrete time model of the grid connected system shown in Fig. 1 in space vector form is given by: vk = Rik +
L [ik − ik−1 ] + vsys k Ts
(3)
where, vk is the converter output voltage at time k, ik is the inductor current at time k, ik−1 is the inductor current at time k − 1, vsys k is the system voltage at time k and Ts is the control interval. The converter output voltage is given by: vk = VDC Sk
(4)
where, VDC is the DC link voltage and√Sk = (Sa + aSb + a2 Sc ) where Sa , Sb , Sc represent the state of the switches in each phase and a = − 21 + j 23 . 2
Fig. 2: System timing diagram The inductor current is generally the primary control objective in power converter systems, so rearranging (3) gives: ik =
1 [Lik−1 + TS vk − TS vsys k ] RTS + L
(5)
Advancing by one time step: ik+1 =
1 [Lik + TS vk+1 − TS vsys k+1 ] RTS + L
(6)
The prediction horizon can be increased to base the desired control action on more information of the future system behaviour. The predicted current equation then becomes: ik+N =
1 [Lik+N−1 + TS vk+N − TS vsys k+N ] RTS + L
(7)
where N is the prediction horizon. The cost function becomes: g = |i∗k+1 − ik+1 |2 + λSW 1 nSW 1 + |i∗k+2 − ik+2 |2 + λSW 2 nSW 2 + |i∗k+N − ik+N |2 + λSW N nSW N
(8)
∗ The predicted values of the grid voltage, vsys k+N , and the reference currents, ik+N , can be obtained via a digital phase locked loop (PLL) at the grid frequency and advancing the phase by the required number of time steps. A digital PLL has been used in this paper and produces very accurate forward predictions.
To implement FCS-MPC in a real system, the calculation time must be taken into account as the measurement of variables and the prediction calculations cannot be performed instantaneously [2]. To accomodate this, the timing structure shown in Fig. 2 has been used [5, 9]. The initial sample time is (k − 1)T where the three phase inverter currents and system voltages are measured and the previously calculated switch combination is applied. A PLL is utilised to synchronise the control system with the fundamental grid voltage and i∗k+1 and vsys k+1 , are obtained via forward estimation within the PLL. At (k − 0.5)T , system variables are measured again and the FCS-MPC algorithm is executed. The inverter voltage, vk , is known and (5) is used to predict 3
Fig. 3: Simulation waveforms of ’a’ phase current using three phase vector algorithm (left) and per phase algorithm (right)
Fig. 4: Block diagram of system topology ik . Equation (7) is then used in conjunction with (8) for each switching combination sequence available over the prediction horizon, N, with the most desirable sequence minimising (8). The desired switching combination is then applied at k and the process is repeated with Fig. 2 shifting forward one time step. Increasing the prediction horizon has the potential to increase the performance of the system [8]. The main drawback of increasing the prediction horizon is the geometric increase in the number of computations that are required [10, 11]. For a full evaluation of all switching combinations, each combination must be tested for each step. The number of computations becomes very large for converters with a high level number [5, 12, 13]. High level numbers can be a problem even for a prediction step of one and a number of papers have focused on ways of reducing computation time or reducing the number of combinations evaluated [5, 10, 11, 13]. Parallel processing has also been proposed as a solution [14, 15]. Some recent literature has focused on extending the prediction horizon in standalone systems and drives [10, 16, 17]; however, there has not been an explicit study of higher prediction horizons for grid connected converters. The aim of this paper is to determine whether there is a benefit to increasing the prediction horizon to justify the higher cost of high performance microprocessors or parallel computing. The FCS-MPC algorithm for a three-phase, multilevel system can be executed in a number of ways. The first method considers the 125 three phase space vectors as available control inputs, and the second considers a sensible subset of the available switching states on a per phase basis, as suggested in [5]. The evaluation of all 125 vectors at each time step leads to a large number of calculations for higher prediction horizons and higher level number whereas the advantage of the per phase algorithm is the lower computational load, making the method more suitable for extension to a larger number of levels. Examples of the current reference tracking ability of each method are shown in Fig. 3. The per phase implementation has larger ripple since there is less switching combination redundancy. The total harmonic distortion (THD) of the ’a’ phase current is 9.92% 4
Table I: System Parameters Parameter
Specification
System Voltage
240Vrms (l−n)
System Frequency
50 Hz
Total DC Link Voltage1φ
360V
Control Frequency
10 kHz
pu
Sbase3φ
10 kVA
pu Vbase
240Vrms (l−n)
LC
0.05 pu
for the vector algorithm and 10.88% for the per phase algorithm. The per phase algorithm has reasonable performance with a significant reduction in the number of switching combination evaluations.
