3.4. Other Approaches: Synchronous Frames

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3.4. Other Approaches: Synchronous Frames

Previous sections have presented two formulations of the instantaneous reactive power developed coordinate axes considered a “fixed”, that is, nonrotating. Specifically, we used the 0, α, β, coordinates or the 1, 2, 3 phase coordinates. In this section other formulations of the instantaneous reactive power in rotating coordinate systems, either in solidarity with the vector of voltage, or at no. This requires performing the appropriate coordinate transformation [26–29].

Mathematical transformations to perform coordinate changes are often used in the field of electric power systems. In general, when a coordinate transformation is achieved, all electrical variables refer to a common frame of reference, and usually a decoupling of the variables is achieved. A singular case is Park’s transformation for the study of electrical machines and power systems. This is a transformation from a three-phase system to a two-phase system rotating at angular velocity ω (synchronous speed). Typically, a third variable known as the zero-sequence component is added to determine the inverse transformation matrix, Figure 3.27. The matrix determines from Park’s transformation 1, 2, 3 phase variables to the 0, d, q variables is given by (3.112),

$[T] = \sqrt{\frac{2}{3}} [\begin{array}{c} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \cos θ_{1} (t) & \cos θ_{2} (t) & \cos θ_{3} (t) \\ - \sin θ_{1} (t) & - \sin θ_{2} (t) & - \sin θ_{3} (t) \end{array}]$

(3.112)

where

$θ_{k} (t) = θ (t - \frac{2 π}{3} (k - 1)), k = 1,2,3$

(3.113)

and θ(t) is an angular arbitrary function. For example, Park processing for analysis of the synchronous machine is a choice of

$θ_{k} (t) = ω t - \frac{2 π}{3} (k - 1), k = 1,2,3$

(3.114)

where ω is the synchronous speed of the machine. If, on the other hand the following is chosen

$θ_{k} (t) = - \frac{2 π}{3} (k - 1), k = 1,2,3$

(3.115)

coordinate transformation to 0, α, β, fixed axes are obtained (3.1)–(3.2), that is, the Concordia–Clarke transformation. The choice of the coefficient √(2/3) ensures that the transformation is a power transformation invariant.

Figure 3.27 Coordinate system 0dq; 0 axis is perpendicular to the plane formed by the dq axes and having direction perpendicular out of the plane.

3.4.1. dq Formulation

The application of Park’s transformation (3.112) to the set of a system of three-phase currents of a power system is what has come to be called the dq formulation of the instantaneous reactive power [26]. This formulation is taken as the angular velocity of rotation of the 0, d, q axes the fundamental harmonic pulsation value of the supply voltage. Thus the transformation (3.112) emerges as a double transformation of axes. First, a transformation from a 1, 2, 3 phase coordinates system, to a fixed coordinate system, 0, α, β. Then, a transformation of 0, α, β coordinate system to a 0, d, q rotating coordinate system at constant angular velocity corresponding to the frequency of the fundamental harmonic of the voltage, ω.

$\begin{array}{l} [\begin{array}{c} i_{0} \\ i_{d} \\ i_{q} \end{array}] = [\begin{array}{c} 1 & 0 & 0 \\ 0 & \cos ω t & \sin ω t \\ 0 & - \sin ω t & \cos ω t \end{array}] \sqrt{\frac{2}{3}} [\begin{array}{c} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 1 & - \frac{1}{2} & - \frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & - \frac{\sqrt{3}}{2} \end{array}] [\begin{array}{c} i_{1} \\ i_{2} \\ i_{3} \end{array}] = \\ = \sqrt{\frac{2}{3}} [\begin{array}{c} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ co s (ω t) & co s (ω t - \frac{2 π}{3}) & co s (ω t + \frac{2 π}{3}) \\ - si n (ω t) & - si n (ω t - \frac{2 π}{3}) & - si n (ω t + \frac{2 π}{3}) \end{array}] [\begin{array}{c} i_{1} \\ i_{2} \\ i_{3} \end{array}] \end{array}$

(3.116)

The coordinate system of 0, d, q axes rotates at an angular velocity of ω = 2π/T (T is the period of the voltage waveform) for fixed axes 1,2,3 or 0, α, β. The angular position of the axis “d” from the axis “α” is determined by the angle θ(t) = ωt that changes with time. If the voltages are balanced and sinusoidal, the voltage vector rotates at the same speed as the axes dq. If voltages are unbalanced and nonsinusoidal, the speed of rotation of the voltage vector is not constant, it becomes influenced by the imbalance and distortion of the voltage. In any case, the dq formulation chosen for rotational speed of dq plane taking as pivot axis 0, the frequency of the fundamental harmonic of the supply voltage.

Example 3.7

A three-phase voltage system whose line-to-neutral voltages include a negative sequence component at the fundamental frequency, and harmonic order h of negative sequence is considered.

$\begin{array}{l} u_{1} (t) = \sqrt{2} V^{+} \cos (ω t + ψ_{1}^{+}) + \sqrt{2} V^{-} \cos (ω t + ψ_{1}^{-}) + \sqrt{2} V_{h}^{-} \cos (h ω t + ψ_{h}^{-}) \\ u_{2} (t) = \sqrt{2} V^{+} \cos (ω t + ψ_{1}^{+} - 120) + \sqrt{2} V^{-} \cos (ω t + ψ_{1}^{-} + 120) + \sqrt{2} V_{h}^{-} \cos (h ω t + ψ_{h}^{-} + 120) \\ u_{3} (t) = \sqrt{2} V^{+} \cos (ω t + ψ_{1}^{+} + 120) + \sqrt{2} V^{-} \cos (ω t + ψ_{1}^{-} - 120) + \sqrt{2} V_{h}^{-} \cos (h ω t + ψ_{h}^{-} - 120) \end{array}$

(3.117)

Determine the 0, d, q coordinates of voltages service indicated.

