664 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 3, MARCH 2011

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1 664 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 3, MARCH 2011 Conservative Power Theory, a Framework to Approach Control and Accountability Issues in Smart Microgrids Paolo Tenti, Fellow, IEEE, Helmo Kelis Morales Paredes, Student Member, IEEE, and Paolo Mattavelli, Member, IEEE Abstract Smart microgrids oer a new challenging domain or power theories and compensation techniques, because they include a variety o intermittent power sources, which can have dynamic impact on power low, voltage regulation, and distribution losses. When operating in the islanded mode, low-voltage smart microgrids can also exhibit considerable variation o amplitude and requency o the voltage supplied to the loads, thus aecting power quality and network stability. Due to limited power capability in smart microgrids, the voltage distortion can also get worse, aecting measurement accuracy, and possibly causing tripping o protections. In such context, a reconsideration o power theories is required, since they orm the basis or supply and load characterization, and accountability. A revision o control techniques or harmonic and reactive compensators is also required, because they operate in a strongly interconnected environment and must perorm cooperatively to ace system dynamics, ensure power quality, and limit distribution losses. This paper shows that the conservative power theory provides a suitable background to cope with smart microgrids characterization needs, and a platorm or the development o cooperative control techniques or distributed switching power processors and static reactive compensators. Index Terms Conservative power theory (CPT), distributed control, electronic power processors, power measurement, revenue metering, smart microgrids. I. INTRODUCTION SMART grids represent one o the grand challenges at planetary level. The inusion o inormation technology throughout the electric grid creates new capabilities, with potential impact on environment, science and technology, economics, and liestyle. The term smart grid outlines the evolution o electrical grids and a change o paradigm in the electric market organization and management [30], [31]. In a global perspective, implementation o smart grids and microgrids on a large scale will result in dramatic improvement o electrical services and considerable market increase. Manuscript received June 30, 2010; revised October 8, 2010; accepted November 8, Date o current version May 13, Recommended or publication by Associate Editor R. Burgos. P. Tenti is with the Department o Inormation Engineering, University o Padova, Padova 35131, Italy ( tenti@dei.unipd.it). H. K. Morales Paredes is with the Department o Electrical Energy Systems, University o Campinas, Campinas, São Paulo ,Brazil ( hmorales@dsee.ee.unicamp.br). P. Mattavelli is with the Center o Power Electronics Systems (CPES), Virginia Polytechnic Institute and State University, Blacksburg, VA USA ( pmatta@vt.edu). Color versions o one or more o the igures in this paper are available online at Digital Object Identiier /TPEL Technically speaking, smart grids include a number o distributed energy resources and electronic power processors, which must be ully exploited to reduce carbon ootprint, improve power quality, and increase distribution eiciency [22], [32], [34]. The smart-grid paradigm is thereore dierent rom the traditional one, based on the assumption o ew power sources with large capacity and sinusoidal supply. Especially in smart microgrids (low-voltage smart grids with installed power not exceeding the megawatt range), energy sources can be small, distributed and interacting, and supply voltages can be asymmetrical and distorted. From the earlier considerations, it ollows that acing the problems o smart grids, and in particular o smart microgrids, requires a revision o traditional power theories and a comprehensive approach to cooperative operation o distributed electronic power processors. This paper shows that the conservative power theory (CPT) oers a consistent ramework to approach smart microgrid characterization and control problems. In particular, the inluence o requency variation and voltage distortion can be taken into account, and the load and supply responsibility or reactive power, asymmetry, and distortion can be analyzed, thus setting the basis or a revision or metering and billing procedures. II. CONSERVATIVE POWER TERMS UNDER PERIODIC NONSINUSOIDAL OPERATION The problem o deining