Cooperative Control of Active Power Filters in Power Systems without Mutual Communication

The procedure for calculating controller parameters of the APFs implemented into a multibus industrial power system for harmonic voltage mitigation is presented. The node-voltage-detection control strategy is applied and the basic controller parameters are found under the condition that the demanded THD factors at the buses where the APFs are placed will be obtained. A cooperative control of several APFs without mutual communication is proposed, simulated, and experimentally verified. By tuning the controller gains without considering the power circuit parameters, all APFs used tend to share harmonic load currents approximately equally regardless of the operation modes of the nonlinear loads in different parts of the power system.


Introduction
The performance and effectiveness of the active power filter APF depends on the point of its connection in the multibus power system 1-3 .
The effectiveness of the bus voltage and branch current detection control strategies of the APFs used in multibus power systems was analyzed and compared in 4 .In 5 the single APF-multiple harmonic optimization problem was formulated in such a way that its solution yields optimum PI controller parameters of the APF for both the control methods.
In 6 , an analytical solution of the multiple APF-single harmonic problem in the frequency domain was formulated.In the case study presented in 6 optimal compensator currents for individual harmonics in case of two APFs were calculated, interaction of the APFs analyzed and, finally, the best places for two APFs recommended.In 7 , an automatic gain adjustment of the APFs was introduced to make each APF damp out harmonic propagation throughout power distribution systems without considering the circuit parameters.The gain adjustment can reduce the APF compensating currents and losses.

Mathematical Problems in Engineering
In 8 , a cooperative control of several APFs with the help of a communication system was proposed.Again, the APFs damp out harmonic propagation throughout power distribution systems.The APFs control gains are tuned by a computer on the basis of voltage total harmonic distortion THD at each APF place of installation.The control reduces the voltage THD at buses of the APFs installation and balances the compensating currents as well, irrespective of the power system conditions.
An instantaneous current control loop to ensure proper sharing of harmonic components of nonlinear loads among inverters in distributed generation systems is included in wireless controllers of these inverters 9 .Contrary to 7, 8 , the output-voltage THD of distributed inverters are not directly included into the gain adjusting mechanism, although the algorithm helps to hold the voltage THD at a reasonable level.
Also in 10 , a harmonic droop scheme for sharing harmonic content of the load currents among distributed generation systems is proposed, being a part of a load-sharing controller that uses only low-bandwidth data communication signals among individual generation systems.
Finally, a distributed APF system for mitigating the harmonic distortion in power systems is proposed in 11 .A droop control strategy based on the VA rating of individual APFs ensures the distribution of the harmonic filtering workload in proportion to the rated capacity of every APF unit.The algorithm does not need the real-time communication among APF units.Again, the output-voltage THD of distributed inverters are not directly included into the gain-adjusting mechanism in 10, 11 .
So, no wireless cooperative control strategy considering both the output-voltage THD of distributed inverters and the rated capacity of every APF has been reported till now.
Here, the multiple APF-multiple harmonic problem in case of applying the node-voltagedetection control strategy is formulated.A cooperative control without any communication of several APFs is proposed, simulated, and experimentally verified.The control aims, by tuning the controller gains without considering the power circuit parameters, at attaining the demanded THD of the voltages at the buses where the APFs are placed, provided that the VA ratings of individual APFs are not exceeded.If the load of an APF tends to be higher than its rated capacity, the algorithm tunes the APF controller gain further to hold the APF below its nominal rating.Additionally, the APF is prevented from a too low load as well by applying this procedure.Thus, all APFs used tend to share harmonic load currents approximately Mathematical Problems in Engineering equally regardless of the operation modes of the nonlinear loads in different parts of the power system.

Multibus Industrial Power System
An example of the multibus industrial power system shown in Figure 1 consists of overhead lines, cables, transformers, induction motors, rectifiers, and passive filters.Some simplifications will be considered: the transformers and induction motors will be represented by inductances only, passive filters considered as ideal ones, without resistances, and ground capacitances of the lines and cables will be neglected as well.The rectifiers will be represented by ideal harmonic current sources I IL with harmonic magnitudes determined by respective apparent powers, overload capacities, and by the so-called magnitude law.On the basis of the diagram of this system an equivalent circuit may be determined and the parameters of circuit elements calculated.

Source voltage
Source + line impedance Linear load 1 Downstream loads The potential places for APFs are bus-bars with the voltage level of 6 kV, that is the bus-bars 2-6.APFs will be represented here by voltage-controlled current sources, that is by the vector I IF of reference currents of the individual APFs.The agreement between the real and reference APF currents depends on an inner control strategy of the real APF that is not analyzed here.Especially for dominant harmonic orders this agreement is reported to be quite well, though.
In the following analysis all quantities will be recalculated to the 6 kV level and expressed in the p.u. system with S B 1 MVA the maximum power of the major rectifier, which is connected to the bus 6 , V B 6 kV, 50 Hz.

