9.8.1 Conventional Compensation Devices

The conventional approaches for voltage regulation and power compensation at fundamental frequency are relatively simple and inexpensive. They include:

• shunt compensators (capacitors, reactors),

• series compensators (series capacitors), and

• phase shifters.

These devices are mechanically controlled at low speed and there are wear-and-tear problems resulting in greater operating margins and redundancies. If conventional compensation devices are employed, the system is not fully controlled and optimized from a dynamic and steady-state point of view.

Conventional devices used for power quality improvement are passive and active power filters, as discussed in Sections 9.1 to 9.7. Passive filters are simple and inexpensive. However, they have fixed (tuned) frequencies and will not perform properly under changing system configurations and/or variable (nonlinear) load conditions. Active filters eliminate harmonics at the point of installation without considering the power quality status of the entire system. In addition, they do not solve the stability problem, and are not generally capable of optimizing the performance (e.g., voltage regulation, reactive power compensation) of power systems.

9.8.2 Flexible AC Transmission Systems (FACTS)

In order to meet stringent specifications and the established contractual obligations set out in supply agreements, various technologies have been used. Among the newly developed technical solutions are FACTS, which were introduced by GyuGyi [18,19]. They offer a variety of options including series and shunt compensation of power at fundamental frequency to achieve instantaneous and individual or simultaneous control of V, P, and Q. FACTS devices are designed to improve the reliability and the quality of power transmission. However, they do not consider the quality of power that is delivered to customers. This section briefly introduces different types of FACTS controllers [19].

9.8.3 Custom Power Devices

The introduction of power electronic loads has raised much concern about power quality problems caused by harmonics, distortions, interruptions, and surges. Equipment such as large industrial drives (e.g., cycloconverters) generate significantly high voltage and current (inter-, sub-) harmonics and create extensive voltage fluctuation. This is more evident in power systems that make use of long transmission lines with limited reactive capacities.

The application of harmonic filters and SVCs to radial transmission systems can offer partial solution to high THD levels and voltage fluctuations. Yet, the lack of dynamic capabilities of these devices limits them to bulk correction. In addition, they might be effective in one application but fail to correct other power quality issues.

The concept of custom power was introduced by Hingorani [20] as the solution to V, P, and Q compensation and power quality problems at the expense of high cost and network complexity. As FACTS controllers improve the reliability and quality of power transmission – by simultaneously enhancing both power transfer capacity and stability – custom power devices enhance the quality and reliability of power delivered to the customer. With a custom power device, a customer (e.g., a sensitive load) will be able to receive a prespecified quality of electric power with a combination of specifications including but not limited to:

• magnitude and duration of over- and under-voltages with specified limits,

• low harmonic distortion in the supply, load voltages, and currents,

• small phase imbalance,

• low flicker in the supply voltage,

• control of power interruptions, and

• control of supply voltage frequency within specified limits.

Custom power devices are classified based on their power electronic controllers, which can be either of the network-reconfiguration type or of the compensation type. The network-reconfiguration devices –also called switchgear – include the solid-state and/or static versions of current limiting, current breaking, and current transferring components. The compensation-type custom power devices either compensate a load (e.g., correct its power factor, imbalance) or improve the quality of the supply voltage (e.g., eliminate its harmonics). They are either connected in shunt or in series or a combination of both. Custom power devices are classified as follows [21]:

• Network-reconfiguration custom power devices include:

• Solid-state current limiter (SSCL),

• Solid-state breaker (SSB), and

• Solid-state transfer switch (SSTS).

• Compensation-custom power devices include:

• Distribution STATCOM (DSTATCOM),

• Dynamic voltage restorer/regulator (DVR), and

• UPQC

Custom power devices are designed to improve the quality of power at their point of installation of the power distribution systems. They are not primarily designed to improve the power quality of the entire system.

9.8.4 Active Power Line Conditioner (APLC)

The APLC is a converter-based compensation device designed to improve the power quality of the entire distribution system by injecting corrective harmonic currents at selected (sensitive) buses. It is usually necessary to use more than one APLC unit to improve the power quality of the entire distribution system. Therefore, APLC units can be considered as a group of shunt active filters; their placement, sizing, and compensation levels (e.g., orders, magnitudes, and phases of injected current harmonics) are optimally designed to improve the power quality of the entire distribution system. The number of required APLC units depends on the severity of distortion, the nature of the distribution system, and types of nonlinear loads, as well as the required quality of electric power (e.g., as specified by IEEE-519). At present, the design of APLCs does not consider transient distortions and stability issues.

9.8.5 Remarks Regarding Compensation Devices

In the absence of saturation and nonlinearity, power system voltage and current waveforms are sinusoidal and FACTS controllers are employed to solve the associated steady-state, transient, and stability problems. Custom power devices are designed to improve the steady-state and transient problems of distorted distribution systems at the point of installation. Power filters and APLCs mitigate power quality problems at a given location and throughout the power system, respectively. However, at present, none of them are designed to simultaneously consider stability and power quality issues. This is highly related to the fact that in conventional centralized power systems, power quality problems are limited to the low- and medium-voltage distribution systems, whereas stability issues are mainly occurring in high-voltage transmission systems. With the introduction of distributed generation (DG) and the fast growing rate of renewable energy sources in today’s power system, there will be an overlap between stability and power quality issues. More attention and further research is required to analyze and examine stability issues in decentralized distorted networks, as well as isolated and DG power systems.