The transformation matrix (3.116) applies to three-phase voltages service (3.117) for the components 0, d, q of the voltage.

$[\begin{array}{c} u_{0} \\ u_{d} \\ u_{q} \end{array}] = T [\begin{array}{c} u_{1} \\ u_{2} \\ u_{3} \end{array}] = \sqrt{3} V^{+} [\begin{array}{c} 0 \\ \cos ψ_{1}^{+} \\ \sin ψ_{1}^{+} \end{array}] + \sqrt{3} V^{-} [\begin{array}{c} 0 \\ \cos (2 ω t + ψ_{1}^{-}) \\ - \sin (2 ω t + ψ_{1}^{-}) \end{array}] + \sqrt{3} V_{5}^{-} [\begin{array}{c} 0 \\ \cos ((h + 1) ω t + ψ_{5}^{-}) \\ - \sin ((h + 1) ω t + ψ_{5}^{-}) \end{array}]$

(3.118)

That is, the fundamental harmonic of positive-sequence phase becomes a constant term, and the negative sequence of the fundamental harmonic and higher harmonics of any sequence phases become oscillatory terms with a frequency of at least twice the fundamental frequency.

The conclusions drawn from Example 3.7 can be applied to the active compensation of a nonlinear unbalanced load. Thus, if we consider any load, applying (3.116) allows the waveforms of load current i_d(t) and i_q(t) to be derived. The mentioned i_d(t) and i_q(t) signals are time variable, and their average values are I_d and I_q, respectively. So, each one can be divided into two terms, a constant value and a variable component.

$i_{d} (t) = I_{d} + {\tilde{i}}_{d} (t)$

(3.119)

$i_{q} (t) = I_{q} + {\tilde{i}}_{q} (t)$

(3.120)

The constant I_d and I_q components are, respectively, the fundamental harmonic positive sequence of i_d(t) and i_q(t). The other components are the fundamental harmonic negative sequence and the rest of i_d(t) and i_q(t) harmonics. The average values of these last components are null [27].

This current decomposition is used to generate the compensation current reference in shunt active power filter applications. So, the source current will include the constant components I_d and I_q, and the shunt active conditioner will inject the variable components. In practical implementation, it is possible to separate both parts with a low-pass filter. In 1–2–3 coordinates, the source currents will be (3.121):

$[\begin{array}{c} i_{S 1} \\ i_{S 2} \\ i_{S 3} \end{array}] = \sqrt{\frac{2}{3}} [\begin{array}{c} \frac{1}{\sqrt{2}} & co s θ & - \sin θ \\ \frac{1}{\sqrt{2}} & co s (θ - 120) & - \sin (θ - 120) \\ \frac{1}{\sqrt{2}} & co s (θ + 120) & - \sin (θ + 120) \end{array}] [\begin{array}{c} 0 \\ I_{d} \\ I_{q} \end{array}]$

(3.121)

and the compensation currents will be:

$[\begin{array}{c} i_{C 1} \\ i_{C 2} \\ i_{C 3} \end{array}] = [\begin{array}{c} i_{L 1} \\ i_{L 2} \\ i_{L 3} \end{array}] - [\begin{array}{c} i_{S 1} \\ i_{S 2} \\ i_{S 3} \end{array}] = \sqrt{\frac{2}{3}} [\begin{array}{c} \frac{1}{\sqrt{2}} & co s θ & - \sin θ \\ \frac{1}{\sqrt{2}} & co s (θ - 120) & - \sin (θ - 120) \\ \frac{1}{\sqrt{2}} & co s (θ + 120) & - \sin (θ + 120) \end{array}] [\begin{array}{c} i_{0} (t) \\ {\tilde{i}}_{d} (t) \\ {\tilde{i}}_{q} (t) \end{array}]$

(3.122)

In conclusion, this theory allows sinusoidal and balanced currents to be obtained, but independent of a symmetrical or distorted voltage supply. Moreover, the active power supplied by the compensator only be null if the voltages are balanced and sinusoidal [58].

3.4.2. i_d–i_q Method

A variant of the formulation dq is called the i_d–i_q method [27]. This is based on the following consideration: the speed of rotation of the system 0dq about the axis 0 is the pulsation of the fundamental harmonic of the supply voltage; thus this synchronous rotating system rotates at the same speed as the voltage vector only in the case of balanced sinusoidal positive-sequence voltages. In general, the rotation speed of the system 0dq and voltage vector will be different. The method i_d–i_q places the axis d in the direction of the projection of the voltage vector on the αβ-plane. This ensures that the dq axes rotates in solidarity with the projection voltage vector, u_αβ, Figure 3.28.

Figure 3.28 System of coordinates for method i_d–i_q; d axis is matched with u_αβ vector.

