Control of Parallel-connected Inverters to Achieve Proportional Load Sharing

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1 Preprnts of the 8th IFAC World Congress Mlano (Italy) August 28 September 2, 2 Control of Parallelconnected Inverters to Acheve Proportonal Load Sharng QngChang Zhong Loughborough Control Centre, Loughborough Unversty Lecestershre LE 3TU, Unted Kngdom Tel: , fax: , Emal: zhongqc@eee.org. Abstract In ths paper, a completely new concept has been brought to the controller desgn for nverters. The capactor C of the nverter LC flter s regarded as a part of the load and, hence, the controlplant for the nverter controllers just the nductor L. Ths reduces the order of the control plant to be, reduces the varables to be measured for feedback to two, and consderably smplfes the desgn and analyss of the controller. The nherent lmtatons of the conventonal droop control scheme are then revealed and t s proved that parallelconnected nverters should have the same perunt mpedance n order for them to share the load accurately n proportonalto ther power ratngs f the conventonaldroop controlscheme s adopted. The droop controllers should also generate the same voltage setpont for the nverters. Both of these two are mpractcal and dffcult to meet, whch results n errors n proportonal load sharng. An mproved droop controller s then proposed to acheve accurate proportonal load sharng. It s robust aganstparameterdrfts and component msmatches. The strategyalso reduces the output voltage drop due to the effect of loadng and droop control so that the output voltage can be mantaned wthn the desred range around the rated value. Expermental results are provded to verfy the analyss and desgn. Copyrght c 2 IFAC. Keywords: Mcrogrds, droop control, parallel operaton of nverters, proportonal load sharng. INTRODUCTION Nowadays, more and more dstrbuted generaton and renewable energy sources, e.g. wnd, solar and tdal power, are connected to the publc grd va power nverters. They often form mcrogrds before beng connected to the publc grd (Guerrero et al., 29). Due to the avalablty of hgh current power electronc devces, t s nevtable that several nverters are needed to be connected n parallel for hghpower and/or lowcost applcatons. Another reason s that parallelconnected nverters provde system redundancy and hgh relablty, whch s mportant for crtcal customers. A natural problem for parallelconnected nverters s that how the load s shared among them. A key technque s to use the droop control (Guerreroet al., 25, 27, 2; Tuladhar et al., 997; Brabandere et al., 27; Zhong and Wess, 2), whch s wdely used n conventonal power generaton systems. The advantage s that no external communcaton mechansm s needed among the nverters. Ths enables good sharng for ether lnear or nonlnear loads (Guerrero et al., 25; Tuladhar et al., 997; Borup et al., 2; Tuladhar et al., 2). The equal sharng of lnear and nonlnear loads has been ntensvely nvestgated (Guerrero et al., 25, 27; Borup et al., 2) and hgh accuracy of equal sharng can be acheved. A voltage bandwdth droop control was used to share nonlnear loads n (Tuladhar et al., 997) and a small sgnal njecton method was proposed to mprove the reactve power sharng accuracy n (Tuladhar et al., 2), whch can also be extended to harmonc current sharng. It s ponted out n (Guerrero et al., 25) that the output mpedance of the nverters plays a crtcal role n power sharng and a droop controller for nverters wth resstve output mpedances s proposed for sharng lnear and nonlnear loads (Guerrero et al., 27). Although sgnfcant progress has been made for the equal sharng of lnear and nonlnear loads, t s stll a problem to share loads accurately n proportonal to the power ratngs of the nverters. In partcular, the accuracy of reactve power sharng (for the Q E and P ω droop) s not hgh (L and Kao, 29). Moreover, some approaches developed for equal sharng cannot be drectly appled to proportonal sharng. Another ssue s that the output voltage drops due to the ncrease of the load and also due to the droop