9.9.1 UPQC Structure

The detailed structure of the right-shunt UPQC is shown in Fig. 9.31. Assuming a three-phase, four-wire configuration with an unbalanced and distorted system voltage (v_s) and nonsinusoidal load current (i_L), the UPQC shall perform the following functions:

• convert the feeder (system) current i_s to balanced sinusoids through the shunt compensator,

• convert the load voltage v_L to balanced sinusoids through the series compensator,

• ensure zero real power injection (and/or absorption) by the compensators, and

• supply reactive power to the load (Q compensation).

UPQC employs two converters that are connected to a common DC link with an energy storage capacitor. The main components of a UPQC are series and shunt power converters, DC capacitors, low-pass and high-pass passive filters, and series and shunt transformers:

• Series converter is a voltage-source converter connected in series with the AC line and acts as a voltage source to mitigate voltage distortions. It eliminates supply voltage flickers or imbalance from the load terminal voltage and forces the shunt branch to absorb current harmonics generated by the nonlinear load. Control of the series converter output voltage is usually performed using pulse-width modulation (PWM). The gate pulses are generated by the comparison of a fundamental voltage reference signal with a high-frequency triangular waveform.

• Shunt converter is a voltage-source converter connected in shunt with the same AC line and acts as a current source to cancel current distortions, to compensate reactive current of the load, and to improve the power factor. It also performs the DC-link voltage regulation, resulting in a significant reduction of the DC capacitor rating. The output current of the shunt converter is adjusted (e.g., using a dynamic hysteresis band) by controlling the status of semiconductor switches such that output current follows the reference signal and remains in a predetermined hysteresis band.

• Midpoint-to-ground DC capacitor bank is divided into two groups, which are connected in series. The neutrals of the secondary transformers are directly connected to the DC link midpoint. As the connection of both three-phase transformers is Y/Y₀, the zero-sequence voltage appears in the primary winding of the series-connected transformer in order to compensate for the zero-sequence voltage of the supply system. No zero-sequence current flows in the primary side of both transformers. It ensures the system current to be balanced even when the voltage disturbance occurs.

• Low-pass filter is used to attenuate high-frequency components at the output of the series converter that are generated by high-frequency switching.

• High-pass filter is installed at the output of shunt converter to absorb current switching ripples.

• Series and shunt transformers are implemented to inject the compensation voltages and currents, and for the purpose of electrical isolation of UPQC converters.

The UPQC is capable of steady-state and dynamic series and/or shunt active and reactive power compensations at fundamental and harmonic frequencies. However, the UPQC is only concerned about the quality of the load voltage and the line current at the point of its installation, and it does not improve the power quality of the entire system.

9.9.2 Operation of the UPQC

The shunt converter forces the feeder (system) current to become a balanced (harmonic-free) sinusoidal waveform, while the series converter ensures a balanced, sinusoidal, and regulated load voltage.

9.10.1 Pattern of Reference Signals

Provided unbalanced and distorted source voltages and nonlinear load currents exist, then – based on Eqs. 9-23 to 9-39 – the output voltage of the series converter (v_c(t), Fig. 9.31) and the output current of shunt converter (i_c(t), Fig. 9.31) must obey the following equations:

$\left\{\begin{array}{l}\begin{array}{l}{v}_c\left(t\right)=\left(V_L-V_{s+}\right)sin\left(\omega t+{\phi }_{+}\right)-v_{s-}\left(t\right)-v_{s0}\left(t\right)-{\displaystyle \sum v_{sh}\left(t\right)}\hfill \end{array}\hfill \\ \begin{array}{l}{i}_c\left(t\right)=I_{L+}cos\left(\omega t+{\delta }_{+}\right)sin{\theta }_L+i_{L-}\left(t\right)+i_{L0}\left(t\right)+{\displaystyle \sum i_{Lh}\left(t\right)\text{,}}\end{array}\hfill \end{array}\right.$

so that the current drawn from the source and the voltage appearing across the load will be sinusoidal.

Generation, tracking, and observation of these reference signals are performed by the UPQC control system. There are two main classes of UPQC control: the first one is based on the Park transformation [32], and the second one on the instantaneous real and imaginary power theory [26,30,31]. These will be introduced in the following sections. There are other control techniques for active and hybrid (e.g., UPQC) filters as discussed in Section 9.7.

9.11.1 General Theory of the Park (dq0) Transformation

In the study of power systems, mathematical transformations are often used to decouple variables, to facilitate the solution of time-varying equations, and to refer variables to a common reference frame. A widely used transformation is the Park (dq0) transformation [32]. It is can be applied to any arbitrary (sinusoidal or distorted) three-phase, time-dependent signals.

The Park (dq0) transformation is defined as

$\begin{array}{l}\left[\begin{array}{c}\hfill x_d\hfill \\ \hfill x_q\hfill \\ \hfill x_0\hfill \end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\hfill cos{\theta }_d\hfill & \hfill cos\left({\theta }_d-2\pi /3\right)\hfill & \hfill cos\left({\theta }_d+2\pi /3\right)\hfill \\ \hfill -sin{\theta }_d\hfill & \hfill -sin\left({\theta }_d-2\pi /3\right)\hfill & \hfill -sin\left({\theta }_d+2\pi /3\right)\hfill \\ \hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill x_a\hfill \\ \hfill x_b\hfill \\ \hfill x_c\hfill \end{array}\right]\text{.}\end{array}$

In this equation, θ_d = ω_d t + ϕ, where ω_d is the angular velocity of signals to be transformed and ϕ is the initial angle.

As discussed in Section 9.7, the main advantage of the dq0 transformation is that the fundamental (positive-sequence) component of waveform will be transferred to a constant DC term under steady-state conditions. In addition, a harmonic component with frequency ω_h will appear as a sinusoidal signal with frequency of ω_h – ω_d in the dq0 domain.