power and current terms under nonsinusoidal conditions dates back to some 80 years [1]. While deinition o active power and current terms was developed in early 30s [2], deinition o power and current terms related to reactive and harmonic phenomena is still under discussion and dierent solutions have been proposed depending on the ield o application [2] [12]. The CPT, presented in [5], [25], provides a background to approach the problem. The theory is summarized hereater by making reerence to the deinitions given in Appendix. Consider a polyphase network and let u and i be the vectors o phase voltages and currents measured at a generic network port, and u is the vector o unbiased voltage integrals [deinition (A1d)]. We deine the ollowing conservative quantities (Appendix, Section II): instantaneous power p = u i (1a) instantaneous reactive energy w = u i. (1b) /$ IEEE

2 TENTI et al.: CPT, A FRAMEWORK TO APPROACH CONTROL AND ACCOUNTABILITY ISSUES IN SMART MICROGRIDS 665 The corresponding average values are as ollows: active power P = p = u,i (2a) reactive energy W = w = u,i. (2b) It is worth noting that deinitions (1b) and (2) hold irrespective o voltage and current waveorms, provided that they are periodic. Moreover, computation o the quantities deined by (1) and (2) is directly done in the time domain and requires integration and low-pass iltering only. Finally, since the conservation property holds in general, active power and reactive energy are additive quantities in every electrical network. Application o (2) to basic passive network elements, considering properties (A3a) and (A3b), gives resistor R: u R = Ri R P R = U 2 R R W R =0 inductor L: u L = L i L P L =0 W L = LI 2 L = 1 2 ε L capacitor C : i C = C u C P C =0 W C = CU 2 C = 1 2 ε C. (3a) (3b) (3c) Thus, irrespective o voltage and current waveorms, resistors absorb only active power, while inductors and capacitors take a reactive energy, which is proportional to their average stored energy ε L and ε C, respectively. Due to conservation property, in passive networks the total active power absorption is obtained by adding the power consumption o each resistor, while total reactive energy is computed as total average inductive energy minus total average capacitive energy. Another relevant, though not conservative, power term characterizing the network operation at a given port is the apparent power A, deined by A = UI (4a) where U and I are the collective rms values o phase voltages and currents, according to deinition (A2c). Correspondingly, the power actor λ is deined as ollows: λ = P A. (4b) In polyphase networks, the power and energy terms deined by (1) and (2) do not depend on the voltage reerence, whereas apparent power and power actor depend on this choice. Irrespective o neutral wire, we adopt a voltage reerence, which provides unity power actor when the load is purely resistive and symmetrical. This removes any ambiguity in the apparent power deinition and allows use o the power actor as an index o power quality. To comply with this condition, in ourwire systems, the neutral voltage u 0 must be taken as reerence (u 0 = 0), while in three-wire systems the voltage reerence is set at the center point o the phase voltages. Correspondingly, the collective rms values U and I o phase voltages and currents are expressed irrespective o neutral wire by U = u,u = N Un 2 I = i,i = N In. 2 (5) Note that, in (5), only phase currents are considered, while neutral current (i 0 ) does not contribute to the summation. This does not mean that neutral current is disregarded, inasmuch rms phase currents are aected by homopolar terms, which relect its presence. Note also that, with the proposed selection o voltage reerence, the neutral current is not even aecting active power and reactive energy. III. CURRENT TERMS UNDER PERIODIC NONSINUSOIDAL OPERATION A. Basic Current Terms Consider a generic port o a N-phase network and let u n and i n be the voltage and current measured at phase n terminals. We decompose current i n so as to identiy the terms related to active power P n and reactive energy W n absorbed by phase n at the given port. 1) The active current is the minimum phase current (i.e., with minimum rms value) needed to convey active power P n. It is expressed by i an = u n,i n u n 2 u n = P n Un 2 u n = G n u n,, 2,...,N I a = N Ian 2 = N ( ) 2 Pn. (6) U n In (6), term G n = P n /U 2 n is the equivalent conductance o phase n. The active current has no impact on reactive energy, as it can easily be shown rom (6) and (A.3a). 2) The reactive current is the minimum phase current needed to convey reactive energy W n. It is expressed by u i rn = n,i n u n 2 u n = W n U 2 n I r = N ( Irn 2 = N u n = B n u n,, 2,...,N W n U n ) 2 (7) In (7), term B n = W n / U 2 n is the equivalent reactivity o phase n. The reactive current has no impact on active power, as it can easily be shown rom (7) and (A3a.) 3) The remaining current term is called void current and is not conveying active power and reactive energy. It is deined by i vn = i n i an i rn n =1, 2,...,N. (8) 4) Orthogonality: All the aorementioned current terms are orthogonal, thus I n = I 2 an + I 2 rn + I 2 vn I = I 2 a + I 2 r + I 2 v. (9)

