A direct power control strategy for three level neutral-point-clamped rectifier under unbalanced grid voltage

Electric Power Systems Research 161 (2018) 103–113

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A direct power control strategy for three level neutral-point-clamped rectifier under unbalanced grid voltage Billel Kahia a,b , Abdelouahab Bouafia a , Abdelmadjid Chaoui a , Zhenbin Zhang c,∗ , Mohamed Abdelrahem b , Ralph Kennel b a

Laboratoire Qualité d’Energie dans les Réseaux Electriques (QUERE), Department of Electrical engineering, Universityof Setif 1, 19000, Setif, Algeria Institute for Electrical Drive Systems and Power Electronics, Technical University of Munich (TUM), 80333 München, Germany c Key Laboratory of Power System Intelligent Dispatch and Control of the Ministry of Education (Shandong University), Jinan 250061, China b

a r t i c l e

i n f o

Article history: Received 14 July 2017 Received in revised form 16 January 2018 Accepted 9 April 2018 Keywords: Active and reactive power Neutral-point clamped Switch table Direct power control Unbalanced grid voltage

a b s t r a c t Direct power control (DPC) strategy has attracted wide attention due to its advantages of simple structure, quick response, strong robustness, and elimination of current regulation loops/PWM blocks. Unfortunately, under unbalanced grid voltage, the conventional DPC (CDPC) scheme with the conventional definitions of active and reactive power cannot work well. In order to solve this problem, a new definition of the active power instead of the conventional one is proposed, discussed and used in this paper. As a result, good performance of the system is achieved and neither complicated calculation of a power compensation term nor positive/negative sequence extraction of grid voltages/currents are required. Then, a switching table based DPC strategy is designed based on the new definition of active power and conventional definition of reactive power. The corresponding switching table is suitable to achieve constant active power, constant reactive power and sinusoidal grid currents with very low total harmonic distortions (THDs). Simulation results are presented to confirm the theoretical study and the effectiveness of the proposed DPC with the new definition of active power (DPC-NP). The performance of the proposed DPC-NP is compared with that of the CDPC and that of the DPC with a new definition of reactive power (DPC-NQ). © 2018 Elsevier B.V. All rights reserved.

1. Introduction Recently, the use of three-level neutral-point-clamped (NPC) converters [1–3] (see Fig. 1) has increased rapidly due to their advantages of low total harmonic distortion (THD) of the input currents, low du/dt, low switching voltage stress, etc. Compared to the conventional two level topology, the three-level NPC topology provides important advantages in high-power applications. Unfortunately, some inherent problems still exist when the three-level NPC converters are used (e.g. the voltage drifts and voltage ripples of the neutral-point). As a result, its practical application is limited. Therefore, several studies have been conducted to assure the balancing of neutral-point potential, which are software based [4] and hardware modification based [5] methods. Generally, direct power control (DPC) [6–9] and voltage oriented control (VOC) [10–12] are considered as high-performance

∗ Corresponding author at: Key Laboratory of Power System Intelligent Dispatch and Control of the Ministry of Education (Shandong University), Jinan 250061,China. E-mail addresses: [email protected] (B. Kahia), [email protected] (Z. Zhang). https://doi.org/10.1016/j.epsr.2018.04.010 0378-7796/© 2018 Elsevier B.V. All rights reserved.

control strategies for PWM ac/dc converters. The former (VOC) was developed based on the well-known field oriented control (FOC); whilst the latter has the similar properties of direct torque control (DTC) [13] for AC drives. In VOC, both active and reactive power components in synchronous frame are obtained by decomposing the grid currents. Through inner current loops using PI controller, the active and reactive power can be regulated. Several VOC strategies have been introduced in the literature to cope with the voltage unbalanced [14–22]. In [14], constant active power is achieved by deriving and regulating the positiveand negative-sequence components of grid currents in the synchronous reference frame [15,16]. However, the use of several PI controllers and the extraction of positive- and negative-sequence components significantly increases the computational burden and tuning efforts. In [17], a dual-frequency resonant compensator and PI regulator were employed to enhance the performance of the control system under unbalanced and distorted grid voltages for doubly-fed induction generators (DFIGs) in variable-speed wind turbine applications. However, due to the sequence decomposition of voltage and current, the time delay and control error in VOC are inevitable. Additionally, the reference current computa-