The process is summarized as follows. The coordinate system 1–2–3 is transformed into the coordinate system 0, α, β by the Clarke–Concordia transformation (3.1)–(3.2), and then a transformation is performed, (3.123), to a coordinate system rotating at one speed ω(t) such that the angular position of the d axis relative to the α axis is given by θ,

$[\begin{array}{c} u_{0} \\ u_{d} \\ u_{q} \end{array}] = [\begin{array}{c} 1 & 0 & 0 \\ 0 & \cos θ & \sin θ \\ 0 & - \sin θ & \cos θ \end{array}] [\begin{array}{c} u_{0} \\ u_{α} \\ u_{β} \end{array}]$

(3.123)

Where now θ represents the angle between the projection of the voltage vector in the plane α, β to the α axis, that is,

$θ = t g^{- 1} \frac{u_{β}}{u_{α}}$

(3.124)

thus:

$co s θ = u_{α} / u_{α β}$

(3.125)

$si n θ = u_{β} / u_{α β}$

(3.126)

So, i_d–i_q method may be expressed as follows, (3.127):

$[\begin{array}{c} i_{0} \\ i_{d} \\ i_{q} \end{array}] = \frac{1}{u_{α β}} [\begin{array}{c} u_{α β} & 0 & 0 \\ 0 & u_{α} & - u_{β} \\ 0 & - u_{β} & u_{α} \end{array}] [\begin{array}{c} i_{0} \\ i_{α} \\ i_{β} \end{array}]$

(3.127)

With this transformation, the direct voltage component is

$u_{d} = |u_{α β}| = u_{α β} = \sqrt{u_{α}^{2} + u_{β}^{2}}$

(3.128)

and the q voltage component is always null, u_q = 0. Current components verify the next propositions:

The positive sequence of the first harmonic becomes a dc quantity. All the other higher order current harmonics including the negative-sequence fundamental component become variable along the time quantities and they receive a frequency displacement in the spectrum. So, they comprise the oscillatory current components.

From the viewpoint of the active compensation, a shunt active filter must compensate variables terms of current over time, both the dq current components and the zero-sequence current. The compensation current is in phase coordinates therefore,

$[\begin{array}{c} i_{C 1} \\ i_{C 2} \\ i_{C 3} \end{array}] = \sqrt{\frac{2}{3}} [\begin{array}{c} \frac{1}{\sqrt{2}} & 1 & 0 \\ \frac{1}{\sqrt{2}} & - \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{1}{\sqrt{2}} & - \frac{1}{2} & - \frac{\sqrt{3}}{2} \end{array}] \frac{1}{u_{α β}} [\begin{array}{c} u_{α β} & 0 & 0 \\ 0 & u_{α} & - u_{β} \\ 0 & - u_{β} & u_{α} \end{array}] [\begin{array}{c} i_{L 0} \\ {\tilde{i}}_{L d} \\ {\tilde{i}}_{L q} \end{array}]$

(3.129)

The compensation currents (3.129) can eliminate neutral currents; however, although the source currents present low distortion after compensation, they never achieve a complete elimination of distortion from a practical point of view, except in the case of balanced and sinusoidal voltages. Furthermore, one-unity displacement factor will not be achieved. This latter circumstance can be overcome by a slight modification to the compensation strategy (3.129). Indeed, under the method i_d–i_q, the instantaneous real power is

$p (t) = u_{0 d q} \cdot i_{0 d q} = u_{0} i_{0} + u_{d} i_{d} + u_{q} i_{q} = u_{0} i_{0} + u_{d} i_{d}$

(3.130)

The load current component i_q does not participate in the transfer of instantaneous real power, making their total compensation possible. The compensation currents for a nonlinear load in the method i_d–i_q are given by (3.131),

$[\begin{array}{c} i_{C 0} \\ i_{C α} \\ i_{C β} \end{array}] = \frac{1}{u_{α β}} [\begin{array}{c} u_{α β} & 0 & 0 \\ 0 & u_{α} & - u_{β} \\ 0 & - u_{β} & u_{α} \end{array}] [\begin{array}{c} i_{L 0} \\ {\tilde{i}}_{L d} \\ i_{L q} \end{array}]$

(3.131)

The cancellation of i_Lq allows us to obtain a unity displacement factor, as the average reactive power is eliminated. The advantage of i_d–i_q method over the dq method is that compensating currents are determined directly from supply voltages, therefore, independent of frequency, face difficulties arising from the use of PLLs in the control circuit of the compensator.

3.4.3. p–q–r Formulation

As has been established in a system with unbalanced voltages and distortion, the rotational speed of the voltage vector is not constant but is influenced by harmonic components and negative sequences thereof. In such cases, a formulation of the instantaneous reactive power in a rotating coordinate system attached to the voltage vector is presented, that is the p–q–r system. The resulting transformation of coordinates defines three independent power variables more simply (specifically an instantaneous real power and two instantaneous reactive power) that support its independent compensation [28,29].

In the p–q–r formulation a new coordinate system in which the p axis coincides with the direction of the voltage vector, the q-axis is perpendicular to p and is in the α–β plane is defined, and the r axis is perpendicular to the previous two. In this system, the current vector has three components and only one voltage vector. Thus, only a component of current vector, the component i_p is collinear with the voltage; the aim of compensation consists of compensate remaining components, components i_q and i_r, which do not carry real power. Two compensation strategies, a call reference power control strategy, which also compensates the part of i_p current that does not transport active power, and another strategy called reference current control compensation are distinguished in which it compensates the alternating part of the i_p current. In both cases, supply current carrying the active power, and collinear with the voltage will be achieved. Thus, if they were nonsinusoidal voltages, supply current obtained would not be sinusoidal.