9.11.2 Control of Series Converter Based on the dq0 Transformation

The system (supply) voltage may contain negative-and zero-sequence components as well as harmonic distortions that need to be eliminated by the series compensator. The control of the series compensator based on the dq0 transformation is shown in Fig. 9.32.

The system voltages are first detected and then transformed into the synchronous dq0 reference frame. If the system voltages are unbalanced and contain harmonics, then the Park transformation (Eq. 9-41) results in

$\begin{array}{l}\left[\begin{array}{l}{v}_d\\ v_q\\ v_0\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\hfill cos{\theta }_d\hfill & \hfill cos\left({\theta }_d-2\pi /3\right)\hfill & \hfill cos\left({\theta }_d+2\pi /3\right)\hfill \\ \hfill -sin{\theta }_d\hfill & \hfill -sin\left({\theta }_d-2\pi /3\right)\hfill & \hfill -sin\left({\theta }_d+2\pi /3\right)\hfill \\ \hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill \end{array}\right]\left[\begin{array}{l}{v}_a\\ v_b\\ v_c\end{array}\right]\\ =\sqrt{\frac{3}{2}}\left\{\left[\begin{array}{c}\hfill V_{sp}cos{\varphi }_p\hfill \\ \hfill V_{sp}sin{\varphi }_p\hfill \\ \hfill 0\hfill \end{array}\right]\right.+\left[\begin{array}{c}\hfill V_{sn}cos\left(2\omega t+{\varphi }_n\right)\hfill \\ \hfill -V_{sn}sin\left(2\omega t+{\varphi }_n\right)\hfill \\ \hfill 0\hfill \end{array}\right]+\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill V_{S0}cos\left(2\omega t+{\varphi }_0\right)\hfill \end{array}\right]\\ \left.+\left[\begin{array}{c}\hfill {\displaystyle \sum V_{sh}cos\left[\left(h-1\right)\left(2\omega t+{\varphi }_h\right)\right]}\hfill \\ \hfill {\displaystyle \sum V_{sh}sin\left[\left(\mathrm{h}-1\right)\left(2\omega t+{\varphi }_h\right)\right]}\hfill \\ \hfill 0\hfill \end{array}\right]\right\}=\left[\begin{array}{c}\hfill V_{dp}\hfill \\ \hfill V_{qp}\hfill \\ \hfill 0\hfill \end{array}\right]+\left[\begin{array}{c}\hfill V_{dn}\hfill \\ \hfill V_{qn}\hfill \\ \hfill 0\hfill \end{array}\right]+\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill V_{00}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill V_{dh}\hfill \\ \hfill V_{qh}\hfill \\ \hfill 0\hfill \end{array}\right]\end{array}$

where ϕ_p is the phase difference between the positive-sequence component and the reference voltage. Based on Eq. 9-42, the fundamental positive-sequence component of the voltage is represented by DC terms (e.g., V_sp cos ϕ_p and V_sp sin ϕ_p) in the dq0 reference frame.

The load bus voltage should be maintained sinusoidal with constant amplitude even if the source voltage is distorted. Therefore, the expected load voltage in the dq0 domain will only consist of one value:

$V_{dq0}^{\mathrm{expected}}=\left[\begin{array}{c}\hfill V_{dF}\hfill \\ \hfill V_{qF}\hfill \\ \hfill V_{0F}\hfill \end{array}\right]=\left[T_{dq0}\left({\theta }_d\right)\right]V_{abc}^{\mathrm{expected}}=\left[\begin{array}{c}\hfill V_m\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]\text{,}$

where

$V_{abc}^{\mathrm{expected}}=\left[\begin{array}{l}{V}_{S}cos\left(\omega t\right)\\ V_{S}cos\left(\omega t+120\right)\\ V_{S}cos\left(\omega t-120\right)\end{array}\right]$

are sinusoids and the compensation reference voltage is

$V_{\mathrm{h}\mathrm{d}\mathrm{q}0}^{\mathrm{r}\mathrm{e}\mathrm{f}}=V_{dq0}^{\mathrm{expected}}-V_{\mathrm{s}\mathrm{d}\mathrm{q}0}\text{,}$

where V_sdq0 is the distorted source voltage in the dq0 frame (Eq. 9-42). Therefore, V_dp of Eq. 9-42 should be maintained at V_m and all other terms must be eliminated by the compensation voltage.

The compensation reference voltage of Eq. 9-44 is then inversely transformed into the abc reference frame. Comparing the compensation reference voltage with a triangular wave, the output voltage of the series converter can be obtained by a PWM voltage control unit.

9.11.3 Control of Shunt Converter Relying on the dq0 Transformation

Similar transformation as employed for voltages can be applied to current waveforms as well:

$\begin{array}{l}\left[\begin{array}{c}\hfill i_d\hfill \\ \hfill i_q\hfill \\ \hfill i_0\hfill \end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\hfill cos{\theta }_d\hfill & \hfill cos\left({\theta }_d-2\pi /3\right)\hfill & \hfill cos\left({\theta }_d+2\pi /3\right)\hfill \\ \hfill -sin{\theta }_d\hfill & \hfill -sin\left({\theta }_d-2\pi /3\right)\hfill & \hfill -sin\left({\theta }_d+2\pi /3\right)\hfill \\ \hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill i_a\hfill \\ \hfill i_b\hfill \\ \hfill i_c\hfill \end{array}\right],\\ =\sqrt{\frac{3}{2}}\left\{\left[\begin{array}{c}\hfill I_{LP}cos{\theta }_p\hfill \\ \hfill I_{Lsp}sin{\theta }_p\hfill \\ \hfill 0\hfill \end{array}\right]\right.+\left[\begin{array}{c}\hfill I_{Ln}cos\left(2\omega t+{\theta }_n\right)\hfill \\ \hfill -I_{Ln}sin\left(2\omega t+{\theta }_n\right)\hfill \\ \hfill 0\hfill \end{array}\right]\left.+\left[\begin{array}{c}\hfill {\displaystyle \sum I_{Lh}cos\left[\left(h-1\right)\left(2\omega t+{\theta }_h\right)\right]}\hfill \\ \hfill {\displaystyle \sum I_{Lh}sin\left[\left(\mathrm{h}-1\right)\left(2\omega t+{\theta }_h\right)\right]}\hfill \\ \hfill 0\hfill \end{array}\right]\right\}\text{.}\end{array}$

Unlike the voltage waveforms, which are maintained at their rated values, the load currents will change with the load characteristics at the PCC and it will not be possible to have a fixed expected reference current value. To solve this problem, different techniques are proposed and implemented to extract the fundamental positive-sequence component (which becomes a DC term in the dq0 reference frame) and the corresponding harmonic components, as discussed in Sections 9.7.1 to 9.7.3.

Therefore, the expected current in the dq0 reference frame has only one value

$i_{dq0}^{\mathrm{expected}}=\left[\begin{array}{c}\hfill I_{Lp}cos{\theta }_p\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]\text{.}$

The compensation reference current is

$i_{\mathrm{h}\mathrm{d}\mathrm{q}0}^{\mathrm{r}\mathrm{e}\mathrm{f}}=i_{dq0}^{\mathrm{expected}}-i_{\mathrm{s}\mathrm{d}\mathrm{q}0}\text{,}$

where i_sdq0 is the distorted load current in the dq0 frame (Eq. 9-45).

The shunt converter operates as a controlled current source and different techniques (e.g., hysteresis, predictive, and suboscillation control) are employed to ensure that its output current follows the corresponding reference signal i_hdq0^ref.

9.11.4 Control of DC Link Voltage using the dq0 Transformation

In the UPQC configuration, the DC link voltage is usually divided into two capacitor units C₁ and C₂ connected in series. The voltages of these capacitors are such that V_C1 = V_C2 under balanced operation. Normally, the DC link voltage is maintained at a desired/nominal level for all operating conditions.

9.12.1 Theory of Instantaneous Real and Imaginary Power

The αβ0 transformation applied to abc three-phase voltages in a four-wire system is

$\left[\begin{array}{c}\hfill v_0\hfill \\ \hfill v_{\alpha }\hfill \\ \hfill v_{\beta }\hfill \end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill & \hfill 1/\sqrt{2}\hfill \\ \hfill 1\hfill & \hfill -1/2\hfill & \hfill -1/2\hfill \\ \hfill 0\hfill & \hfill \sqrt{3/2}\hfill & \hfill -\sqrt{3/2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill v_a\hfill \\ \hfill v_b\hfill \\ \hfill v_c\hfill \end{array}\right]\text{.}$

The inverse αβ0 transformation is

$\left[\begin{array}{c}\hfill v_a\hfill \\ \hfill v_b\hfill \\ \hfill v_c\hfill \end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\hfill 1/\sqrt{2}\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1/\sqrt{2}\hfill & \hfill -1/2\hfill & \hfill \sqrt{3/2}\hfill \\ \hfill 1/\sqrt{2}\hfill & \hfill -1/2\hfill & \hfill -\sqrt{3/2}\hfill \end{array}\right]\left[\begin{array}{l}{v}_0\\ v_{\alpha }\\ v_{\beta }\end{array}\right]\text{.}$

Identical relations hold for the currents.

The instantaneous three-phase active power is given by [31]

$\begin{array}{c}{p}_{\mathrm{three}-\mathrm{phase}}\left(t\right)=v_a{i}_a+v_b{i}_b+v_c{i}_c=p_a\left(t\right)+p_b\left(t\right)+p_c\left(t\right)\\ =v_{\alpha }{i}_{\alpha }+v_{\beta }{i}_{\beta }+v_0{i}_0=p_{\alpha }\left(t\right)+p_{\beta }\left(t\right)+p_0\left(t\right)\\ =p\left(t\right)+p_0\left(t\right)\text{,}\end{array}$

where p(t) = p_α(t) + p_β(t) = v_αi_α + v_βi_β corresponds to the instantaneous three-phase real power and p₀(t) = v₀i₀ is the instantaneous zero-sequence power. Note the separation of the zero-sequence component in the αβ0 system.

The instantaneous three-phase reactive power is defined as

$\begin{array}{c}{q}_{\mathrm{three}-\mathrm{phase}}\left(t\right)=q_{\alpha }\left(t\right)+q_{\beta }\left(t\right)=v_{\alpha }{i}_{\beta }-v_{\beta }{i}_{\alpha }\\ =-\left[\left(v_a-v_b\right)i_c+\left(v_b-v_c\right)i_a+\left(v_c-v_a\right)i_b\right]/\sqrt{3}\text{.}\end{array}$

Therefore, the instantaneous real, reactive, and zero-sequence powers are calculated from

$\left[\begin{array}{c}\hfill p_0\hfill \\ \hfill p\hfill \\ \hfill q\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill v_0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill v_{\alpha }\hfill & \hfill v_{\beta }\hfill \\ \hfill 0\hfill & \hfill -v_{\beta }\hfill & \hfill v_{\alpha }\hfill \end{array}\right]\left[\begin{array}{c}\hfill i_0\hfill \\ \hfill i_{\alpha }\hfill \\ \hfill i_{\beta }\hfill \end{array}\right]\text{.}$

Here, p and q (in the αβ0 space) are not expressed in the conventional watts and VArs.