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Fig. 1. Topology of the three-level NPC rectifier.

tion for different control targets is also complicated. Hence, to reduce the control complexity of dual current controller, a number of improved methods have been presented in the literature using PI plus resonant and vector PI controller (see e.g. [18–21]). Unfortunately, the extraction of positive- and negative-sequence components of grid voltages is still required. In [22], a new definition of reactive power (proposed in [23]) is investigated in VOC strategy, sinusoidal input current and accurate regulation of DCvoltage are achieved. However, positive- and negative-sequence extraction of both grid voltage/currents and complicated calculations of current reference are not required any more [24]. Various new methods have been presented in the literature to further improve the steady-state performance of the DPC [25–33], like replacing the conventional two-level hysteresis comparators by multi-level ones (see e.g. [25–28]); using of a modulation stage (see e.g. [29,30]); employing of predictive algorithm instead of the switching table (see e.g. [31–33]). However, only a few publications consider the operation under unbalanced grid voltages [34,35]. Unbalanced grid voltage has already been a very common phenomenon, which can be caused by, e.g. non-ideal three-phase load, asymmetry faults, large capacity single-phase load, asymmetry of power transmission system, etc. In [34], by adding the distorted terms into the power references, three different specialized control targets under unbalanced grid conditions can be achieved in LUT based DPC, the control targets are no negative sequence current, smooth active power, and smooth reactive power. However, synchronous transformation, extraction of both positive/ negative components of grid voltage and current are needed. A simplified power compensation block is proposed in [35], where the negative sequence current is not required and three different targets are implemented. Unfortunately, the extraction of the positive- and negative-sequence components of grid voltages and the positivesequence component of grid currents are required. As a result, the control complexity and computational burden is increased. To entirely eliminate the sequence decomposition and keep the simplicity of DPC scheme as far as possible, the authors in [36] combines the merits of DPC and the new definition of instantaneous reactive power, taking the active power and new reactive power as control variables, which is more suitable for unbalanced grid voltages than the conventional reactive power definition. To simplify the calculation process, DPC with this new definition of the reactive power for two-level voltage source converter based on ˛ˇ frame instead

of dq frame was presented in [37]. However, here is the room for further improvement. In this paper, a DPC with a new definition of instantaneous active power (DPC-NP) expressed in the dq reference frame for threelevel NPC rectifier under unbalanced grid voltages is proposed. By using this new definition of active power instead of conventional one under unbalanced grid voltage, good performance of the DPC scheme is achieved and the decoupling process of positive and negative sequences are not required. As a result, reducing the computational burden and simplifying the control structure. The proposed DPC strategy is much simpler in structure, in comparison with the conventional DPC (CDPC) solutions and the DPC scheme with the new definition of the reactive power (DPC-NQ) that presented in [37]. Furthermore, no compensation block or calculations of the current references are required. The effectiveness of proposed DPC is validated by extensive simulation data in different scenarios. Its performance is compared with that of both CDPC and DPC-NQ schemes. The results illustrate the superior performance of proposed DPC-NP in comparison with CPDP and DPC-NQ.

2. DPC of three-level NPC rectifier In this section, the system description and modeling of the underlying three-level NPC power converter is presented.