In what follows, the new framework will be achieved by two successive rotational transformations. A new system of coordinates 0α′β′ will be obtained from the rotation of the αβ plane through the axis 0 angle θ₁ = tg⁻¹(u_β/u_α) allowing the α axis to be aligned with the projection of the vector of voltage in the αβ plane (transformation of i_d–i_q method), Figure 3.29a shows that the transformation for the current is given by (3.132),

$[\begin{array}{c} i_{0} \\ i_{α'} \\ i_{β'} \end{array}] = [\begin{array}{c} 1 & 0 & 0 \\ 0 & \cos θ_{1} & \sin θ_{1} \\ 0 & - \sin θ_{1} & \cos θ_{1} \end{array}] [\begin{array}{c} i_{0} \\ i_{α} \\ i_{β} \end{array}] = \frac{1}{u_{α β}} [\begin{array}{c} u_{α β} & 0 & 0 \\ 0 & u_{α} & - u_{β} \\ 0 & - u_{β} & u_{α} \end{array}] [\begin{array}{c} i_{0} \\ i_{α} \\ i_{β} \end{array}]$

(3.132)

Figure 3.29 Construction pqr coordinates system. (a) Rotation of αβ plane taking as axis of rotation zero axis, (b) rotation of 0α’ plane taking as the axis of rotation β’≡q-axis. The q-axis enters perpendicular to the pr plane.

The pqr reference frame is formed by a rotation of the plane 0 – α′ angle θ₂ = tg⁻¹(u₀/u_αβ) against rotation β′ axis such that the α′ axis coincides with the direction vector voltage u_0αβ, new p axis Figure 3.29b.

$[\begin{array}{c} i_{p} \\ i_{q} \\ i_{r} \end{array}] = [\begin{array}{c} \sin θ_{2} & \cos θ_{2} & 0 \\ \cos θ_{2} & - \sin θ_{2} & 0 \\ 0 & 0_{1} & 1 \end{array}] [\begin{array}{c} i_{0} \\ i_{α'} \\ i_{β'} \end{array}] = \frac{1}{u_{α β}} [\begin{array}{c} \frac{u_{0}}{u_{0 α β}} & \frac{u_{α β}}{u_{0 α β}} & 0 \\ \frac{u_{α β}}{u_{0 α β}} & - \frac{u_{0}}{u_{0 α β}} & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{c} i_{0} \\ i_{α'} \\ i_{β'} \end{array}]$

(3.133)

The β′ axis will be the new q-axis and the zero axis is the new r axis.

Finally, currents and voltages are translated from 0αβ coordinates to pqr coordinates by means of the next matrix [7, 8]:

$[\begin{array}{c} i_{p} \\ i_{q} \\ i_{r} \end{array}] = \frac{1}{u_{0 α β}} [\begin{array}{c} u_{0} & u_{α} & u_{β} \\ 0 & - \frac{u_{0 α β} u_{β}}{v_{α β}} & \frac{u_{0 α β} u_{α}}{v_{α β}} \\ u_{α β} & - \frac{u_{0} u_{α}}{u_{α β}} & - \frac{u_{0} u_{β}}{u_{α β}} \end{array}] [\begin{array}{c} i_{0} \\ i_{α} \\ i_{β} \end{array}]$

(3.134)

$[\begin{array}{c} u_{p} \\ u_{q} \\ u_{r} \end{array}] = [\begin{array}{c} u_{0 α β} \\ 0 \\ 0 \end{array}]$

(3.135)

Equation (3.135) presents the voltage transformation. Now the voltage vector has only one component that is not null

$u_{p} = u_{0 α β} = \sqrt{u_{0}^{2} + u_{α}^{2} + u_{β}^{2}}$

(3.136)

Once defined, the instantaneous real power p(t) (dot product of voltage/current space vectors) and the instantaneous reactive power vector q(t) (cross product of voltage/current space vectors), three power variables are disposed: instantaneous real power p(t) and two instantaneous reactive powers q_r(t) and q_q(t) (Figure 3.30).

Figure 3.30 Pqr axes arrangement where the voltage vector is in the direction of the axis p.

Current components i_p, i_q, i_r respect to those three power variables are calculated as follows:

$[\begin{array}{c} i_{p} \\ i_{q} \\ i_{r} \end{array}] = \frac{1}{u_{p}} [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{array}] [\begin{array}{c} p \\ q_{r} \\ q_{q} \end{array}]$

(3.137)

The p–q–r formulation presents two control strategies: Reference Current Control and Reference Power Control [28]. Results are very similar in both cases. In the Reference Power Control, the compensation current equation is as follows:

$[\begin{array}{c} i_{C 0} \\ i_{C α} \\ i_{C β} \end{array}] = \frac{1}{u_{0 α β}} [\begin{array}{c} u_{0} & 0 & u_{α β} \\ u_{α} & - \frac{u_{0 α β} u_{β}}{u_{α β}} & - \frac{u_{0} u_{α}}{u_{α β}} \\ u_{β} & \frac{u_{0 α β} u_{α}}{u_{α β}} & - \frac{u_{0} u_{β}}{u_{α β}} \end{array}] [\begin{array}{c} \frac{{\tilde{p}}_{L}}{u_{p}} \\ \frac{q_{L r}}{u_{p}} \\ \frac{- q_{L q}}{u_{p}} + \frac{u_{0}}{u_{α β}} val (\frac{p_{L}}{u_{p}}) \end{array}]$

(3.138)

where val(p_L/u_p) means the numerical value of the ratio of the instantaneous real power load and voltage.

As can be seen in (3.138), the variable instantaneous real power, the whole component q of the instantaneous reactive power and the necessary part of the instantaneous reactive power component r to eliminate the neutral current are compensated.