3 666 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 3, MARCH 2011 B. Balanced Current Terms As or conventional networks under sinusoidal operation, it makes sense to identiy the eects o supply voltage asymmetry and load unbalance, because they both aect power quality. O course, under nonsinusoidal operation, the traditional approach must be revised and this can easily be done based on the earlier deinitions. Notice, irst o all, that we use term unbalance to characterize the load attitude to perorm asymmetrically in the dierent phases. Instead, term asymmetry is associated to an asymmetrical behavior o the supply seen rom load terminals. Irrespective o supply asymmetry, load unbalance, and waveorm distortion, we deine the ollowing current terms. 1) The balanced active currents are the minimum currents (i.e., with minimum collective rms value) needed to convey total active power P absorbed at the port. They are given by i b a = u,i u 2 u = P U 2 u = Gb u I b a = P U. (10) In (10), coeicient G b = P/U 2 is the equivalent balanced conductance. Note that these currents are the same that would be absorbed by a symmetrical resistive load with same active power consumption as the actual load. 2) Similarly, the balanced reactive currents are the currents with minimum collective rms value needed to convey total reactive energy W absorbed at the port. They are given by i b r = u,i u = W u 2 2 u = B b u I b r = W. (11) U U In (11), coeicient B b = W/ U 2 is the equivalent balanced reactivity. Note that these currents are the same that would be absorbed by a symmetrical reactive load taking same reactive energy as the actual load. C. Unbalanced Current Terms 1) Unbalanced active currents are deined by dierence i u an = ( G n G b) u n,, 2,...,N I u a = N ( ) 2 ( ) 2 Pn P. (12a) U Clearly, these currents exist only i the load is unbalanced, i.e., i the equivalent phase conductances dier rom each other. Note that the balanced and unbalanced active currents are collectively orthogonal, i.e., i u a,i b a =0 I u a = I 2 a I b2 a. (12b) 2) Similarly, the unbalanced reactive currents are deined by U n i u rn = ( B n B b) u n,, 2,...,N ( ) I u r = N 2 ( ) 2 W n W. (13a) U n U These currents exist only i the equivalent phase reactivities dier rom each other. The balanced and unbalanced reactive currents are collectively orthogonal, thus, i u r,i b r =0 I u r = I 2 r I b2 r (13b) We collectively reer to unbalance currents as the sum o active and reactive unbalance terms, which are orthogonal each other. Thus, i u = i u a + i u r I u = I u 2 a + I u 2 r. (14) D. Complete Current Decomposition In conclusion, the phase currents can be split as ollows: i = i b a + i b r + i u + i v. (15a) All terms are orthogonal, thus, I = I b2 a + I b2 r + I u 2 + I 2 v. (15b) We emphasize once more that the earlier deinitions are valid, and keep their physical meaning, or whichever voltage and current waveorm, supply asymmetry and load unbalance. IV. POWER TERMS UNDER PERIODIC, NONSINUSOIDAL OPERATION From (15b), the apparent power deined by (4a) is decomposed as ollows: A 2 = U 2 I b2 a +U 2 I b2 r +U 2 I u 2 +U 2 I 2 v = P 2 + Q 2 +N 2 +V 2 (16) where P = UI b a is active power, Q = UI b r is reactive power, N = UI u is unbalance power, and V = UI v is void power. Note rom (11) that the reactive power can be expressed as ollows: THD (u) = Q = U U W = ωw 1+(THD(u)) (THD( u )) 2 (17a) where ω is angular line requency and THD is total harmonic distortion, i.e., U U U U U THD( u )=. U (17b) In (17b) U and U are the collective rms values o undamental phase voltages and their unbiased integrals, which are related by U = ωu. Equation (17a) shows that, unlike reactive energy W, reactive power Q is not conservative. In act, it depends on local voltage distortion. Note inally that all current and power terms deined earlier can be derived rom basic phase quantities, namely, active power P n, reactive energy W n, and rms voltage terms U n and U n.