Simulation Results A simulation has been developed for a three phase, five level cascaded h-bridge multilevel StatCom using the Saber® simulation package. A block diagram of the simulation system is shown in Fig. 4 and the system parameters are shown in Table I. The FCS-MPC algorithm is implemented as a ’C’ code DLL which interfaces with the Saber® platform. The per phase FCS algorithm is utilised with the cost function of (8) implemented with the individual capacitor target voltage equal to 180V. The total stack voltage is controlled via a PI controller which generates a current reference to regulate the stack voltage to 360V. A number of simulation cases have been considered for both N = 1 and N = 2, and varying values of the switching transition coefficient, λSW , from zero to three.
Ideal System Conditions An ideal stiff grid system with fixed DC sources in the DC links of the CHB is considered here. The cost function in this case considers only the error in the current tracking and the number of switch transitions between control periods. Fig. 5 shows results for the per phase algorithm with prediction horizons of N = 1 and N = 2. The number of transitions is generally lower when the prediction horizon is extended. The maximum difference in the number of transitions between the N = 1 and N = 2 plots is 341 at λSW = 2.5, corresponding to a reduction of 24%. This is a significant reduction in the number of switching transitions with only minor deterioration in the output current performance.
Capacitor Voltage Balancing Included in Cost Function The system is modified by removing the fixed DC sources in the DC links of the CHB and a voltage balancing penalisation is included in the cost function for the FCS-MPC scheme as shown in (9). g = |i∗k+1 − ik+1 | + λSW 1 nSW 1 + |i∗k+2 − ik+2 | + λSW 2 nSW 2 + αv vcaperror1 + αv vcaperror2
(9)
It is expected that more transitions will be required relative to the ideal case as the control scheme is forced to regulate the capacitor voltages in order for the system to operate correctly and the reduction in the number of switch transitions as λSW is increased should be lower. 5
Fig. 5: Switching transitions vs. λSW for N = 1 and N = 2 - Ideal system (left), with capacitor balancing (right) Fig. 5 shows the number of transitions for N = 1 and N = 2. These results have been obtained with a voltage penalty coefficient of 0.1. In general as λSW is increased, the number of transitions is lower for N = 2. The largest reduction in the range considered occurs at λSW = 3 where the reduction is 10.8%.
Voltage Sag at Point of Common Coupling Voltage dips and sags can occur in the grid system as a result of large loads such as motors turning on, losing feeders and loss of generation sources. A voltage sag and recovery of approximately one second has been simulated. The voltage sag lasts approximately 0.6 seconds and the recovery is approximately 0.4 seconds. Fig. 6 shows the number of transitions for N = 1 and N = 2 where the voltage penalty coefficient is 0.5. The results show that the reduction in transitions with an increased horizon is not very large. The greatest reduction occurs at λSW = 1.4 which is 8.1%. The lower reduction in the number of transitions as the horizon is increased during a voltage sag is due to the inacccuracy introduced into FCS-MPC. The voltage estimates from the PLL used to generate the current references are not accurate for increased horizons since the voltage reduction that would occur between control intervals is not taken into account. Thus the information the FCS-MPC scheme is using to make the best decision on is not an accurate representation of the system. Therefore the reduction in the number of transitions as the horizon is increased is lower.