The equation of Reference Current Control is as follows:

$[\begin{array}{c} i_{c p} \\ i_{c q} \\ i_{c r} \end{array}] = [\begin{array}{c} {\tilde{i}}_{L p} \\ i_{L q} \\ i_{L r} - \frac{u_{0}}{u_{α β}} val ({\tilde{i}}_{L p}) \end{array}]$

(3.139)

where

val ({\tilde{i}}_{L p})

means the numeric value of the load current component p variable part.

Here, i_cq = i_Lq to compensate its constant part as its variable part and to get a source current in phase to the voltage supplied.

3.5. Dual Instantaneous Reactive Power

In this section the theory of dual formulations of the instantaneous reactive power are developed. They are widespread in the literature, but have traditionally been applied to active power filters parallel connection: the original formulation and modified formulation. Hereinafter it is shown that these developments allow the use of such formulations in designing strategies compensation nonlinear loads of type Harmonic Voltage Source (HVS) using series active power filters.

So far, all formulations have been presented and applied to the case of active power filters in parallel connection. These formulations are not directly applicable to the case of series active filters. Now a dual formulation of the two formulations indicated will be developed and implemented to the compensation of the HVS type that constitutes the most appropriate load for the use of compensation equipment of series connection [4, 30].

3.5.1. Dual Original Instantaneous Reactive Power Formulation

The formulation of the instantaneous reactive power is applied since its inception in controlling compensation equipment in parallel connection. Thus, the components of the compensation currents are obtained from the voltages at 0αβ coordinates, the instantaneous real power, and the instantaneous imaginary power. However, when it comes to equipment of series compensation a problem statement is necessary from the point of dual view. Thus, the voltages are those to be determined from the 0,α,β components of current, and power variables of load [4].

It was initially considered a three-wire three-phase system. Instantaneous real power within the αβ coordinates is calculated by the expression (3.140),

$p_{α β} = u_{α} i_{α} + u_{β} i_{β}$

(3.140)

This power can be written in vector form by the scalar product (3.141),

$p_{α β} = i_{α β}^{T} u_{α β}$

(3.141)

where i^T_αβ is the transposed vector of currents in α–β coordinates,

$i_{α β} = {[\begin{array}{c} i_{α} & i_{β} \end{array}]}^{T}$

(3.142)

In like manner u_αβ is the vector of voltages in the same coordinates

$u_{α β} = {[\begin{array}{c} u_{α} & u_{β} \end{array}]}^{T}$

(3.143)

The instantaneous imaginary power is defined by the expression

$q_{α β} = u_{α} i_{β} - u_{β} i_{α}$

(3.144)

Similar to the instantaneous real power, the instantaneous imaginary power can be expressed by the dot product (3.145),

$q_{α β} = i_{α β ⊥}^{T} u_{α β}$

(3.145)

where i^T_αβ⊥ is the normal or perpendicular vector defined by (3.146),

$i_{α β ⊥} = {[\begin{array}{c} i_{β} & - i_{α} \end{array}]}^{T}$

(3.146)

Thus, the instantaneous real power and instantaneous imaginary power can be expressed in matrix form by (3.147),

$[\begin{array}{c} p_{α β} \\ q_{α β} \end{array}] = [\begin{array}{c} i_{α β}^{T} \\ i_{α β ⊥}^{T} \end{array}] u_{α β}$

(3.147)

In view of the expression (3.147), it is possible to obtain the voltage in αβ coordinates based on the instantaneous power as

$u_{α β} = \frac{1}{i_{α}^{2} + i_{β}^{2}} [\begin{array}{c} i_{α β}^{T} \\ i_{α β ⊥}^{T} \end{array}] [\begin{array}{c} p_{α β} \\ q_{α β} \end{array}]$

(3.148)

or else

$u_{α β} = \frac{p}{i_{α β}^{2}} i_{α β} + \frac{q}{i_{α β}^{2}} i_{α β ⊥}$

(3.149)

where

$i_{α β}^{2} = i_{α}^{2} + i_{β}^{2}$

(3.150)

The analysis in αβ terms of (3.149) allows two coordinate axes defined by the vectors i_αβ e i_αβ⊥. Thus, the voltage vector is decomposed into its orthogonal axes defined by the two current vectors, as shown in Figure 3.31.

Figure 3.31 Split of voltage vector along the axes defined by i_αβ e i_αβ⊥.

This formulation developed part of a voltage vector and a current vector to effect transformation of Clarke–Concordia to put its components along the α and β axes. This development is valid for systems wherein the neutral current is zero because the system is four-wire or three-wire balanced. Analogously to Section 3.2, to take into account the zero-sequence component, it is necessary to add a third coordinate (0 axis) perpendicular to the plane formed by the αβ coordinates. In the new 0αβ reference system of interest, following vectors are defined: the vector of zero-sequence current i_0, which is the only non-zero component of the zero-sequence current, that is,

$i_{0} = {[\begin{array}{c} i_{0} & 0 & 0 \end{array}]}^{T}$

(3.151)

Furthermore, in this situation i_αβ and i_αβ⊥ include a zero-sequence component null, which are both defined in the form

$i_{α β} = {[\begin{array}{c} 0 & i_{α} & i_{β} \end{array}]}^{T}$

(3.152)

and

$i_{α β ⊥} = {[\begin{array}{c} 0 & i_{β} & - i_{α} \end{array}]}^{T}$

(3.153)

Instantaneous real power is decomposed into two terms

$p (t) = p_{α β} + p_{0}$

(3.154)

where p_αβ is the instantaneous real power defined by (3.140) and p₀ is the zero-sequence instantaneous power defined in the usual way,

$p_{0} = u_{0} i_{0}$

(3.155)