From Eq. 9-52, the α and β components of currents may be calculated as

$\left[\begin{array}{c}\hfill i_{\alpha }\hfill \\ \hfill i_{\beta }\hfill \end{array}\right]=\frac{1}{\mathrm{\Delta }}\left[\begin{array}{cc}\hfill v_{\alpha }\hfill & \hfill -v_{\beta }\hfill \\ \hfill v_{\beta }\hfill & \hfill v_{\alpha }\hfill \end{array}\right]\left[\begin{array}{c}\hfill p\hfill \\ \hfill q\hfill \end{array}\right]\mathrm{with}\mathrm{\Delta }=v_{\alpha }^2+v_{\beta }^2\text{.}$

These are divided into two types of currents

$\begin{array}{c}\left[\begin{array}{c}\hfill i_{\alpha }\hfill \\ \hfill i_{\beta }\hfill \end{array}\right]=\frac{1}{\mathrm{\Delta }}\left\{\left[\begin{array}{cc}\hfill v_{\alpha }\hfill & \hfill -v_{\beta }\hfill \\ \hfill v_{\beta }\hfill & \hfill v_{\alpha }\hfill \end{array}\right]\left[\begin{array}{c}\hfill p\hfill \\ \hfill 0\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill v_{\alpha }\hfill & \hfill -v_{\beta }\hfill \\ \hfill v_{\beta }\hfill & \hfill v_{\alpha }\hfill \end{array}\right]\left[\begin{array}{c}\hfill 0\hfill \\ \hfill q\hfill \end{array}\right]\right\}=\left[\begin{array}{c}\hfill i_{\alpha p}\hfill \\ \hfill i_{\beta p}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill i_{\alpha q}\hfill \\ \hfill i_{\beta q}\hfill \end{array}\right]\text{,}\end{array}$

where

i_αp = v_αp/Δ is the α-axis instantaneous active current,

i_αq = –v_βq/Δ is the α-axis instantaneous reactive current,

i_βp = v_βp/Δ is the β-axis instantaneous active current, and

i_βq = v_αq/Δ is the β-axis instantaneous reactive current.

Now, the power components of the phases α and β can be separated as

$\left[\begin{array}{c}\hfill p_{\alpha }\hfill \\ \hfill p_{\beta }\hfill \end{array}\right]=\left[\begin{array}{c}\hfill v_{\alpha }{i}_{\alpha }\hfill \\ \hfill v_{\beta }{i}_{\beta }\hfill \end{array}\right]=\left[\begin{array}{c}\hfill v_{\alpha }{i}_{\alpha p}\hfill \\ \hfill v_{\beta }{i}_{\beta p}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill v_{\alpha }{i}_{\alpha q}\hfill \\ \hfill v_{\beta }{i}_{\beta q}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill p_{\alpha p}\hfill \\ \hfill p_{\beta p}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill p_{\alpha q}\hfill \\ \hfill p_{\beta q}\hfill \end{array}\right]\text{,}$

where

$\begin{array}{c}{p}_{\alpha p}=v_{\alpha }{i}_{\alpha p}=v_{\alpha }^{2}p/\mathrm{\Delta },\\ p_{\alpha q}=v_{\alpha }{i}_{\alpha q}=-v_{\alpha }{v}_{\beta }q/\mathrm{\Delta },\\ p_{\beta p}=v_{\beta }{i}_{\beta p}=v_{\beta }^{2}p/\mathrm{\Delta },\\ p_{\beta q}=v_{\beta }{i}_{\alpha q}=v_{\alpha }{v}_{\beta }q/\mathrm{\Delta }\text{,}\end{array}$

because p_αq + p_βq = 0, and thus p_{three–phase} = p_α + p_β + p_o = p_αp + p_βp + p₀.

All these instantaneous quantities are valid under steady-state, transient, and nonsinusoidal conditions of voltage and/or current waveforms. Furthermore,

• The conventional concept of reactive power corresponds to the peak value of p(t) that has a zero average value. However, the new concept of instantaneous reactive power in the αβ0 space corresponds to the parts of p(t) that are dependent on the instantaneous imaginary power q, which independently exist in each phase but vanish when added (e.g., p_αq + p_βq = 0). If these powers (p_αq and p_βq) are transformed back to the original abc system, they add up to zero.

• The instantaneous real power (p) represents the net energy per second being transferred from source to load and vice versa at any time. It depends only on the voltages and currents in phases α and β. It does not have any zero-sequence power components.

• The imaginary power (q) does not contribute to the total instantaneous flow of energy. It relates to a block of energy exchanged between the phases. For example, consider a balanced three-phase capacitor bank connected to a balanced three-phase voltage source. The energy stored in each capacitor changes with time W(t) = Cv²(t)/2. However, $v_a^2\left(t\right)+v_b^2\left(t\right)+v_c^2\left(t\right)$ is constant. Therefore,

$\left\{\begin{array}{c}{p}_{\mathrm{three}-\mathrm{phase}}=p+p_0=0\\ q=\overline{q}=-3V_{+}{I}_{+}sin\left({\varphi }_{+}-{\delta }_{+}\right)\\ =3V_{+}{I}_{+}=-Q_{\mathrm{three}-\mathrm{phase}}\text{.}\end{array}\right.$