2.1. Topology and mathematical modeling of the three-level NPC rectifier The basic topology of a three-level NPC rectifier is shown in Fig. 1, where, Sin (n = 1, 2, 3, 4 and i = a, b, c) are the 12 semiconductor switches of the rectifier, C1 and C2 are capacitors in dc-bus and L, R represent inductance and resistance of the input filter, respectively. ei , ii and ui (i = a, b, c) are the three-phase grid voltages, currents and ac-side voltages of rectifier, respectively. Udc1 and Udc2 are the voltages of C1 and C2 , respectively. i0 is the current of neutral point. The mathematical model of three-level NPC rectifier in the stationary reference frame ˛ˇ can be expressed as follows [36]: di˛ 1 = (e˛ − u˛ − Ri˛ ) L dt

(1)

B. Kahia et al. / Electric Power Systems Research 161 (2018) 103–113

diˇ dt

1 (e − uˇ − Riˇ ) L ˇ

=

(2)

According to the instantaneous power theory [38], the complex power, active power, and reactive power of the NPC rectifier can be calculated respectively as 3 S = (i∗ e) 2 P = Re(S) = Q = Im(S) =

(4)

3 3 Im(i∗ e) = (i ⊗ e) 2 2

(5)

where * signifies the conjugate of a complex vector,  is dot product and ⊗ is cross product. In [36,37], an improved definition of reactive power from the extended pq theory is used, which is expressed as 3 Re(i∗ e ) 2

Q nov =

=

3 (i  e ) 2

(6)

where e

signifies the variable that lags e by 90 electrical degrees in the time domain [38]. In this paper, a new definition of active power is proposed, which is expressed as P nov =

⎧ 3 + + − − ⎪ Qo = (idq ⊗ edq + idq ⊗ edq ) ⎪ 2 ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

Qc2 =

3 + − − + (i ⊗ edq + idq ⊗ edq ) 2 dq

Qs2 =

3 + − − + (−idq  edq + idq  edq ) 2

(3) 3 3 Re(i∗ e) = (i  e) 2 2

3 Im(i∗ e ) 2

=

+ +jwt − −jwt e = e+ + e− = edq e + edq e

(8)

+ +jwt − −jwt i = i+ + i− = idq i + idq i

(9)

+

⎧ nov nov cos(2ωt) + P nov sin(2ωt) P = Ponov + Pc2 s2 ⎪ ⎪ ⎨ nov = Po

⎪ ⎪ ⎩

=

−

+

−

+ − + − = ed+ + jeq , edq = ed− + jeq , idq = id+ + jiq , idq = id− + jiq . ω where edq is the grid frequency in [rad/s]. The lagging vector e of unbalanced grid voltage can be expressed as

− Qs2 cos(2ωt) + Qc2 sin(2ωt)

Ponov

+



2 (Qc2

2 ) sin + Qs2



 2ωt +  − 2



Q = Qnov o + Qc2 cos(2ωt) + Qs2 sin(2ωt) = Qo +

(7)

Under unbalanced grid voltage condition, the sum of positive- and negative-sequence vectors represents respectively the grid voltage and grid current as [36]:

(16)

where (Po , Qo ), (Pc2 , Qc2 ), and (Ps2 , Qs2 ) are respectively the average value of active and reactive power, the harmonics peak of active and reactive power cosine phase, and the harmonics peak of active and reactive power sine phase, respectively. nov = −Q From Eqs. (15) and (16), it is appeared that Pc2 s2 and nov = Q . The active power and reactive power in Eqs. (12) and Ps2 c2 (13) can be rewritten, respectively, as

{

3 (i ⊗ e ) 2

105



(18)

2 + Q 2 ) sin(2ωt + ) (Qc2 s2



The ripple amplitude of active power (  )) 2

(17)

2 + Q 2 ) sin(2ωt +  − (Qc2 s2



2 + Q 2 ) sin(2ωt + is the same as that of reactive power ( (Qc2 s2 )), where  = arctan(Qc2 /Qs2 ). Thus, if the oscillation in the active power is nullified (i.e. the ripple amplitude of the active power is null), the oscillation in reactive power should be null. Therefore, the active power and reactive power will be constant without oscillations, which make both the active and reactive power track their respective references.