4 TENTI et al.: CPT, A FRAMEWORK TO APPROACH CONTROL AND ACCOUNTABILITY ISSUES IN SMART MICROGRIDS 667 These quantities are evaluated directly in the time domain and can be measured by simple instrumentation. Note also that the power terms deined by the CPT are generally dierent rom those considered in traditional metering approaches. However, they provide an insight on relevant physical phenomena, i.e., power consumption and energy storage, which should be taken into account or revenue metering. V. ACCOUNTABILITY The CPT clariies the meaning o active, reactive, unbalance and void current, and power terms, and sets the basis or consistent power measurements under asymmetrical and distorted operation. In this section, we approach the accountability problem, i.e., the problem o separating supply and load responsibility on the generation o unbalance and distortion. Consider that, in general, current harmonics can be caused by inherent load nonlinearity and/or by supply voltage distortion. Similarly, asymmetrical currents can be generated by load and/or supply voltage asymmetry. In general terms, it is very diicult to develop a theory, which is able to separate source and load contributions to distortion and unbalance, when only measurements at the point o power delivery [point o common coupling (PCC)] are available. There are previous papers on the separation o load and source responsibility on distortion [26] [29], but it was not possible to establish a theoretical background, which is valid in every operating condition. The CPT and related decompositions oer a viable tool to establish an accountability approach, which gives some meaningul inormation under practical operating conditions. For this aim, let us assume a three-phase system and measure phase quantities (P n, W n, U n, and U n ) at PCC. We then estimate the power terms (i.e., active, reactive, unbalance, and void power), which would be absorbed by the load i it was ed by purely sinusoidal and symmetrical voltages. This portion is accounted to the load, while the rest is accounted to the supply. This simple procedure is based on two assumptions. 1) First, we assume that supply voltage asymmetry and distortion are not caused by the load. This is true only i rated load power is much smaller than grid power capability at PCC. 2) Second, we assume that equivalent phase parameters G n and B n keep the same value irrespective o voltage asymmetry and distortion. This corresponds to a very rough approximation o load operation, which can, however, be reined i a more accurate load modeling is available. The undamental positive-sequence voltages u p at PCC can be computed according to the procedure described in [24], which only requires elementary operations in the time domain. Since such voltages are sinusoidal and symmetrical, their rms value U p is the same in all phases, thus: U p = 3U p. Moreover, U p = U p /ω. According to the aorementioned assumptions, the active power and current terms accountable to the load are as ollows: P ln = G n U p2 P l = 3 P ln = U p2 3 G n I aln = G n U p I al = 3 Ialn 2 = U p 3 G 2 n. (18a) Similarly, or reactive energy and current, the terms accountable to the load are as ollows: p 2 W ln = B n U W l = 3 W ln = U p2 3 B n = p I rln = B n U I rl = 3 Irln 2 = U p 3 Bn 2 = U p ω ( ) U p 2 3 B n ω 3 Bn. 2 (18b) From (18), we derive parameters G b l and Bb l o the equivalent balanced load and the corresponding balanced active and reactive currents G b l = B b l = P l 3U p2 W l 3 U p2 I b al = P l 3U p I b rl = W lω 3U p. (19) From (12b) and (13b), we can then compute the collective rms values o the active and reactive unbalance currents. Given the current components, we immediately derive the corresponding power terms according to (16). Note that also the void power should be revised or accountability, according to the procedure described in [25]. However, this is needed only in those situations, not requent in practice, where the void power has a considerable amount compared to other power terms. In all other cases, we assume I vl I v. VI. REMOTE CONTROL OF DISTRIBUTED COMPENSATORS The CPT sets also a basis or remote control o every type o compensator, i.e., switching power compensators (SPC) and static VAR compensators (SVC). SVC include low-requency compensators, like thyristor-switched capacitors (TSC), thyristor-controlled reactors (TCR), and static compensator (STATCOM). SPC include high-requency compensators, like active power ilters (APF) and switching power interaces (SPI), connected between energy sources and grid. The compensation capabilities are dierent, since TSC are mostly used or reactive compensation, while TCR allow reactive and unbalance compensation [23]. STATCOM allow all types o low-requency compensation, while SPC operate in the multikilohertz range and provide harmonic mitigation too [13], [21], [35].