control. Hence, the proportonal sharng problem needs to be nvestgated n a systematcal way. It has been recognsed that addng an ntegral acton to the droop controller s able to mprove the accuracy of load sharng for grdconnected nverters; see (Marwal et al., 24; L et al., 24; Da et al., 28). However, t s stll a problem for nverters operated n the standalone mode and also there s an ssue assocated wth the change of the operaton mode. A strategy, whch nvolves addng a vrtual nductor and estmatng the effect of the lne mpedance, was proposed n (L and Kao, 29) to mprove the stuaton but the strategy s qute complcated and there s stll room for mprovements. All these strateges are senstve to numercalcomputatonalerrors, parameter drfts and component msmatches, to the best knowledge of the author. In ths paper, t s proved that, n order for the parallelconnected nverters to share the load n Copyrght by the Internatonal Federaton of Automatc Control (IFAC) 2785

2 Preprnts of the 8th IFAC World Congress Mlano (Italy) August 28 September 2, 2 proportonal to ther power ratngs, the nverters should have the same perunt mpedance. It alsorequres thatthe RMS voltage setponts for the nverters to be the same. Both are verystrongcondtons. A robustdroop controller s then proposed to acheve accurate proportonal load sharng among nverters connected n paralleln mcrogrds operated n the standalone mode. The accuracy of sharng s no longer dependent on the output mpedance of the nverters orgnally desgned nor on the RMS voltage setpont. Moreover, the controller s able to regulate the output voltage to reduce the effect of the load and droop control on the output voltage. In ths paper, the robust droop controller s proposed for nverters wth resstve output mpedances and t can be appled to nverters wth nductve output mpedances as well, by usng the Q E and P ω droop. An approach s also proposed n ths paper to desgn an nverter to have a resstve output mpedance. Snce there s normally an LC flter n an nverter to reduce the swtchng nose n the output voltage, the approaches proposed n the lterature all treat the LC flter as the control plant and the controllers are all desgned based on ths fact, to the best knowledge of the author. Most of them adopt the nductor current, the output current and the output voltage as feedback sgnals. Some adopt the capactor current as feedback. In ths paper, a completely new concept s brought to the controller desgn for nverters, based on the observaton that the capactor can be regarded as a part of the load, nstead of a part of the controlplant. Hence, the controllercan be desgned accordngto the flter nductor only. Ths reduces the order of the control plant to one and smplfes the control desgn and system analyss. Moreover, only two sensors (for the nductor current and the output voltage) are needed for feedback, whch reduces the cost of the controller. Because of ths, the nverter can be desgned to have resstve output mpedance overa wde range of frequences, whch consderablyfacltates the sharngof nonlnearloads. The combnaton of the above leads to a very neat strategy to acheve accurate proportonal load sharng. 2. CONTROLLER DESIGN FOR INDIVIDUAL INVERTERS TO ACHIEVE RESISTIVE OUTPUT IMPEDANCE The crcut of a snglephase nverter under consderaton s shown n Fg. (a). It conssts of a snglephase Hbrdge nverter powered by a DC source, and an LC flter. The nverter s connected to the AC bus va a crcut breaker CB and the load s assumed to be connected to the AC bus. The control sgnal u s converted to a PWM sgnal to drve the Hbrdge so that the average of u f over a swtchng perod s the same as u,.e. u u f. Hence, the PWM block and the Hbrdge can and wll be gnored n the controller desgn. The nductor current s measured to construct a controller so that the output mpedance of the nverter s forced to be resstve and that t domnates the mpedance between the nverter and the AC bus. Moreover, the output voltage s measured, together wth the nductor current, for proportonalload sharng. Ths avods measurng the load current o and