+ +j(wt−(/2)) − −j(wt−(/2)) e = edq e + edq e = −jedq e+j(wt) + jedq e−j(wt) (10)

2.2. Comparison between conventional and new active power

Considering the grid voltages and currents in Eqs. (8), (9) and (10); (4), (5) and (7) can be developed as

The use of the extended pq theory to compute the active power under unbalanced grid voltage allows getting constant active and reactive power under the assumption that the grid currents contain only both positive- and negative-sequence components (i.e. no zero-sequence component is exist). According to Eqs. (17) and (18), constant active and reactive power means that

+

  

P=

−

3 3 + +j(wt) − −j(wt) ∗ + +j(wt) − −j(wt) Re(i∗ e) = Re(idq e + idq e ) (edq e + edq e ) 2 2

(11)

= Po + Pc2 cos(2ωt) + Ps2 sin(2ωt) P nov =

3 3 + − + +j(wt) − −j(wt) ∗ e + idq e ) (−jedq e+j(wt) + jedq e−j(wt) ) Im(i∗ e ) = Im(idq 2 2

(12)

nov nov cos(2ωt) + Ps2 sin(2ωt) = Ponov + Pc2

Q =

3 3 + +j(wt) − −j(wt) ∗ + +j(wt) − −j(wt) e + idq e ) (edq e + edq e ) Im(i∗ e) = Im(idq 2 2

(13)

= Qo + Qc2 cos(2ωt) + Qs2 sin(2ωt)

⎪ ⎪ ⎪ ⎪ ⎩

Pc2

Ps2 =

(14)

3 + − − + (i ⊗ edq − idq ⊗ edq ) 2 dq

⎧ 3 + + − − nov ⎪ ⎪ Po = 2 (−idq  edq + idq  edq ) ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

nov = Pc2

3 + − − + − idq  edq ) (i  edq 2 dq

nov = Ps2

3 + − − + + idq ⊗ edq ) (i ⊗ edq 2 dq

c2

nov = Q ⎪ Ps2 ⎪ c2 ⎪ ⎪ ⎩

(19)

Q0 = 0

with

⎧ 3 ⎪ P = (i+  e+ + i−  e− ) ⎪ ⎪ o 2 dq dq dq dq ⎪ ⎨

3 + − − + = (idq  edq + idq  edq ) 2

⎧ nov Po = P ref ⎪ ⎪ ⎪ ⎪ ⎨ P nov = −Qs2

(15)

where Pref is the reference value of active power. The four values of the reference currents (id+ iq+ id− iq− ) can be obtained by solving the four equations of (19). Thus, the injection of negative-sequence current is sufficient and necessary to obtain constant active and reactive power under unbalanced grid voltage. The grid current would be not contain harmonics where both active and reactive power accurately track their respective reference values. Hence, by using the new active power definition instead of the conventional one, the grid current will be sinusoidal but still unbalanced. Thus, the new active power in the extended pq theory is more suitable than the conventional one under unbalanced grid voltage.

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Fig. 2. Variation of instantaneous active power and reactive power for various rectifier voltage vectors under (a, b) balanced grid voltages, (c, d) unbalanced grid voltages.

In contrast, if conventional active power is employed, constant active and reactive power requires that

⎧ Po = P ref ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Pc2 = 0 ⎪ ⎪ ⎨ Ps2 = 0 ⎪ Q0 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Qc2 = 0 ⎪ ⎪ ⎩

de = jωe+ − jωe− = −ωe dt (20)

(22)



de = ωe+ + ωe− = ωe dt

(23)

Taking Eqs. (1), (2), (4), (7), and (23) into account, the differentiations of active power and new active power are obtained as

Qs2 = 0

From Eq. (20), the four values of the reference currents (id+ iq+ id− iq− ) cannot be obtained by solving the six equations of (20). As a result, harmonics components must be added to the grid to obtain constant active and reactive power, as shown in [36]. The grid current amplitude can be calculated from Eqs. (4) and (5) as P + jQ |i| = 1.5e

of DPC. To the best of the authors knowledge, the new active power has not been reported in the frame of DPC. From Eqs. (8) and (10), the lagging value of grid voltage vector and its differentiation can be written as

(21)

Constant values of P and Q lead to get variable amplitude of grid current with time. As a result, the harmonics would appear.