In matrix form the three power variables can be expressed by (3.156),

$[\begin{array}{c} p_{0} \\ p_{α β} \\ q_{α β} \end{array}] = [\begin{array}{c} i_{0}^{T} \\ i_{α β}^{T} \\ i_{α β ⊥}^{T} \end{array}] u_{0 α β}$

(3.156)

In (3.156) the matrix including the components of currents is given by

$[\begin{array}{c} i_{0} & 0 & 0 \\ 0 & i_{α} & i_{β} \\ 0 & i_{β} & - i_{α} \end{array}]$

(3.157)

It is a square matrix whose inverse is

$\frac{1}{i_{0} (i_{α}^{2} + i_{β}^{2})} [\begin{array}{c} (i_{α}^{2} + i_{β}^{2}) & 0 & 0 \\ 0 & i_{0} i_{α} & i_{0} i_{β} \\ 0 & i_{0} i_{β} & - i_{0} i_{α} \end{array}]$

(3.158)

Thus, in this situation you can obtain the vector of voltages u_0αβ in the form given by (3.159),

$u_{0 α β} = [\begin{array}{c} u_{0} \\ u_{α} \\ u_{β} \end{array}] = [\begin{array}{c} i_{0}^{T} / i_{0}^{2} \\ i_{α β}^{T} / i_{α β}^{2} \\ i_{α β ⊥}^{T} / i_{α β}^{2} \end{array}] [\begin{array}{c} p_{0} \\ p_{α β} \\ q_{α β} \end{array}]$

(3.159)

or alternatively expressed

$u_{0 α β} = \frac{p_{0}}{i_{0}^{2}} i_{0} + \frac{p_{α β}}{i_{α β}^{2}} i_{α β} + \frac{q_{α β}}{i_{α β}^{2}} i_{α β ⊥}$

(3.160)

The vectors i₀, i_αβ, e i_αβ⊥ define three orthogonal coordinate axes. The expression (3.160) determines the components of the u_0αβ voltage vector from the projections on these axes.

The different dual formulations allow instantaneous reactive power compensation to set different strategies. The decomposition of the power variables gets these terms of power with one or other compensation objectives. The dual original instantaneous reactive power formulation is applied to the series constant power compensation.

The objective of the constant power strategy is that the source supplies a constant instantaneous power. The instantaneous powers p₀(t), p_αβ(t), y q_αβ(t) defined in (3.156), can be divided by a constant term and a variable term over time. The first is obtained as the integral average of the instantaneous value; the second term will have an average value of zero (on this we discussed in Subsection 3.2.3). Thus, the instantaneous power p_αβ(t) can be expressed in the form

$p_{α β} (t) = P_{α β} + {\tilde{p}}_{α β} (t)$

(3.161)

where P_αβ is the average value and

{\tilde{p}}_{α β} (t)

is the variable term.

The zero-sequence instantaneous power may also decompose in a similar way, that is,

$p_{0} (t) = P_{0} + {\tilde{p}}_{0} (t)$

(3.162)

where, P₀ is the constant term and

{\tilde{p}}_{0} (t)

is the variable term.

Similarly, the instantaneous imaginary power is decomposed into a constant term Q_αβ and other variable

{\tilde{q}}_{α β} (t)

, that is,

$q_{α β} (t) = Q_{α β} + {\tilde{q}}_{α β} (t)$

(3.163)

Thus, for an arbitrary load their powers can be expressed in the form (3.164),

$\begin{array}{l} p_{L 0} (t) = P_{L 0} + {\tilde{p}}_{L 0} (t) \\ p_{L α β} (t) = P_{L α β} + {\tilde{p}}_{L α β} (t) \\ q_{L α β} (t) = Q_{L α β} + {\tilde{q}}_{L α β} (t) \end{array}$

(3.164)

The compensation objective is to set the system so as to supply a constant instantaneous real power and q_αβ(t) = 0. Therefore, references to voltage and current shown in Figure 3.32, the source power will be

$p_{S} (t) = - P_{L α β} - P_{L 0}$

(3.165)

Figure 3.32 Single-phase equivalent circuit for series compensation where voltage and current references are indicated.

So that the series compensation equipment must supply the rest of the powers of (3.164), that is,

$\begin{array}{l} p_{C 0} (t) = - {\tilde{p}}_{L 0} (t) \\ p_{C α β} (t) = - {\tilde{p}}_{L α β} (t) \\ q_{C L α β} (t) = - Q_{L α β} - {\tilde{q}}_{L α β} (t) = - q_{L α β} (t) \end{array}$

(3.166)

In matrix form, (3.166) is

$[\begin{array}{c} p_{C 0} (t) \\ p_{C α β} (t) \\ q_{C α β} (t) \end{array}] = [\begin{array}{c} - {\tilde{p}}_{L 0} (t) \\ - {\tilde{p}}_{L α β} (t) \\ - q_{L α β} (t) \end{array}]$

(3.167)

In this, expression (3.159) is applied, allowing determination of the vector of voltages, u_C0αβ the compensator must generate in 0αβ coordinates, that is,

$u_{C 0 α β} = [\begin{array}{c} u_{C 0} \\ u_{C α} \\ u_{C β} \end{array}] = [\begin{array}{c} i_{0}^{T} / i_{0}^{2} \\ i_{α β}^{T} / i_{α β}^{2} \\ i_{α β ⊥}^{T} / i_{α β}^{2} \end{array}] [\begin{array}{c} - {\tilde{p}}_{L 0} (t) \\ - {\tilde{p}}_{L α β} (t) \\ - q_{L α β} (t) \end{array}]$