Solution to Application Example 9.6

$\begin{array}{ccc}\hfill \left\{\begin{array}{l}{v}_{\alpha }=\sqrt{3}{V}_{S}sin\omega t\hfill \\ v_{\beta }=\sqrt{3}{V}_{S}cos\omega t\hfill \\ v_0=0\hfill \end{array}\right.\hfill & \hfill \mathrm{and}\hfill & \hfill \left\{\begin{array}{l}{i}_{\alpha }=\sqrt{3}{I}_{L}sin\left(\omega t-\mathrm{\varphi }\right)\hfill \\ i_{\beta }=\sqrt{3}{I}_{L}cos\left(\omega t-\mathrm{\varphi }\right).\hfill \\ i_0=0\hfill \end{array}\right.\hfill \end{array}$

Therefore, from Eq. 9-52

$\begin{array}{l}p=v_{\alpha }{i}_{\alpha }+v_{\beta }{i}_{\beta }=3V_s{I}_{L}cos\mathrm{\varphi }=P_{3-\mathrm{phase}}\hfill \\ q=v_{\alpha }{i}_{\beta }-v_{\beta }{i}_{\alpha }=3V_s{I}_{L}sin\mathrm{\varphi }=Q_{3-\mathrm{phase}}.\hfill \end{array}$

Equation E9.6-3 is equivalent to the traditional active and reactive power concept.

Solution to Application Example 9.7

$\left\{\begin{array}{l}{i}_{\alpha }={\displaystyle {\sum }_{h=1}^{\infty }\frac{2}{\sqrt{3}}{I}_{\mathit{hL}}sin\left(h\omega t-{\mathrm{\varphi }}_h\right)\left[1-cos\left(h120\right)\right]}\hfill \\ i_{\beta }={\displaystyle {\sum }_{h=1}^{\infty }2I_{\mathit{hL}}cos\left(h\omega t-{\mathrm{\varphi }}_h\right)sin\left(h120\right)}\hfill \\ i_0=\frac{1}{\sqrt{3}}\left(i_{\mathit{aL}}+i_{bL}+i_{\mathit{cL}}\right)={\displaystyle {\sum }_{h=1}^{\infty }\sqrt{6}{I}_{3\mathit{hL}}sin\left(3h\omega t-{\mathrm{\varphi }}_{3h}\right)\text{.}}\hfill \end{array}\right.$

It is interesting to note that only the triplen harmonics (e.g., h = 3k) are present in i₀. The power components are

$\left\{\begin{array}{c}p\left(t\right)=v_{\alpha }{i}_{\alpha }+v_{\beta }{i}_{\beta }=\overline{p}+\tilde{p}\\ =3V_S{I}_{1L}cos{\varphi }_1-3V_S{I}_{2L}cos\left(3\omega t-{\varphi }_2\right)+\\ 3V_S{I}_{4L}cos\left(3\omega t+{\varphi }_4\right)-3V_S{I}_{5L}cos\left(6\omega t-{\varphi }_5\right)\\ +3V_S{I}_{7L}cos\left(6\omega t+{\varphi }_7\right)-\dots \\ q\left(t\right)=v_{\alpha }{i}_{\beta }-v_{\beta }{i}_{\alpha }=\overline{q}+\tilde{q}\\ =3V_S{I}_{1L}sin{\varphi }_1-3V_S{I}_{2L}sin\left(3\omega t-{\varphi }_2\right)+\\ 3V_S{I}_{4L}sin\left(3\omega t+{\varphi }_4\right)-3V_S{I}_{5L}sin\left(6\omega t-{\varphi }_5\right)\\ +3V_S{I}_{7L}sin\left(6\omega t+{\varphi }_7\right)-\dots \\ p_0=v_0{i}_0\end{array}\right.$

where the bar (–) and the tilde (~) correspond to the average value and the alternating components, respectively. In this example, the following relationships exist between the conventional and the instantaneous power theories:

• The average values of power ($\overline{p}$ = P_{three–phase} and $\overline{q}$ = Q_{three–phase}) correspond to the conventional active and reactive powers, respectively.

• The alternating components of the real power ($\tilde{p}$) represent the energy per second that is being transferred from source to load or vice versa at any time. This is the energy being stored (or released) at the three-phase (abc) or two-phase (αβ) load or source.

• The alternating (fluctuating) component of the imaginary power ($\tilde{q}$) is responsible for the harmonic reactive power (due to the nonlinear nature of the load) in each phase but will vanish when added.

• The instantaneous energy transferred does not depend on the imaginary power ($q=\overline{q}+\tilde{q}$), although reactive currents exist in each phase and flow in the conductor.

• The power H including harmonics can be expressed as $H=\sqrt{{\overline{P}}^2+{\overline{Q}}^2}$ where $\overline{P}$ and $\overline{Q}$ are the root-mean-square values of $\overline{p}$ and $\overline{q}$, respectively.