3 dP R = [|e|2 − Re(v∗ .e)] − P − ωQ, 2L L dt nov

dP dt

=

3 R Im[(e∗ − v∗ )e ] − P nov − ωQ. 2L L

(24) (25)

Similarly, the differentiation of reactive power can be obtained from Eqs. (1), (2), (5) and (22) as 3 dQ R = Im[(e∗ − (v∗ )e)] − Q + ωP nov . 2L L dt

(26)

The grid voltage vector and rectifier voltage vector can be expressed under balanced grid voltage as

2.3. Design of the new switching table

e = |e|ejωt ,

In this paper, a more suitable look up table is newly established to achieve simultaneous control of both new active power and reactive power under unbalanced grid voltage, which is the key aspect

where E is the RMS value of line to line grid voltage and (n = 1, . . ., 12), and |vi |max is the amplitude of small or medium or large voltage vectors.

|e| = E,

v = |vi |max |e|j(/3)(n−1)

(27)

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107

Fig. 3. Space voltage vectors of a three-level NPC rectifier.

In order to simplify the equations, the resistance R and the coupling terms are ignored. Thus, Eqs. (24), (25) and (26) are rewritten as



dP dt

(28)





3 2 3   |e| − |v |max |e| sin ωt − − (n − 1) 2L 2L i 2 3

=−



(29)



dQ 3  = − |vi |max |e| sin ωt − (n − 1) 2L 3 dt





3Ts 2 3Ts  |e| − |v |max |e| cos ωt − (n − 1) , 2L 2L i 3



Q = −

(31)



3Ts 2 3Ts   |e| − |v |max |e| sin ωt − − (n − 1) , 2L 2L i 2 3

P nov = −



(32)



3Ts  |v |max |e| sin ωt − (n − 1) , 2L i 3

(33)

In order to normalize the power variation, both sides of Eqs. (31), s (32) and (33) are divided by the constant 3T |v | |e|, yielding 2L i max



¯ i= P





 3 |e| − cos ωt − (n − 1) 2 Udc 3



P¯inov = −



(34)



  3 |e| − sin ωt − − (n − 1) 2 Udc 2 3



(35)



¯ i = − sin ωt −  (n − 1) Q 3

LVV MVV SVV ZVV

¯ i q

>0

<0

>0

v8 v14 v11 v6 v15 v9 v12 v1 v4 v16 v7 v13 v10 v0

v17 v5 v2 v18 v3

v11 v5 v8 v3 v9 v6 v10 v4 v7

=0

<0

v2 v14 v17 v12 v18 v15 v1 v13 v16 v0

(30)

In each sampling period, the influence of each switching space vector on active and reactive power are calculated by P =

−p¯nov i



dP 3 2 3  = |e| − |v |max |e| cos ωt − (n − 1) 2L 2L i 3 dt nov

Table 1 Variations of active power and reactive power in sector  1 .

(36)

¯ i= From the equations in (34), (35) and (36), one can draw that: P −P¯inov . The influence of each switching space vector on the normalized value of variation of the instantaneous new active power and reactive power under unbalanced grid voltage can be determined from Eqs. (35) and (36), which are illustrated in Fig. 2. According to the magnitude of each space vector, the switching space vectors are classified to four groups (see Fig. 3): zero voltage vector (ZVV) v0 , which produces null output voltage and it