reduces the cost and complexty of the controller. As s now well known, t s advantageous to force the output mpedance of parallelconnected nverters to be resstve (Guerrero et al., 25). The nverter conssts of an LC flter and, to the best knowledge of the author, all control strateges proposed n the lterature have adopted a secondorder model for the nverter. Here, an mportant step forward has been made, that s to regard the capactor C as a part of the load nstead of a part of the nverter. Ths reduces the control plant to an Hbrdge and an nductor, as shown n Fgure (b). The advantages of ths are: ) t reduces the order of the control plant to be ; 2) t reduces the sgnals to be measured for feedback to one(excludng the feedback for voltage/power control); and 3) t consderably smplfes the desgn and analyss of the controller, whch facltates the understandng of the nature of nverter control. Snce the control plant s now of the frst order, the controller can be desgned wth ease. The proposed controller, as shown n Fgure 2, nvolves the feedback of the nductor current wth a proportonal gan. u u PWM + V DC IGBT brdge u f L o C CB (a) Used for physcal mplementaton PWM + V DC IGBT brdge u f (b) Used for controller desgn Fgure. A sngephase nverter u K L v r C CB AC bus AC bus Fgure 2. The proposed controller to acheve a resstve output mpedance The followng two equatons hold for the closedloop system consstng of Fgure (b) and Fgure 2: u = v r K, u f = sl +. Snce the average of u f over a swtchng perod s the same as u, there s (approxmately) v r K = sl +, 2786

3 Preprnts of the 8th IFAC World Congress Mlano (Italy) August 28 September 2, 2 whch gves wth = v r Z o (s) Z o (s) = sl + K. If the gan K s chosen bg enough, the effect of the nductance s not sgnfcant and the output mpedance can be made nearly purely resstve over a wde range of frequences. Then, the output mpedance s roughly Z o (s) K, whch s ndependent of the nductance. Wth the above control strategy, the nverter can be approxmated as a controlled deal voltage supply v r cascaded wth a resstve output mpedance R o descrbed as wth = v r R o () R o = K. Note that u = v r f no load s connected. 3. INHERENT LIMITATIONS OF THE CONVENTIONAL DROOP CONTROL SCHEME Fg. 3 shows two nverters wth resstve output mpedances connected n parallel. The lne mpedances are omtted because the output mpedances of the nverters are desgned to domnate the mpedance from the nverter to the ACbus. The reference voltages of the two nverters are, respectvely, v r = 2E sn(ω t + δ ), v r2 = 2E 2 sn(ω 2 t + δ 2 ). The power ratngs of the nverters are S = E I and S 2 = E I 2. They share the same output voltage. Snce the output mpedances of the nverters are desgned to be resstve (constant) over a wde range of frequences, all the harmonc current components can be shared among the nverters n proporton to ther power ratngs. Hence, proportonal sharng can be acheved for lnear and nonlnear loads and the followng analyss s applcable to both lnear and nonlnear loads. ~ S = + P jq R o E δ o V o S + Z 2 = P2 jq2 R o2 E2 δ 2 Fgure 3. Two nverters wth resstve output mpedances connected n parallel The actve and reactve power of each nverter njected nto the bus (Guerrero et al., 25) are ~ v r E ω t+δ s E * * ω Fgure 4. The conventonal droop control scheme n m P = E V o cosδ Vo 2, (2) R o Q = E V o R o snδ. (3) In order for the nverters to share the load, the conventonal droop controller E = E n P, (4) ω = ω + m Q, (5) as shown n Fgure 4, s wdely used to generate the ampltude and frequency of the voltage reference v r for each nverter (Guerrero et al., 27), where ω s the rated frequency. Note that the P E and Q ω droop s used because the output mpedances are resstve. Otherwse, the P ω and Q E droop should be used when the output mpedances are nductve. The droopng coeffcents n and m are normally determned by the desred voltage and frequency drops, respectvely, at the rated actve power and reactve power. The frequency ω s ntegrated to form the phase of the voltage reference v r. In order for the nverters