(3.168)

From (3.30) vector components of voltages in phase coordinates are obtained. It is applied to become (3.169),

$u_{C} = [\begin{array}{c} u_{C 1} \\ u_{C 2} \\ u_{C 3} \end{array}] = \sqrt{\frac{3}{2}} [\begin{array}{c} \frac{1}{\sqrt{2}} & 1 & 0 \\ \frac{1}{\sqrt{2}} & - \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{1}{\sqrt{2}} & - \frac{1}{2} & - \frac{\sqrt{3}}{2} \end{array}] [\begin{array}{c} u_{C 0} \\ u_{C α} \\ u_{C β} \end{array}]$

(3.169)

3.5.2. Dual Modified Instantaneous Reactive Power Formulation

The modified formulation begins with the 123 phase coordinate transformation to 0αβ coordinates given by (3.1)–(3.2). Thus voltage vector (u) and current (i), and power variables defined in (3.79) and (3.88) are obtained. However, this section will have for convenience instantaneous reactive power in the form given in (3.170),

$[\begin{array}{c} q_{0} \\ q_{α} \\ q_{β} \end{array}] = [\begin{array}{c} 0 & i_{β} & - i_{α} \\ - i_{β} & 0 & i_{0} \\ i_{α} & - i_{0} & 0 \end{array}] [\begin{array}{c} u_{0} \\ u_{α} \\ u_{β} \end{array}]$

(3.170)

To obtain the components of the voltage as a function of the four variables of power (instantaneous real power and three instantaneous reactive power) a dual reasoning set out in Section 3.3 shall be followed.

The square matrix (3.170) is of rank 2, whereby a row is a linear combination of the other two. Thus, by multiplying q₀ for i₀, q_α for i_α, and q_β for i_β is obtained (3.171),

$[\begin{array}{c} i_{0} q_{0} \\ i_{α} q_{α} \\ i_{β} q_{β} \end{array}] = [\begin{array}{c} 0 & i_{0} i_{β} & - i_{0} i_{α} \\ - i_{α} i_{β} & 0 & i_{0} i_{α} \\ i_{α} i_{β} & - i_{0} i_{β} & 0 \end{array}] [\begin{array}{c} v_{0} \\ v_{α} \\ v_{β} \end{array}]$

(3.171)

From (3.171) on verification

$0 = i_{0} q_{0} + i_{α} q_{α} + i_{β} q_{β}$

(3.172)

The different powers introduced here can be grouped in matrix form, this is

$[\begin{array}{c} p \\ q_{0} \\ q_{α} \\ q_{β} \end{array}] = [\begin{array}{c} i_{0} & i_{α} & i_{β} \\ 0 & i_{β} & - i_{α} \\ - i_{β} & 0 & i_{0} \\ i_{α} & - i_{0} & 0 \end{array}] [\begin{array}{c} u_{0} \\ u_{α} \\ u_{β} \end{array}]$

(3.173)

From (3.173) it can always find the inverse transformation of dual form as proposed in Section 3.3.1; so you get to the expression (3.174),

$[\begin{array}{c} u_{0} \\ u_{α} \\ u_{β} \end{array}] = \frac{1}{i_{0 α β}^{2}} [\begin{array}{c} i_{0} & 0 & - i_{β} & i_{α} \\ i_{α} & i_{β} & 0 & - i_{0} \\ i_{β} & - i_{α} & i_{0} & 0 \end{array}] [\begin{array}{c} p \\ q_{0} \\ q_{α} \\ q_{β} \end{array}]$

(3.174)

where

i_{0 α β}^{2} = i_{0}^{2} + i_{α}^{2} + i_{β}^{2}

. The relation (3.174) gives the vector voltages depending on the variables of power, with the only restriction that is verified

i_{0 α β}^{2} \neq 0

Then the dual modified formulation for nonlinear loads compensation by series active filter is applied; namely the constant power strategy will be discussed.

According to the formulation of modified instantaneous reactive power, for any load it is possible to define an instantaneous power (p_L) and a vector of instantaneous reactive power (q_L), whose components are q_L0, q_Lα, q_Lβ. As in the original formulation, instantaneous active power can be divided into a constant term (P_L) given by the average value of p_L and a variable term

{\tilde{p}}_{L}

, that is

$p_{L} = P_{L} + {\tilde{p}}_{L}$

(3.175)

Compensation objectives to constant power are established:

• The source instantaneous reactive power vector must be zero, that is q_S = 0. Therefore for this goal to be met, the instantaneous reactive power vector (q_C) the compensator should be

$q_{C} = - q_{L}$

(3.176)

that is, the compensator must transfer the instantaneous reactive power of the load.

• The source must transfer a constant instantaneous active power (p_S). Considering that the average power transferred by the compensator is zero, the following applies

$p_{S} = - P_{L}$

(3.177)

then attending to this criterion, the compensation equipment must provide an instantaneous power (p_C) given by (3.177),

$p_{C} = - {\tilde{p}}_{L}$

(3.178)

whose average value is zero.