Control Strategies
The voltage and current detection feedback control strategies belong among control strategies very often used for parallel APFs applied in simple power distribution systems.We proposed taken into consideration as well, with regard to problems with stability of the whole control system.
We will analyze an application of the voltage detection feedback control strategy in multibus industrial power systems with a few harmonic power sources and several APFs.
Let the current vector I IF of APFs be generated by using a feedback of the node voltage vector where Y G is the feedback gain matrix.

Multiple Harmonic Problem
For the multiple APF-multiple harmonic problem the objective function may be written where the weighting factor a i 0 ≤ a i ≤ 1, i 1, . . ., n − 1 reflects the different importance of the harmonic voltage mitigation at different nodes.
The formulae for the hth harmonic of the voltage V h N i at the node i, which include the parameters of the controllers of APFs, should be formulated.The optimum values of the controller parameters may be determined by finding the minimum of g H 4.1 for V h N i declared.
Here, the objective function 4.1 will be used for solving the multiple APF-multiple harmonic problem in case of applying the node-voltage-detection control strategy.For finding optimum parameters of the controllers of the APFs, the relation for the hth harmonic of the voltage V h N i , determined in 5 for the single APF, must be reformulated to be valid for more than only one APF placed at the node f.Because where Z h NIF is a transfer impedance matrix of the injected APF currents, V h N0 is the node voltage vector without any compensation, and determines the node-voltage-detection control strategy, the node voltage vector may be expressed by The elements of this voltage vector enter the objective function 4.1 .
In the process of minimization of the objective function the optimum parameters Y Gi , i 1, . . ., m F for the m F controllers can be found.
But, in most of real cases the minimum value of the objective function, which is the power of the global value of THD of voltages in p.u. system and over the whole number of bus-bars in question , is not necessary to reach.The attaining of predefined values of THDs of the voltages at the bus-bars where the APFs are placed seems to be a more practical aim.Then, the harmonic currents generated by the APFs may be remarkably lower than those needed in case of the mentioned global optimization obtained by the minimization of the objective function 4.1 .
But, a question still remains whether THDs of voltages at the other bus-bars where the APFs are not placed will be sufficiently low to conform to our demands.Thus, comparing these THDs of voltages at remaining bus-bars of the industrial power system for different combinations of APFs places may be a key to find the best places for the APFs in this case.
For finding the parameters of the controllers for two APFs, which ensure that the predefined values of THD * V N i , i f1, f2 at the bus-bars f1, f2 are attained, the objective function 4.1 should be modified as follows 4.5

Self-Tuning Controller of APF
By using the strategy presented in the previous paragraph we can find a proper place and controller parameters, or at least to identify places unsuitable for the APF.The analysis assumes that the system parameters and harmonic current sources are known.Such an analysis may be done as the first step of the solution of the problem with some nominal system parameters and operating modes of harmonic current sources mostly rectifiers .But, the parameters in a real system are not precisely known and, especially, the harmonic spectra of the current sources are changing due to changes in a manufacturing process.Thus, a simple and efficient way to tune the controller parameters of the APF is very desirable.
We have two indicators to be checked, namely the current I IFRMS that should not exceed its nominal value for the APF used and the real value of THD V N f that should not be much higher than its prescribed limit THD * V N f , because in such a case there is a real danger of an increase of THD V N i at the remaining bus-bars and of the global index THD V N all over the system too.On the other hand, an operation of the APF with , is not very economical either, and it is possible either to switch the APF off or to decrease THD * V N f and, as a consequence, the values of THD V N i and THD V N as well.Figure 2 shows the scheme of a self-tuning controller of the APF.The input variables are the reference THD * V N f and I * IFRMS .The required value of THD * V N f is compared with the real value THD V N f , and the error enters a block where the gain Y GP of the APF controller is determined.
A possible configuration is that the basic value Y * GP of the gain is modified by the output of the I-type controller with fairly low gain.Because the higher the negative value of the control error − THD * V N f − THD V N f is, the higher the absolute values of the gain Y GP and I IF are; a limiter of the presented type should be used.
The real value I IFRMS of the AFP current is compared with its required nominal value I * IFRMS and the error is proceeded in a nonlinear block, whose output let us call it as constant C modifies the reference value THD * V N f of the voltage at the bus where the APF is placed.Thus, the reference THD * V N f is modified in order that it may be obtained without exceeding the required nominal value I * IFRMS of the current generated by the APF.If the error I * IFRMS − I IFRMS is positive up to some specific value, the reference THD * is not modified because the output of the nonlinear block has the value 1.But, for I * IFRMS − I IFRMS < 0 the output of the block is going up quickly and increases the value of THD * V N f , which leads to a decrease of the APF current I IFRMS and the error I * IFRMS − I IFRMS becomes lower.On the other hand, a big positive error I * IFRMS − I IFRMS can result in that the lower value THD * V N f is required and achieved.Another option is that this big positive-error value may be the impetus for switching the APF off.
It is evident that the control loop of the APF current I IFRMS is superior to the control loop of THD V N f of the voltage where the APF is placed.