corresponds to three different configurations, small voltage vectors (SVVs) √ v1 , v4 , v7 , v10 , v13 and v16 with has an amplitude equal to Udc / 6, medium voltage √ vectors (MVV) v3 , v6 , v9 , v12 , v15 and v18 with amplitude of Udc / 2 and the large voltage vectors (LVVs) v2 , v5 , v8 , v11 , v 14 and v17 that generate a space vector with ampli2/3Udc . The unbalance of grid voltage is initially tude equal to set to 15% to consider more rigorous grid unbalance condition. As shown in Fig. 2, the variation of instantaneous active and reactive power are symmetrical with the same peak value when the grid voltage is balanced, and are also symmetrical but with different peak value when the grid voltage is unbalanced. Therefore, the classification of the impact of each voltage vectors on active and reactive power is the same under both balanced and unbalanced grid voltages. As a results, the new switching table is the same under balanced or unbalanced grid voltages. The basic idea for synthesizing the new DPC switching table is to select the best rectifier voltage vector among all the possible vectors in order to restrict the instantaneous active and reactive power tracking errors, simultaneously. The new switching table synthesis is based on the sign and magnitude of variation of instantaneous active and reactive power for each sector as shown in Fig. 2. For example, the influence of voltage vectors provided by the rectifier on the active and reactive power is shown in Table 1 considering the first sector  1 , the table is classified according to the LVV, MVV, SVV and ZVV. The vectors of the same type are listed in an ascending order according to the impact degree on the power. As an example, firstly, the influence of LVVs on the active power, v11 can increase the active power more than v14 , and v14 can increase the active power more than v8 . v2 can decrease the active power more than v5 , and v5 can decrease the active power more than v17 . Secondly, the influence of LVVs on the reactive power, v8 can increase

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Table 2 Switching table relative to the proposed DPC under balanced or unbalanced grid voltages. P/Pnov

Q

1

2

3

4

5

6

7

8

9

 10

 11

 12

↑ ↑ ↓ ↓

↓ ↑ ↓ ↑

16 4 2 3

16 4 3 5

1 7 5 6

1 7 6 8

4 10 8 9

4 10 9 11

7 13 11 12

7 13 12 14

10 16 14 15

10 16 15 17

13 1 17 18

13 1 18 2

the reactive power more than v5 , and v5 can increase the reactive power more than v11 . v17 can decrease the reactive power more than v14 , and v14 can decrease the reactive power more than v2 . The same principal for the other voltage vectors. Therefore, from this analysis and from Table 1, it appears that all voltage vectors have influence on active power and reactive power, but the effected extent was diverse. Thus, the switching table suitable to achieve simultaneous control of both Pnov and Q can be obtained in Table 2. We see from Table 1 that the switching table derived under balanced grid voltage is the same as that under unbalanced grid voltage. Figs. 4–6 show the block scheme of the conventional DPC, the new DPC and the simulation block of the conventional and new DPC for three-level NPC rectifier, respectively. The neutral point potential balancing control is one of the most important requirements for the three-level NPC rectifier, the aim here is have the voltage of both capacitors C1 and C2 equal to each other as far as possible. The imbalance of neutral point potential is fundamentally caused by the non-zero current of neutral point (i0 ). For three-level NPC rectifier as mentioned above, there are four kinds of vectors. In [39], the authors presented the relationships between every vector and the current of neutral point, both zero and large vectors have no any affect on the neutral potential (the neutral current is equal to zero). With the medium vectors, only one

Fig. 4. Block scheme of the conventional DPC for three-level NPC rectifier.

Fig. 5. Block scheme of the new DPC for three-level NPC rectifier.

Fig. 6. Simulation block scheme of the conventional and new DPC for three-level NPC rectifier.

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Fig. 7. Simulation results of the CDPC strategy under balanced and unbalanced grid voltages (from top to bottom): (a) grid voltages, (b) input currents, (c) active power, (d) reactive power, (e) new active power, (f) new reactive power, (g) DC voltages, and (h) current of neutral point.