Accordingly, the compensator transfer power space vector (3.179),

$[\begin{array}{c} p_{C} \\ q_{C_{0}} \\ q_{C_{α}} \\ q_{C_{β}} \end{array}] = [\begin{array}{c} - {\tilde{p}}_{L} \\ - q_{L_{0}} \\ - q_{L_{α}} \\ - q_{L_{β}} \end{array}]$

(3.179)

After obtaining the components of current vector in 0–α–β coordinates, the vector of voltages that must generate the compensation equipment in the same coordinates is given by (3.180),

$[\begin{array}{c} u_{C 0} \\ u_{C α} \\ u_{C β} \end{array}] = \frac{1}{i_{0 α β}^{2}} [\begin{array}{c} i_{0} & 0 & - i_{β} & i_{α} \\ i_{α} & i_{β} & 0 & - i_{0} \\ i_{β} & - i_{α} & i_{0} & 0 \end{array}] [\begin{array}{c} - {\tilde{p}}_{L} \\ - q_{L_{0}} \\ - q_{L_{α}} \\ - q_{L_{β}} \end{array}]$

(3.180)

The compensation voltages in phase coordinates are obtained from the application of the inverse coordinates transformation (3.180).

This strategy is now applied to a practical case.

Example 3.8

Figure 3.33 shows a nonlinear load compensated by a series active filter, which is considered ideal for the purposes of this exercise. The active filter applies the strategy of constant power control based on the formulation of the dual instantaneous reactive power. This strategy is applied to a load of HVS type, that is, an uncontrolled three-phase rectifier with a 2200 μF capacitor in parallel with a 50/3 Ω resistor in the dc side. This is a three-wire nonlinear balanced load. As in previous examples, the system is simulated in Matlab–Simulink environment using Simulink SymPowerSystem toolbox.

The block diagram for calculating the compensation reference voltage is shown in Figure 3.34. The voltage vector components in the load side and current of source in coordinates are obtained 0αβ. The matrix of currents is constructed and multiplied by the vector of voltages u_0αβ. The result of this product provides the vector of instantaneous reactive power and instantaneous real power. The latter is decomposed into a variable term and constant term through a low-pass filter (LPF). The vector of compensation powers is formed and multiplied by the matrix currents defined in (3.174), which compensates the voltages vector in 0αβ coordinates are obtained as defined in (3.180). The transformation to phase coordinates enables the components of the compensation voltage to be obtained.

Figure 3.34 Block diagram for calculating the voltage reference constant power compensation for a series active filter when the formulation of the dual modified instantaneous reactive power is applied.

When a series active filter is connected (set at t = 0.08 s) to a load with this compensation strategy, the waveforms of the voltage at the point of common coupling, PCC, before and after compensation shown in Figure 3.35a are obtained. Figure 3.35b shows the voltage of phase 1 in the PCC and the load current before and after compensation. The waveforms presented correspond to the sums of compensation voltage reference and load voltage considering an ideal active filter; in Chapter 5 the results of the operation with series active power filter are analyzed.

Figure 3.35 (a) Voltages in the PCC before and after compensation to constant power compensation with series active filter, (b) voltage of phase 1 in the PCC and current of phase 1 (multiplied by 5) for a HVS load before and after compensation.

Finally, Figure 3.36 shows the waveforms of power variables in the PCC. Figure 3.36a shows the waveform of the instantaneous real power before and after compensation; the instantaneous power is constant after compensation. Figure 3.36b shows the instantaneous reactive power (3.108) before and after compensation; instantaneous reactive power associated with each coordinate axis is zero after compensation. Figure 3.36 verifies the compensation objectives originally specified.

Figure 3.36 (a) Instantaneous real power in the PCC before and after constant power compensation with series active filter, (b) instantaneous reactive power at the PCC before and after constant power compensation.

3.6. Summary

This chapter has described the theory of instantaneous reactive power for three/four-wire three-phase systems in the most general terms of asymmetry and distortion. The theory of instantaneous reactive power today corresponds to a set of formulations with the same common base, that is, the partition of current in components of only instantaneous real power transfer, and one or more components that do not transfer instantaneous real power. Thus began stating the basis for the formulation of Akagi–Nabae [1,2], whose original formulation was designated as the first in the world to introduce the concept of instantaneous reactive power in three-phase systems. He followed this with an analysis of the formulation’s greatest impact, which was subsequently published as Refs [6–30]. In particular, a comprehensive analysis of the so-called modified instantaneous reactive power formulation, and formulations that require a rotating framework of reference, dq formulation, i_d–i_q method, and pqr formulation was made. From the presentation made in this chapter, one can deduce that the formulation of modified instantaneous reactive power or using rotating reference frames from the transformation of Park, essentially have the same pattern as the original formulation: a mathematical transformation that enables finding three independent power variables from which it is possible to explain the flow of supply source and load. Over the years there have been other formulations that have had less impact and can be found in references [31–33]. It has also paid special attention to the developments of the original and modified formulations in phase coordinates allowing a description of the theory and its subsequent application in load compensation more simply.

The formulation of the instantaneous reactive power arose from finding compensating control strategy-based power converters (active power filters), so this chapter has presented the basis for obtaining the currents of reference in designing a shunt APLC with and without energy storage elements. In that order of things, the theory of instantaneous reactive power can also be applied in the design of series active compensators control. One section was dedicated to the dual formulation of the original instantaneous reactive power and the modified formulation and its application in load compensation of HVS type using series active filter. Several application examples have been included throughout the chapter.

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Table of Contents for 3.4. Other Approaches: Synchronous Frames

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3.4. Other Approaches: Synchronous Frames

3.4.1. dq Formulation

3.4.2. id–iq Method

3.4.3. p–q–r Formulation

3.5. Dual Instantaneous Reactive Power

3.5.1. Dual Original Instantaneous Reactive Power Formulation

3.5.2. Dual Modified Instantaneous Reactive Power Formulation

3.6. Summary

Table of Contents for
3.4. Other Approaches: Synchronous Frames

3.4.2. i_d–i_q Method