to share the load n proportonal to ther power ratngs, the droop coeffcents of the nverters should be n nverse proportonal to ther power ratngs (Tuladhar et al., 997),.e., n and m should be chosen to satsfy n S = n 2S 2 = = n ns n, (6) m S = m 2S2 = = m nsn. (7) It s easy to see that n and m also satsfy n = n 2 = = n n. m m 2 m n 3. Actve power sharng Substtutng (4) nto (2), the actve power of the two nverters can be obtaned as P = E cosδ V o. (8) n cosδ + R o /V o Substtutng (8) nto (4), the voltage ampltude devaton of the two nverters s E = E 2 E = E cosδ V o cosδ + Ro n V o P Q E cosδ 2 V o cosδ 2 + Ro2 n 2V o. (9) It s known from (Tuladhar et al., 997) that the voltage devaton of the two unts leads to consderable errors n load sharng. In order for n P = n 2 P 2 or P S = P 2 S

4 Preprnts of the 8th IFAC World Congress Mlano (Italy) August 28 September 2, 2 to hold, the voltage devaton E should be accordng to (4). Ths s a very strct condton because there are always numercal computatonal errors, dsturbances, parameterdrfts and componentmsmatches. Ths condton s satsfed f n = n 2 () R o R o2 and δ = δ 2. () In other words, n should be chosen to be proportonal to ts output mpedance R o. Takng (6) nto account, n order to acheve accurate sharng of actve power, the (resstve) output mpedance should be desgned to satsfy R o S = R o2s 2 = = R ons n. (2) Snce the perunt output mpedance of Inverter s there s R o γ = E /I = R os (E ) 2, γ = γ 2 = = γ n. = γ. Ths smply means that the perunt output mpedances of all nverters should be the same n order to acheve accurate proportonal actve power sharng. Recall that power transformers wth dfferent power ratngs have more or less the same perunt output mpedances (although not resstve). 3.2 Reactve power sharng When the system s n the steady state, the two nverters work under the same frequency,.e., ω = ω 2. It s well know that ths guarantees the accuracy of reactve power sharng for nverters wth resstve output mpedances (or the accuracy of actve power sharng for nverters wth nductve output mpedances); see e.g. (L and Kao, 29). Indeed, from (5), there s m Q = m 2 Q 2. Snce the coeffcents m are chosen to satsfy (7), reactve power sharng proportonal to ther power ratngs s (always) acheved,.e., Q S Accordng to (3), there s = Q 2 S2. m E V o R o snδ = m 2 E 2 V o R o2 snδ 2. (3) If δ = δ 2 and E = E 2 then m = m 2. (4) R o R o2 Theorem For nverters desgned to have resstve output mpedances, f the system s stable, then the followng two sets of condtons are equvalent: { E = E 2 n = n 2 R o R o2 { δ = δ 2 m = m 2. R o R o2 Proof. If () and E = E 2 hold, then proportonal actve power sharng s acheved accordng to (4). As a result, () holds accordngto (9) and (3). Furthermore, reactve power sharng proportonalto ther ratngs s acheved and (4) holds. Conversely, f () and (4) hold, then E = E 2 accordng to (3). Furthermore, () holds accordng to (9). Ths completes the proof. Ths theorem ndcates that f nverters wth resstve output mpedances are desgned to acheve accurate proportonal actve power sharng, then they also acheve proportonal reactve power sharng n the deal case. The converse s also true. However, t s almost mpossble n realty f ths strategy s used. It s dffcult to mantan E = E 2 = = E n because there are always numercal computatonal errors, dsturbances and noses. It s also dffcult to mantan γ = γ 2 = = γ n because of parameter drfts and component msmatches. A mechansm s needed to guarantee that accurate proportonal load sharng can be acheved. 