Fig. 8. Simulation results of the DPC-NQ strategy under balanced and unbalanced grid voltages (from top to bottom): (a) grid voltages, (b) input currents, (c) active power, (d) reactive power, (e) new active power, (f) new reactive power, (g) DC voltages, and (h) current of neutral point.

of the 3 phases currents is connected to the neutral point. Finally, with the small vectors, two of the 3 phases currents are connected to the neutral point. As a result, small vectors with their two different configurations (see Fig. 3) can produce the same line to line voltages with opposite influence on the neutral point potential. As shown in Figs. 4 and 5, the switching table selecting block can identify which one will be finally employed. The block has one output and three inputs as follows: (i) the output of switching table, which may be a large or a middle vector or a pair of small vectors; (ii) three input phases current which are used to judge the polarity of the neutral point current [40]; (iii)  i , the angular position of space voltage vector, which changes from 1 to 12; (viii) S, the digitized error between Udc1 and Udc2 , which are equal to:

Table 3 System and control parameters.



Sdc =

if

Udc1 − Udc2 > Hdc

1 if

Udc1 − Udc2 < Hdc

0

(37)

where 1 means neutral point potential should be increased, 0 means neutral point potential should be decreased, and Hdc is the hysteresis band of neutral-point potential comparator. 3. Simulation results In order to verify the behavior of the proposed control strategies, comparative simulations have been carried out. The system

Name

Nomenclature/unit

Value

AC line voltage (RMS) Line resistance Load resistance Frequency AC-side inductance DC-bus capacitors DC-bus voltage

e [V] R [] RL [] f [Hz] L [H] C1 , C2 [F] Udc [V]

170 0.3 80 50 0.01 680 × 10−6 500

and control parameters are listed in Table 3; the grid voltage is unbalanced with an amplitude of the negative sequence grid voltage equal to 15%. The performance of the proposed DPC strategy with the new active power definition (DPC-NP) is compared with that of the following two control schemes: (1) the conventional DPC (CDPC) method with the traditional definition of the active and reactive power, (2) the DPC scheme with the new definition of the reactive power (DPC-NQ) that presented in [36]. With the CDPC strategy, the control targets are the conventional active power and conventional reactive power. With the DPC-NQ strategy, the control targets are the conventional active power and new reactive power. With the DPC-NP strategy, the control targets are the new active power and conventional reactive power.

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Fig. 11. Simulation results of the DPC-NP strategy under step change in the DC Load RL (from top to bottom): (a) grid voltages, (b) input currents, (c) active power, (d) reactive power, (e) new active power, and (f) new reactive power.

Fig. 9. Simulation results of the DPC-NP strategy under balanced and unbalanced grid voltages (from top to bottom): (a) grid voltages, (b) input currents, (c) active power, (d) reactive power, (e) new active power, (f) new reactive power, (g) DC voltages, and (h) current of neutral point.

Figs. 7–10 show the simulation results of those three control schemes (i.e. CDPC, DPC-NQ, and proposed DPC-NP) under balanced and unbalanced grid voltages. At the time instant t = 0.2 s, a 15% negative-sequence voltage is imposed on the grid. The response of the CDPC scheme is illustrated in Fig. 7, this figure shows that the grid currents are highly distorted when the grid voltages become unbalanced and the new active and new reactive power, computed according to the new definition, start to oscillate. The DC-link volt∗ = 500 V, and the balancing age Udc tracks its reference value Udc of the upper and lower capacitor voltages is achieved. The THD of the grid currents using the conventional DPC is 5.42%; see Fig. 10a.

Fig. 10. Harmonic spectrum and THD of input currents for three-level NPC rectifier, with the aforementioned control methods.

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Fig. 12. Simulation results of the DPC-NP strategy under step change in the reactive power reference (from top to bottom): (a) grid voltages, (b) input currents, (c) active power, (d) reactive power, (e) new active power, and (f) new reactive power.

Fig. 13. Simulation results of the DPC-NP strategy under step change in the value of grid unbalanced (from top to bottom): (a) grid voltages, (b) input currents, (c) active power, (d) reactive power, (e) new active power, and (f) new reactive power.