5 668 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 3, MARCH 2011 Fig. 1. Microgrid with distributed compensators. In this section, we discuss the control methods applicable to the various compensators under the assumption that they are driven by conservative power and energy commands [16]. The use o control commands based on conservative quantities allows them to be partitioned and addressed to a variety o remote compensators, with the intent to take ull advantage o the compensation capability distributed over the network. Fig. 1 gives a microgrid representation, where distributed compensators are shown separately, while the other networks components (distribution lines, transormers, loads, energy sources, etc.) are included in box π. In this section, we separately analyze the control techniques applicable to SVC and SPC [20]. A. Control o TSC Each phase o a TSC is made up o a set o parallel branches, each one including a capacitor in series with thyristor switch. By switching ON and OFF the various branches, the total capacitance can be varied stepwise in the range 0 C C max. Usually TSC operation is symmetrical and all phases show the same capacitance C. Assuming a three-phase TSC with deltaconnected capacitors, according to (3c), the total reactive energy absorbed by the compensator is as ollows: W TSC = C ( U U U31 2 ) (20) where U 12, U 23, and U 31 are the rms line-to-line voltages. When the TSC receives a reactive energy command, its controller determines the needed capacitance rom (20). B. Control o TCR A three-phase TCR is made up o three delta-connected branches, each including an inductor L max and a thyristor switch. By phase-controlling the thyristors, the equivalent phase inductances are continuously regulated in the range 0 L L max. The TCR can, thereore, be represented by three controllable reactivities B 12,B 23, and B 31. According to (3b), the total reactive energy taken by the TCR is as ollows: W TCR = B 12U 12 + B 23U 23 + B 31U 31. (21) I the TCR is required to provide reactive energy compensation only, the three reactivities are set equal and their value is determined rom (21) based on line-to-line voltage measurement. A more complex situation occurs i the TCR perorms unbalance compensation too. In act, the unbalance power deined by (16) is not conservative, and is thereore not suitable or remote control o TCR. A solution can be ound by making reerence to the sequence components deined in [24]. Consider, in act, that unbalance is considerably attenuated i the TCR compensates or negative-sequence currents absorbed by the load, and this can be achieved by a suitable choice o parameters B 12,B 23, and B 31. The problem is greatly simpliied by considering only the undamental positive-sequence component o the supply voltages (u p ). In act, the other voltage components are useless or reactive power and unbalance compensation and can be neglected or TCR control purposes. With this assumption, the negative-sequence currents absorbed by the TCR become B 23 B 12 B 31 i n = B 12 B 31 B 23 B 31 B 23 B 12 u p. (22) Observing that currents i n do not change by adding a common quantity to all rectivities, we can arbitrarily assume B 31 =0. Let U p and In be the rms values o voltages up and currents i n, respectively, the undamental unbalance power is deined as ollows: N =3U p In. (23a) From easy computations, we derive the relation between N and parameters B 12 and B 23 N cos ϕ n = 3 U p2 2ω B 12 N sin ϕ n = U p2 2ω (2B 23 B 12 ) (23b) where ϕ n is the phase displacement between voltage u p 1 and current i n 1. It is worth noting that unbalance power terms N cos ϕ n and N sin ϕ n are conservative, thus they can be used or remote control o SVC. The unbalance power command can be expressed in complex terms as ollows: Ṅ = N cos ϕ n + jn sin ϕ n. (23c) When the TCR receives an unbalance power command, its controller determines the values o B 12 and B 23 rom (23b). I they are both positive, the solution is acceptable. Otherwise, a common quantity is added to B 12,B 23, and B 31 to make two o them positive and the third zero. It can be demonstrated that unbalance compensation limits the reactive power compensation capability o the TCR by an amount up to 3N.