Simulation
Let us verify some key results, conclusions, and also the function of the controller proposed in the previous chapter by the simulation in the time domain.
The industrial power system shown in Figure 1 was simulated by using Matlab-Simulink.
The APFs were placed at the bus-bars 3 and 4 and equipped with the controller presented in Figure 2. Time and all variables are expressed in p.u. related to the nominal frequency 50 Hz, the voltage 6 kV, and to the maximum power of the biggest rectifier connected at the bus-bar 6.
The reference variables were set like this: THD * Figure 3 a shows transient processes following after increasing twice the currents withdrawn by the rectifiers at the buses 7 and 8 and decreasing to zero the currents withdrawn by the rectifiers at the buses 9, 10, 11, and 6.The responses of the following variables are shown from up to bottom: I IFiRMS , constant C i at the output of the nonlinear block modifying the reference value THD * V N j , THD V N j , Y GPi , i 1, 2 number of the APF , and j 4, 3 the place of the APF .The control loops of I IFiRMS are disconnected now, so changes of the constants C i do not affect THD * V N j 0.009.We can see that the APF2 placed at the bus 3, which lies upstream the buses 9-12, goes out of the operation I IF2RMS Y GP2 0, while the current I IF1RMS 0.181 p.u. of the APF 1 becomes higher than its nominal value.The reason is that the value of THD V N 3 0.0081 is lower than its reference THD * V N 3 0.009 in the final steady state.

Experimental Verification
The function of the proposed control strategy of several APFs in the power system without their mutual communication has been experimentally verified.Figure 4 shows an experimental setup that consists of two pairs of a nonlinear and linear load.
These loads are connected to the source voltage via different impedances Z LINE1 and Z LINE2 that represent impedances of the lines, cables, and transformers of a real power system.Two APFs are placed in parallel with these load pairs.The control algorithm presented in Figure 2 is implemented into both dSPACE DS1103 control systems of APFs.
Figures 5-7 demonstrate the function of the control loop of THD V N 1 of the APF1. Figure 8 shows how the superior current control loops prevent overloading of the APFs.
Figure 5 shows the waveforms of the load current a , grid current in the branch 1 b , and line-to-line voltage c after switching the load in this branch 1 Figure 4 on.
Figure 6 presents the respective details of the load current a , grid and APF1 currents b , and THD of line-to-line voltage c after switching the load in the branch 1 on.The current i su i fu in Figure 6 b before the load in the branch 1 is switched on is generated by the APF1 to hold THD V N 1 equal to THD * V N 1 0.06, because the full load in the branch 2 is applied.In Figure 6 c two waveforms are shown: the first one is the signal THD V N 1 in the APF digital controller, the second one is a result of analysis of the captured voltage v luv from Figure 5 c made a posteriori the experiment.Both the curves feature a very good agreement.
Figure 7 shows the waveforms of the load, APF, and grid current in the branch 1 in steady state.It is evident that load current harmonics are effectively mitigated by the respective APF1.
Figure 8 shows transient processes following after: 1 increasing nonlinear load 1 by 50% and 2 disconnecting nonlinear load 2.
It is evident that the APF 1 is overloaded and the APF 2 is switched-off if the APF control loops of I IFiRMS , i 1, 2 are disconnected and only the control loops of THD V N 1 , THD V N 2 are in function.Contrary to that, both the APFs remain in operation if the superior control loops of I IFiRMS , i 1, 2 of both the APFs are in function.The values of I * IF1RMS I * IF2RMS were set at 1 A and THD V N 1 THD V N 2 1.5%.The experimental results are in a good agreement with those obtained by simulation, Figure 3.

Conclusion
The procedure for calculating controller parameters of the APFs implemented into the multibus power system has been presented.The node-voltage-detection control strategy was applied and the controller parameters were found by solving the modified multiple APFmultiple harmonic problem under the condition that demanded THD factors at the buses where the APFs are placed will be obtained.
A cooperative control of several APFs without any communication has been proposed and simulated as well.By tuning the controller gains without considering the power circuit parameters, all APFs used tend to share harmonic load currents approximately equally regardless of the operation modes of the nonlinear loads in different parts of the power system.

Figure 3 :
Figure 3: Transient processes following after increasing twice the current withdrawn by the rectifiers at the buses 7 and 8 and decreasing to zero the currents withdrawn by the rectifiers at the buses 9, 10, 11, and 6; a control loops of I IFiRMS are disconnected, b control loops of I IFiRMS in function.

Figure 4 :
Figure 4: Experimental setup with two APFs working without communication.

Figure 5 :
Figure 5: Waveforms of the load current a , grid current in branch 1 b , and line-to-line voltage c after switching load in this branch 1 on.

1 TimeFigure 6 :Figure 7 :Figure 8 :
Figure 6: Details of the load current a , grid and APF currents b , and THD of line-to-line voltage c after switching load in this branch 1 on.

Figure 3 b
shows the similar transient processes, but the I IFiRMS control loops of the APFs are in function now.The result is that both the APFs remain in operation with I IF1RMS 0.133 p.u. and I IF2RMS 0.060 p.u., that is under the nominal value I * IFRMS 0.15 p.u., and with THD V N 4 0.0090 and THD V N 3 0.0078.