Then, according to the performance of the DPC-NQ which is shown in Fig. 8, the active power and the new reactive power are constant without oscillations and the grid currents are almost sinusoidal. ∗ = 500 V, The DC-link voltage Udc follows its reference value Udc and the balancing of capacitor voltages Udc1 and Udc2 is fulfilled. The THD of the grid currents using DPC-NQ is 1.23%; see Fig. 10b. Finally, Fig. 9 illustrates the response of the proposed DPC-NP. It can be observed that both the new active power and the reactive power are constant without any oscillations. Also, the tracking ∗ = 500 V and of the DC-bus voltage Udc to its reference value Udc

the balancing of upper and lower capacitor voltages (i.e. Udc1 and Udc2 , respectively) is achieved. Furthermore, the grid currents are almost sinusoidal and the THD of these currents is 1.14%; see Fig. 10c. In order to further investigate the performance of the proposed DPC-NP, the dynamic response of the proposed DPC-NP under step change in the DC load RL with unbalanced grid voltages is investigated. At the time instant t = 0.2 s, a step change in the DC load RL from 80  to 55  is applied to the control system. It can be seen from Fig. 11 that the proposed DPC-NP achieves good dynamic per-

Fig. 14. Harmonic spectrum and THD of input currents for three-level NPC rectifier, with the aforementioned control methods.

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formance. Furthermore, the THD of the grid currents is very low (0.99%); see Fig. 14a. Furthermore, the performance of the proposed DPC-NP is tested under step change in the reactive power from 0 Var to 200 Var at time instant t = 0.2 s. According to Fig. 12, it is clear that the system works with good dynamic performance (the input current is almost sinusoidal with THD equal to 1.17%; see Fig. 14b) and both new active power and reactive power follow respectively their reference values. Finally, at the time instant t = 0.2 s, 30% negative sequence voltage is utilized instead of the first value (i.e. 15%). According to Fig. 13, the performance of the proposed DPC-NP is still good. Only, very small oscillations in the new active power appeared. However, the input current waveforms are almost sinusoidal with a THD of 1.07%; see Fig. 14c. 4. Conclusion The present study was designed to solve the drawbacks of the conventional direct power control (CDPC) while the grid voltage is unbalanced. Therefore, a new definition of active power based on the extend pq theory is proposed, and discussed in detail in the first and second subsection of Section 2. This new definition of active power was used instead of the conventional definition. Then, based on theory of the instantaneous power, a novel table-based DPC suitable to operate a three-level NPC rectifier under unbalanced grid voltage condition is proposed, the process of constructed this new switching table is discussed in the third subsection of Section 2. The new switching table is validate under balanced and unbalanced grid voltage. The performance of the proposed DPC-NP have been compared with that of both CDPC strategy and DPC scheme with new definition of reactive power (DPC-NQ). The results have validated that the proposed DPC-NP considerably outperformed the CDPC and DPC-NQ methods, in terms of better dynamic and steady-state performance. Furthermore, the waveforms of the grid side currents are almost sinusoidal with very low THDs under both balanced and/or unbalanced grid voltages. The conducted studies conclude the following important features of the proposed NP-DPC strategy under unbalanced grid voltage: (1) The proposed NP-DPC strategy does not need the extraction of the positive and/or negative components of grid voltage and current to achieve a sinusoidal input currents with low total harmonic distortion (THD). (2) The proposed NP-DPC strategy does not need the extraction of the positive and/or negative components of grid voltage and current to achieve smooth active and reactive power waveforms. (3) The proposed NP-DPC strategy is much simpler than the improved methods that proposed to solve the drawbacks of the conventional DPC (CDPC) under unbalanced grid voltage. (4) Compared to DPC scheme with new definition of reactive power (DPC-NQ), the proposed NP-DPC strategy shows good performance with very low THDs of input currents under different step changes in the DC-bus load, reactive power reference and grid voltage unbalanced value. Finally, the proposed NP-DPC strategy is also attractive for three phase PWM converter applications, and the proposed method can be applied into wind converter systems. References [1] G. Hu, X. Yu, D.G. Holmes, W. Shen, Q. Wang, F. Luo, N. Liu, An improved virtual space vector modulation scheme for three-level active neutral-point-clamped inverter, J. Power Electron. 15 (2015) 106–115.

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