6 TENTI et al.: CPT, A FRAMEWORK TO APPROACH CONTROL AND ACCOUNTABILITY ISSUES IN SMART MICROGRIDS 669 Fig. 2. Concept scheme o cooperative control or distributed compensators. Fig. 3. Experimental three-phase our-wire circuit. C. Control o SPC SPC are capable o high-requency operation, i.e., they can compensate or every power or current term (reactive, unbalance, and void) also in presence o asymmetrical and distorted supply [19]. For this purpose, they are driven by instantaneous power and energy commands (p SPC and w SPC ). In three-phase SPC, these commands are transormed into current commands i SPC, considering that u i SPC = p SPC u i SPC = w SPC u 1 u 2 u i SPC p SPC = u 1 u 2 u 3 w SPC i SPC n = (24) Current commands i SPC are then executed according to usual current control techniques. VII. COOPERATIVE CONTROL PRINCIPLE The implementation o cooperative control is widely discussed in literature [14] [19], [32] [34]. Here, we do summarize the principle o operation based on the application o CPT [20]. The concept control scheme or distributed compensators is shown in Fig. 2. It reers to a situation, where control aims at improving the power actor at PCC by properly driving all available compensators. The controller perorms as ollows. 1) First o all, the balanced active currents absorbed at PCC are determined, according to (10), and taken as reerences (i re ). In act, all remaining current terms (reactive, unbalance, and void) should be suppressed by the compensation system. 2) Reerences i re are then compared with actual currents absorbed at PCC (i PCC ) to generate error signals ε, which are processed by error ampliier A ε to generate internal current reerences i. 3) Reerences i, together with PCC voltages u PCC,are then ed to sequence processor SP to extract undamental positive-sequence voltages u p and reerence currents i p r (undamental positive-sequence reactive currents) and i n (undamental negative-sequence currents), to be compensated by the SVC system. 4) The reerences or the SPC system are determined by dierence as: i s = i ir p i n. TABLE I FUNDAMENTAL VOLTAGES FOR CASES I AND II 5) The SVC control generates total reactive power command W and unbalance power command Ṅ or the SVC system. These commands are then distributed by power sharing unit PS SVC among the various SVC according to their type, compensation capability, and distance rom PCC. 6) Similarly, the SPC control unit generates instantaneous active power and reactive energy commands p s and ws or the SVC system. These commands are then distributed by power sharing unit PS SPC among the various SPC, according to their compensation capability and distance rom PCC. VIII. APPLICATION EXAMPLES A. Example 1 Revenue Metering In order to analyze the proposed accountability approach, a general purpose virtual instrument has been implemented, based on dual-core PC and eight-channel acquisition board (DAQmx PCI-6143-S, by National Instruments, Austin, TX). The analog signals are measured by means o Hall-eect voltage and current transducers (LV-25P and LA-55P, by LEM). The acquisition routines are implemented by means o a graphical programming language (LabView), while CPT calculations are implemented in C++ and compiled into dynamic link libraries accessed by LabView. Fig. 3 shows the experimental setup, including programmable ac voltage source, line impedance, linear, and nonlinear loads. The measures are done at PCC. The selected line impedance causes a voltage drop around 10% or rated load, corresponding to a weak power grid, e.g., a smart microgrid. Four dierent supply situations have been considered. 1) Case I Symmetrical sinusoidal voltages. 2) Case II Asymmetrical sinusoidal voltages. 3) Case III Symmetrical nonsinusoidal voltages. 4) Case IV Asymmetrical nonsinusoidal voltages.

7 670 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 3, MARCH 2011 TABLE II POWER TERMS MEASURED AT PCC AND ACCOUNTED TO THE LOAD FOR THE CIRCUIT OF FIG. 3 are slightly dierent (less than 2%) rom the measured ones. 4) Case IV: This is the worst case, where phase voltages are asymmetrical and distorted. The apparent and active power accounted to the load are signiicantly lower than those measured at PCC, due to the depuration o the eects o voltage asymmetry and distortion. In particular, the active power accounted to the load is 12% lower than that measured at PCC. Fig. 4. Simulated network. Table I shows the rms value and phase o undamental voltages applied in cases I and II. The phase voltages or cases III and IV have the same undamental component o cases I and II, with the addition o third, ith, seventh, and ninth harmonics (the amplitude o each harmonic is 5% o undamental component). The accountability approach o Section V was applied, and Table II shows the results o power measurements at PCC and the corresponding power terms accounted to the load. 1) Case I: In spite o the sinusoidal and symmetrical voltage supply, the power terms accounted to the load are slightly dierent rom the measured ones. This is the eect o the voltage drops across line impedances, which relect the asymmetry and distortion o load currents and aect the voltages at PCC. Also, the computation o parameters G n and B n is inluenced by these voltage drops, thus contributing to the dierences between power terms. In practice, since the THD o PCC voltages is less than 6%, the power terms accounted to the load are very close to the measured quantities. 2) Case II: The power terms measured to the PCC and accounted to the load are now dierent, especially the active power, which diers about 10%. This is the eect o asymmetrical supply voltages and voltage drops on line impedances, which cause asymmetry and distortion o the voltages at PCC (phase voltages distortion is 9.7%, 5%, and 8.5%, respectively). 3) Case III: In this case, the THD o phase voltages is about 13%, while asymmetry is negligible (less than 1%). Since terms G n and B n are moderately inluenced by voltage distortion, the power terms accounted to the load B. Example 2 Cooperative Control As an example o application o cooperative control, the network o Fig. 4 was simulated. It includes unbalanced and distorting loads, transmission lines, transormers, and compensation units (ixed capacitor bank, TCR, and APF). The supply voltages ed at PCC are asymmetric and distorted. The network operation is irst analyzed without any type o compensation (t <0.2 s). At t =0.2 s, the SVC is turned ON and compensates or reactive power and load unbalance. At t =0.6s, also the APF is turned ON and compensates or the remaining unwanted current terms. Finally, at t =1.2s, a sudden load change is introduced (control angles o thyristor rectiier delayed by 50 ) to analyze control dynamics. The control perorms according to the principle discussed in the Section VI. The system operation is described by making reerence to the voltage and current waveorms at PCC. The asymmetry and distortion o supply voltages is appreciable in Fig. 5, while Fig. 6 shows the current waveorms in the initial situation, when all compensators are OFF. The high current distortion is due not only to the thyristor rectiier but also to the capacitor bank, which is supplied by distorted voltages. The current asymmetry is due to load unbalance and voltage asymmetry. Fig. 7 shows the currents at PCC ater turning ON the SVC. Although distorted, the currents are now smaller than beore, due to the reduction o reactive and unbalance terms. Fig. 8 shows the currents at PCC ater turning ON the APF too. Within the control bandwidth, the input currents now include balanced active terms only and track the supply voltages with good accuracy. Finally, Fig. 9 shows the behavior o power actor λ at PCC. Initially, the power actor is very low, due to unbalance, distortion, and reactive power. The intervention o the SVC reduces reactive and unbalance currents, thus the power actor steps up.

8 TENTI et al.: CPT, A FRAMEWORK TO APPROACH CONTROL AND ACCOUNTABILITY ISSUES IN SMART MICROGRIDS 671 Fig. 5. Line voltages at PCC. Fig. 8. Currents at PCC with SVC and APF turned ON. Fig. 6. Line currents at PCC with SVC and APF turned OFF. Fig. 9. Time behavior o power actor. The power actor improves urther, approaching unity, ater intervention o the APF, which removes every residual reactive and unbalance currents together with void currents, including those generated by the SVC. The eect o the load transient on the power actor is minimal, showing that the system reacts properly to a sudden active and reactive power variation. Fig. 7. Line currents at PCC with SVC turned ON. IX. CONCLUSION The paper shows that CPT provides a platorm to approach the problems o power measurement and compensation in smart microgrids, where supply voltage distortion and requency variation can considerably aect system operation. Active and reactive current and power terms have been initially revisited to suit conditions, where voltages and currents can be severely distorted. An extension to polyphase circuits has also been discussed to identiy the eects o supply asymmetry and load unbalance. An accountability approach has then been introduced, which provides a criterion to separate load and supply responsibility on the generation o active, reactive, and unbalance power.