Harmonic Pollution in Grid Voltage The grid voltage can contain harmonic components resulting from the use of power electronic devices such as drives. The peformance of the FCS-MPC has been investigated when the harmonic components are not explicitly considered. The 5th and 7th harmonics have been included in the grid voltage at 5% of the fundamental magnitude. Fig. 6 shows the number of transitions for N = 1 and N = 2 where the voltage penalty coefficient is 0.5. The general trend for greater penalisation on the switching is a reduction in the number of transitions when the 6
Fig. 6: Switching transitions vs. λSW for N = 1 and N = 2 - Voltage sag (left), harmonics in voltage (right) horizon is increased. For smaller penalisations (λSW < 1), there is no benefit to increasing the horizon. The greatest reduction in transitions is 12.1% and occurs at λSW = 2.5. The harmonic content of the grid voltage manifests itself in the converter current and the DC link voltages. However, the reference current for the converter is kept relatively free of harmonic content as a fundamental frequency PLL is used to generate the reference. The impact of harmonics in the converter current is a larger number of transitions clustered around the positive and negative amplitudes as the 5th and 7th harmonic distortion is most prevalent there. It appears that as the switching penalisation is increased, some of these transitions are neglected and even more so when the prediction horizon is increased.
Experimental Results In order to validate the simulation results, the ’C’ code DLL is modified to execute in the real-time environment. This system is a 415VAC 19 level H-Bridge prototype for an 11kV direct connect StatCom and a block diagram is shown in Fig. 7. The system has been augmented to operate with a five level output voltage and uses the per phase algorithm to reduce the number of computations. The timing structure shown in Fig. 2 is used with a control frequency of 2.5kHz. Experimental results have been obtained using a 28V grid. The control algorithm implements the cost function shown in (9) with the capacitor voltage target equal to the average of the two capacitor voltages in each phase. The total stack voltage is controlled via a PI controller to a reference of 1.5 times the supply peak voltage. The number of transitions is measured as the change in the the number of capacitors and as such the reduction in transitions exhibited is expected to be lower. Fig. 9 and Fig. 10 show the capacitor voltages and the number of transitions for N = 1 and N = 2 with λSW = 0 respectively. The capacitors are regulated to approximately 29.5V as expected. The capacitor voltage ripple is larger for N = 2 but the number of transitions is lower, 5675 compared to 5720 for N = 1. This confirms the trade-off between the control objectives and also shows that when the prediction horizon is increased, the number of transitions can be reduced. Fig. 8 shows the measured and reference currents for N = 1 and N = 2. There is no noticeable difference in the tracking performance since the penalisation on the transitions is zero and the capacitor voltage penalty 7
Fig. 7: Block diagram of experimental system
Fig. 8: Experimental results showing ’a’ phase current tracking: N = 1 (left) and N = 2 (right) is low compared to the current penalty. As such, the current tracking performance is not expected to change noticeably when increasing the prediction horizon in this case.
Contributions and Conclusions This paper has presented an explicit investigation of increasing the prediction horizon from one to two in a FCS-MPC scheme applied to a 5 level, cascaded h-bridge StatCom. The purpose was to determine the worth in using an increased horizon by quantifying the benefit in terms of switching transition reduction. The investigation suggested that in an ideal system the number of transitions could be reduced by as much as 24%. However, when the scheme is applied to non-ideal system conditions such as voltage sag and harmonic pollution, and when the DC link capacitors are required to be regulated, the reduction is much lower. Under these conditions, the maximum reduction is approximately 10%. Experimental results showing the implementation of FCS-MPC for both N = 1 and N = 2 have been presented. They confirm the correct operation of the scheme and that a small reduction in transitions is achieved when the prediction horizon is increased. The results show that the transition reduction benefit is minor and especially so when the system is not ideal. This investigation is not definitive or exhaustive and further study will be performed to determine the conditions where extending the horizon can provide significant benefit, as well as determining how far the horizon should be extended. 8
Fig. 9: Experimental results showing ’a’ phase capacitor voltages (top) and ’a’ phase transitions (bottom) for N = 1
Fig. 10: Experimental results showing ’a’ phase capacitor voltages (top) and ’a’ phase transitions (bottom) for N = 2 9
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