4. ROBUST DROOP CONTROLLER TO ACHIEVE ACCURATE PROPORTIONAL LOAD SHARING As a matter of fact, the voltage droop (4) can be rewrtten as E = E E = n P, and the voltage E can be mplemented va ntegratng E, that s, E = t E dt. Ths works for the grdconnected mode where E s eventually (so that the desred power s sent to the grd wthout error), as proposed n (Marwal et al., 24; L et al., 24; Da et al., 28). However, t does not work for the standalone mode because the actual power P s determned by the load and E cannot be. Ths s why dfferent controllers had to be used for the standalone mode and the grdconnected mode. Accordng to (), the output voltage drops when the load ncreases. It also drops due to the droop control, accordng to (4). In order to make sure that the output voltage remans wthn a certan requred range, the output voltage drop E V o can be added to E va an amplfer K e. Ths actually results n an mproved droop controller shown n Fgure 5. It s able to elmnate (at least consderably reduce) the mpact of computatonal errors, noses and dsturbances. As to be explaned below, t s also able to mantan accurate proportonal load sharng and hence robust wth respect to parameter drfts, component msmatches and dsturbances. In the steady state, the nput to the ntegrator should be. Hence, n P = K e (E V o ). (5) The rghthand sde of the above equaton s always the same for all nverters connected n parallel as long as K e s chosen the same, whch can be easly met. Hence, accurate real power sharng can be acheved wthout havng the same E, whch s more natural. The actve power sharng 2788

5 Preprnts of the 8th IFAC World Congress Mlano (Italy) August 28 September 2, 2 s nolongerdependenton the nverteroutputmpedances and s alsommune to the numercal computatonalerrors and dsturbances, whch guarantees the accuracy of real power sharng. Moreover, from (5), there s V o = E n K e P. The output voltage drop s no longer determned by the output mpedance orgnally desgned but by the parameters n, K e and the actual power P. It can be consderably reduced by usng a large K e. Ths easly solves the compromse between the voltage drop and the speed of dynamc responses (Tuladhar et al., 997). The droop coeffcent n can be chosen bg to speed up the dynamc response whle the voltage drop can be kept small by usng a large K e. Although the output mpedance of Inverter s ntally desgned as R o, e.g., as desgned usng the approach presented before, whch could be bg, the overall voltage drop could be made small. v r E ω t+δ s s * ω K e n m E * P Q RMS Fgure 5. The proposed robust droop controller to obtan accurate proportonal load sharng 5. EXPERIMENTAL RESULTS The above strategy has been verfed n a laboratory setup. It conssts of two snglephase nverters controlled by dspace kts and powered by separate 42V DC power supples. The values of the nductors and capactors are 2.35mH and 22µF, respectvely. The swtchng frequency s 7.5kHz and the frequency of the system s 5Hz. The nomnal output voltage s 2V RMS and K e =. The droop coeffcents are: n =.4 and n 2 =.8; m =. and m 2 =.2. Hence, t s expected that P = 2P 2. In the experments, K was chosen as 4 for both nverters. Due to the confguraton of the hardware setup, the voltage for Inverter 2 was measured by the controllerof Inverter and then sent out va a DAC channel, whch was then sampled by the controller of Inverter 2. Ths brought some latency nto the system but the effect was not notceable. 5. Wth a lnear load A lnear load of about 9Ω was connected to Inverter 2 ntally. Inverter was connected to the system at around t = 2 second and was then dsconnected at around t = 7.5 second. Fgure 6 shows the relevant curves of the P P 2 Q 2 Q Tme (sec) (a) Actve power and reactve power Tme (sec) (b) Currents Fgure 6. Expermental results wth a lnear load experment. It can be seen that the two nverters shared the load very accurately n the rato of 2 :, although E E 2. A promnent feature s that the voltage E and E 2 are dfferent and both are hgher than the rated value 2V. There was no need to change the operaton mode of Inverter 2 when connectng or dsconnectng Inverter. 5.2 Wth a nonlnear load A nonlnear load, consstng of a rectfer loaded wth an LC flter and the same rheostat used n the prevous experment, was connected to Inverter 2 ntally. Inverter was connected to the system at around t = 2.7 second and was then dsconnected at around t = 9.7 second. Fgure 7 shows the relevant curves of the experment. It can be seen that the two nverters were stll able to share the load very accurately n the rato of 2 :, although E E 2. The dynamc performance dd not change much ether CONCLUSIONS In ths paper, t has been shown that the capactor of the nverter LC flter can be regarded as a part of the load nstead of a part of the nverter, whch consderablyreduces the complexty of system analyss and controller desgn and reduces the cost of the controller. The, the nherent lmtatons of the conventonal droop control scheme has 2789