Solution to Application Example 9.8

Expressing these instantaneous voltages in terms of the αβ0-coordinate system yields

$\left\{\begin{array}{l}{v}_{\alpha }=\sqrt{3}{V}_{s+}sin\left(\omega t+{\varphi }_{+}\right)+\sqrt{3}{V}_{s-}sin\left(\omega t+{\varphi }_{\_}\right),\hfill \\ v_{\beta }=-\sqrt{3}{V}_{s+}sin\left(\omega t+{\varphi }_{+}\right)+\sqrt{3}{V}_{s-}sin\left(\omega t+{\varphi }_{\_}\right),\hfill \\ v_0=\sqrt{6}{V}_{\mathit{so}}sin\left(\omega t+{\varphi }_0\right)\text{.}\hfill \end{array}\right.$

Similarly, the load currents may be expresses in terms of their instantaneous αβ0 components

$\left\{\begin{array}{l}{i}_{\alpha }=\sqrt{3}{I}_{L+}sin\left(\omega t+{\delta }_{+}\right)+\sqrt{3}{I}_{L-}sin\left(\omega t+{\delta }_{-}\right),\hfill \\ i_{\beta }=-\sqrt{3}{I}_{L+}sin\left(\omega t+{\delta }_{+}\right)+\sqrt{3}{I}_{L-}sin\left(\omega t+{\delta }_{-}\right)\hfill \\ i_0=\sqrt{6}{I}_{L0}sin\left(\omega t+{\delta }_0\right).\hfill \end{array}\right.$

For this case, the instantaneous active, reactive, and zero-sequence powers are

$\left\{\begin{array}{l}p\left(t\right)=\overline{p}+\tilde{p},\hfill \\ q\left(t\right)=\overline{q}+\tilde{q},\hfill \\ p_0\left(t\right)={\overline{p}}_0+{\tilde{p}}_0\text{,}\hfill \end{array}\right.$

where:

• Instantaneous average real power is

$\overline{p}=3V_{+}{I}_{L+}cos\left({\varphi }_{+}-{\delta }_{+}\right)+3V_{-}{I}_{L-}cos\left({\varphi }_{-}-{\delta }_{-}\right)·$

• Instantaneous average imaginary power is

$\overline{q}=-3V_{+}{I}_{L+}sin\left({\varphi }_{+}-{\delta }_{+}\right)+3V_{-}{I}_{L-}sin\left({\varphi }_{-}-{\delta }_{-}\right)·$

• Instantaneous alternating real power is

$\begin{array}{l}\tilde{p}=-3V_{+}{I}_{L-}cos\left(2\omega t+{\varphi }_{+}+{\delta }_{-}\right)\\ -3V_{-}{I}_{L+}cos\left(2\omega t+{\varphi }_{-}+{\delta }_{+}\right)·\end{array}$

• Instantaneous alternating imaginary power is

$\tilde{q}=3V_{+}{I}_{L-}sin\left(2\omega t+{\varphi }_{+}+{\delta }_{-}\right)-3V_{-}{I}_{L+}cos\left(2\omega t+{\varphi }_{-}+{\delta }_{+}\right)·$

• Instantaneous average zero-sequence power is

${\overline{p}}_0=3V_0{I}_{L0}cos\left({\varphi }_0-{\delta }_0\right)·$

• Instantaneous alternating zero-sequence power is

${\tilde{p}}_0=-3V_0{I}_{L0}cos\left(2\omega t+{\varphi }_0+{\delta }_0\right)·$

9.12.2 UPQC Control System Based on Instantaneous Real and Imaginary Powers

Operation of the UPQC isolates the utility voltage from current quality problems of the load and, at the same time, isolates load from voltage quality problems of the utility. The most important subject in the UPQC control is the generation of reference compensating currents of the shunt converter (e.g., signal $i_C^{*}\text{≡}\left[i_{ca}^{*},i_{cb}^{*},i_{\mathit{cc}}^{*}\right]$ Fig. 9.31) and the reference compensation voltages of the series converter (e.g., signal $v_C^{*}\text{≡}\left[v_{ca}^{*},v_{cb}^{*},v_{\mathit{cc}}^{*}\right]$ Fig. 9.31). PWM control blocks will use these references to generate corresponding switching signals of the shunt and series converters.

In this section, the concept of instantaneous real and imaginary power will be relied on to generate reference signals i_C* and v_C*. The three-phase voltage and current signals are transformed from the abc reference frame to the αβ0 frame (Eq. 9-48) and the instantaneous real, imaginary, and zero-sequence powers are calculated (Eq. 9-52). The decomposition of voltage and current harmonic components into symmetrical components shows that the average of real ($\overline{p}$) and imaginary ($\overline{q}$) powers consists of interacting components of voltage and current that have the same frequency and sequence order (Eq. E9.8-6). Therefore, with the calculation of instantaneous powers and the derivation of their oscillation parts (e.g., $\tilde{p}$ and $\tilde{q}$), the voltage and current reference signals can be obtained and applied to the PWM controls of power electronic converters (Fig. 9.31). Before analyzing the control systems of the shunt and series converters, the circuits of the phase-lock loop (PLL) and the positive-sequence voltage detector (PSVD) will be introduced.

9.12.2.1 Phase-Lock Loop (PLL) Circuit

The phase-lock loop (PLL) circuit (Fig. 9.33) is one of the components of the positive-sequence voltage detector (Fig. 9.34). It detects the fundamental angular frequency (ω₁) and generates synchronous sinusoidal signals that correspond to positive-sequence auxiliary currents under sinusoidal as well as highly distorted and unbalanced system voltages. The inputs of the PLL are v_ab = v_a – v_b and v_cb = (v_c – v_b).

Operation of PLL is based on the instantaneous active three-phase power theory ($p_{\mathrm{three}-\mathrm{phase}}=\overline{p}+\tilde{p}=v_{\mathit{ab}}{i}_a+v_{cb}{i}_c$):

• The current feedback signals i_a(ωt) = sin (ωt) and i_c(ωt) = sin (ωt – 2π/3) are generated by sine generator circuits through integration of ω at the output of a PI-controller. These currents are 120 degrees out of phase and represent a feedback from a positive-sequence component at angular frequency ω.