9 672 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 3, MARCH 2011 The application o CPT to remote control o compensators has also been approached, showing that the theory is applicable to every type o compensator and can be the basis or cooperative control o distributed compensators in microgrids. The accountability and control approaches have inally been tested experimentally and by simulation, to show their capabilities in a working scenario o practical interest. APPENDIX PERIODIC VARIABLES AND THEIR PROPERTIES For periodic quantities (period T, requency =1/T, and angular requency ω =2π), we deine the operators as ollowing: Average value : x = x = 1 T xdt T 0 (A1a) Time derivative: x = dx (A1b) dt t Time integral: x = x (τ) dτ (A1c) 0 Unbiased time integral: x = x x (A1d) Internal product: x, y = 1 T xydt T 0 (A1e) Norm (rms value): X = x = x, x (A1) Orthogonality: x, y =0. (A1g) For vector quantities x and y o size N, we deine Scalar product: x y = Internal product: N x n y n (A2a) N x,y = x n,y n (A2b) Norm: X = x = N x n,x n = N Xn. 2 (A2c) The vector norm is also called collective rms value. The earlier quantities have the ollowing properties: x, x =0 x, x =0 (A3a) x, y = x,y x, y = x,y (A3b) x, y = x, y = x, y. (A3c) CONSERVATIVE POWER AND ENERGY TERMS Considering network Π with L branches, a set o voltages {u l } L l=1 and currents {i l} L l=1 is said to be consistent with the network i they satisy the Kircho s Law or voltages (KLV) and currents (KLC), respectively. It is easy to show that i branch voltages u l are consistent with the network, the same happens or quantities u l and u l. Similarly or branch currents i l and related quantities i l and i l. According to the Tellegen s theorem, we can thereore airm that every scalar product o KLVconsistent terms u l, u l, and u l and KLC-consistent terms i l, i l, and i l is a conservative quantity. 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Power Electron., vol. 21, no. 4, pp , Jul Helmo Kelis Morales Paredes (S 09) was born in Puno-Perú, Peru. He received the B.S. degree in electrical engineering rom Arequipa National University, Arequipa, Perú, in 2002,and the M.Sc. degree in electrical engineering in 2006 rom the University o Campinas (UNICAMP), Campinas, São Paulo, Brazil, where he is currently working toward the Ph.D. degree. From October 2009 to June 2010, he was a Visitor Student in the Department o Inormation Engineering, University o Padova, Padova, Italy, where he was engaged on Conservative Power Theory and their application to smart microgird. His current research interests include power quality evaluation and power deinitions under nonsinusoidal and/or asymmetries conditions and its application in power conditioning and revenue metering. Mr. Morales Paredes is a Student Member o the Brazilian Power Electronics Society. Paolo Mattavelli (M 00) received the Ph.D. degree in electrical engineering in 1995 rom the University o Padova, Padova, Italy, in From 1995 to 2001, he was a Researcher at the University o Padova. In 2001, he joined as an Associate Proessor at the University o Udine, Udine, Italy, where he was leading the Power Electronics Laboratory. In 2005, he joined the University o Padova, Vicenza with the same duties. In 2010, he joined the Bradley Department o Electrical and Computer Engineering at Virginia Tech, Blacksburg, VA as a Proessor, where he is currently a member o the Center o Power Electronics Systems. His research interests include analysis, modeling and control o power converters, digital control techniques or power electronic circuits, and gridconnected converters or power quality and renewable energy systems. In these research ields, he was leading several industrial and government projects. Pro. Mattavelli is currently an Associate Editor or IEEE TRANSACTIONS ON POWER ELECTRONICS, and as an Industrial Power Converter Committee Technical Review Chair or the IEEE TRANSACTIONS ON INDUSTRY APPLICA- TIONS. During and , he was a member-at-large o the IEEE Power Electronics Society s Administrative Committee. He was also the recipient o the Prize Paper Award in the IEEE TRANSACTIONS ON POWER ELECTRONICS and in 2007, a second place in the Prize Paper Award at the IEEE Industry Application Annual Meeting. Paolo Tenti (F 99) graduated with honors in electrical engineering rom the University o Padova, Padova, Italy, in He is currently a Proessor o electronics in the Department o Inormation Engineering, University o Padova, Padova, Italy. His research interests include industrial and power electronics, electromagnetic compatibility (EMC), application o modern control methods to power electronics, EMC analysis o electronic equipment, and cooperative control o distributed electronic power processors in smart grids. Pro. Tenti was a member o the Executive Board o the IEEE Industry Applications Society and a Chair o various society committees rom 1991 to In 1997, he served as the IEEE Industry Applications Society (IAS) President. In 2000, he was a Chair o the IEEE World Conerence on Industrial Applications o Electrical Energy, Rome. During , he was appointed the IEEE IAS Distinguished Lecturer on EMC in industrial equipment. From 2002 to 2008, he served as the Department Director with the University o Padova and a Chairman o the Board o Directors. He is also President o CREIVen, an industrial consortium or research in industrial electronics with special emphasis on EMC.