6 Preprnts of the 8th IFAC World Congress Mlano (Italy) August 28 September 2, P 9 7 P Q 2 Q Tme (sec) (a) Actve power and reactve power Tme (sec) (b) Currents Fgure 7. Expermental results wth a nonlnear load been exposed. In order to acheve accurate proportonal load sharng among parallelconnected nverters, the nverters should have the same perunt resstve output mpedances and the voltage setpont (E ) should be the same. These are almost mpossble n realty. An mproved droop control strategy s then proposed to obtan accurate proportonal load sharng for mcrogrds workng n the standalone mode (and naturally also n the grdconnected mode). Ths strategy does not requre that the voltage setponts of the nverters to be the same; t does not requre the output mpedance to be the same ether. The proposed strategy s also able to mantan excellent capablty of voltage regulaton. The strategy proposed here works for nverters wth resstve output mpedances but t can be appled to nverters wth nductve output mpedances by usng the Q E and P ω droop. REFERENCES Borup, U., Blaabjerg, F., and Enjet, P. (2). Sharng of nonlnear load n parallelconnected threephase converters. IEEE Transactons on Industry Applcatons, 37(6), do:.9/ Brabandere, K.D., Bolsens, B., den Keybus, J.V., Woyte, A., Dresen, J., and Belmans, R. (27). A voltage and frequency droop control method for parallel nverters. IEEE Trans. Power Electroncs, 22(4), 7 5. Da, M., Marwal, M., Jung, J.W., and Keyhan, A. (28). Power flow control of a sngle dstrbuted generaton unt. IEEE Trans. Power Electroncs, 23(), do:.9/tpel Guerrero, J.M., Vasquez, J.C., matas, J., Garca de Vcuna, L., and Castlla, M. (2). Herarchcal control of droopcontrolled AC and DC mcrogrdsa general approach towards standardzaton. IEEE Trans. Industral Electroncs, PP(99), to appear. do:.9/tie Guerrero, J., GarcadeVcuna, L., Matas, J., Castlla, M., and Mret, J. (25). Output mpedance desgn of parallelconnected UPS nverters wth wreless loadsharng control. IEEE Transactons on Industral Electroncs, 52(4), do:.9/tie Guerrero, J., Matas, J., de Vcuna, L.G., Castlla, M., and Mret, J. (27). Decentralzed control for parallel operaton of dstrbuted generaton nverters usng resstve output mpedance. IEEE Transactons on Industral Electroncs, 54(2), do:.9/tie Guerrero, J., Vasquez, J., Matas, J., Castlla, M., and de Vcuna, L. (29). Control strategy for flexble mcrogrd based on parallel lnenteractve UPS systems. IEEE Trans. Industral Electroncs, 56(3), do:.9/tie L, Y.W. and Kao, C.N. (29). An accurate power control strategy for powerelectroncsnterfaced dstrbuted generaton unts operatng n a lowvoltage multbus mcrogrd. IEEE Trans. Power Electroncs, 24(2), do:.9/tpel L, Y., Vlathgamuwa, D., and Loh, P.C. (24). Desgn, analyss, and realtme testng of a controller for multbus mcrogrd system. IEEE Trans. Power Electroncs, 9(5), do:.9/tpel Marwal, M., Jung, J.W., and Keyhan, A. (24). Control of dstrbuted generaton systems part II: Load sharng control. IEEE Trans. Power Electroncs, 9(6), do:.9/tpel Tuladhar, A., Jn, H., Unger, T., and Mauch, K. (997). Parallel operaton of sngle phase nverter modules wth no control nterconnectons. In Appled Power Electroncs Conference and Exposton, 997. APEC 97 Conference Proceedngs 997., Twelfth Annual, volume, 94 vol.. do:.9/apec Tuladhar, A., Jn, H., Unger, T., and Mauch, K. (2). Control of parallel nverters n dstrbuted ac power systems wth consderaton of lne mpedance effect. Industry Applcatons, IEEE Transactons on, 36(), do:.9/ Zhong, Q.C. and Wess, G. (2). Synchronverters: Inverters that mmc synchronous generators. IEEE Trans. Industral Electroncs, 58(4),

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