• The PLL circuit can reach a stable point of operation only if the input of the PI-controller has a zero average value ($\overline{p}$ = 0) with minimized low-frequency oscillations (e.g., small $\tilde{p}$). Because i_α(ωt) and i_c(ωt) contain only positive-sequence components and have unity magnitudes, the components of power (Eqs. E9.8-6a and E9.8-6c) simplify to

$\begin{array}{c}\hfill \overline{p}=3V_{+}{I}_{L+}cos\left({\varphi }_{+}-{\delta }_{+}\right)=0\hfill \\ \hfill \tilde{p}=-3V_{-}{I}_{L+}cos\left(2\omega t+{\delta }_{+}\right)\approx 0\text{,}\hfill \end{array}$

  (9-56)

where ϕ₊ and δ₊ are the initial phase angles of the fundamental positive-sequence voltage and current i_α(ωt), respectively. According to these equations ϕ₊ – δ₊ = π/2. This means that the auxiliary currents i_α(ωt) = sin(ωt) and i_c(ωt) = sin(ωt – 2π/3) become orthogonal to the fundamental positive-sequence component of the measured voltages v_a and v_c, respectively. Therefore, i_α1(ωt) = sin(ωt – π/2) is in phase with the fundamental positive-sequence component of v_a. Similar relations hold for currents i_b1 and i_c1.

• Finally, signals i_a1, i_b1, and i_c1 are transformed to i_α′ and i_β′ (with unity magnitudes) to the αβ0 domain by using an αβ0 block and magnitude divider blocks (which divide each signal by its magnitude).

9.12.2.2 Positive-Sequence Voltage Detector (PSVD)

The positive-sequence voltage detector (PSVD) circuit (Fig. 9.34) is one of the components of the shunt converter controller (Fig. 9.35). It extracts the instantaneous α and β values (v_α′ and v_β′) of the fundamental positive-sequence components of the three-phase voltages v_α, v_b, and v_c (Fig. 9.34).

The operation of the positive-sequence voltage detector is as follows:

• A phase-lock loop (PLL) circuit (Fig. 9.33) produces i_α′ = sin(ω₁t) and i_β′ = cos(ω₁t) that correspond to an auxiliary fundamental positive-sequence current transformed into the αβ0 reference frame.

• Because i_α′ and i_β′ contain only the positive-sequence component, the average values of real power and imaginary powers comprise only the fundamental positive-sequence components of the voltages, and Eq. E9.8-6 simplifies to

$\begin{array}{l}\overline{p}=3V_{+}{I}_{+}^{\prime }cos\left({\varphi }_{+}-{\delta }_{+}\right)\hfill \\ \overline{q}=-3V_{+}{I}_{+}^{\prime }sin\left({\varphi }_{+}-{\delta }_{+}\right)\text{.}\hfill \end{array}$

  (9-57)

• The average values of instantaneous powers $\overline{p}\prime$ and $\overline{q}\prime$ are extracted using low-pass filters (LPFs). Consequently, the instantaneous values of the positive-sequence voltage can be calculated as

$\left[\begin{array}{c}\hfill v_{\alpha }^{\prime }\hfill \\ \hfill v_{\beta }^{\prime }\hfill \end{array}\right]=\frac{1}{{i_{\alpha }^{\prime }}^2+{i_{\beta }^{\prime }}^2}\left[\begin{array}{cc}\hfill i_{\alpha }^{\prime }\hfill & \hfill i_{\beta }^{\prime }\hfill \\ \hfill i_{\beta }^{\prime }\hfill & \hfill -i_{\alpha }^{\prime }\hfill \end{array}\right]\left[\begin{array}{l}{\overline{p}}^{\prime }\\ {\overline{q}}^{\prime }\end{array}\right]\text{.}$

  (9-58)

9.12.2.4 Control of DC Voltage using Instantaneous Power Theory

Figure 9.36 illustrates the block diagram of the DC voltage control system. There are two reasons for the variation of the DC capacitor average voltages.

• The first reason is the real power losses of power electronic converters and the power injected into the network by the series branch. When a voltage sag or voltage swell appears, the series branch of the UPQC injects a fundamental component voltage into the network. The required (observed) real power to be injected is generated by the converter via the DC link, causing variation of the DC-link average voltage.

• Another factor that causes a variation of the average DC capacitor voltages is the zero-sequence component of the current, which is injected by the shunt converter into the system. These currents have no effect on the DC-link voltage but make the two capacitor voltages asymmetric.

The DC voltage controller performs the following two functions (Fig. 9.36):

• Regulating the DC-link average voltage by adding two capacitor voltages, comparing the sum with the reference DC voltage (V_ref), and applying the average voltage error to a PI controller. The output of this controller (${\overline{p}}_{\mathrm{loss}}$) is used by the shunt converter control system to absorb this power from the network and regulate the DC-link voltage. Note that the deviation of voltages is filtered by a low-pass passive filter and the PI controller matches the desired ${\overline{p}}_{\mathrm{loss}}$ that neutralizes the DC bus voltage variations. The low-pass filter makes the voltage regulator insensitive to fundamental frequency voltage variations, which appear when the shunt active filter compensates the fundamental zero-sequence current of the load.

• The correction of the difference between two capacitor average voltages is performed by calculating the difference voltage signal, passing it through a low-pass filter, applying it to a limiter, and adding it (e.g., signal ɛ in Fig. 9.36 with the upper and the lower hysteresis band limits. This process absorbs a zero-sequence current by the shunt converter and corrects the difference of the capacitor average voltages. Similarly, another low-pass filter is applied to filter the signal V_C2 – V_C1. The filtered voltage difference produces ɛ according to the following limit function:

$\left\{\begin{array}{lll}\varepsilon =-1\hfill & \iff \hfill & \mathrm{\Delta }V<-0.05V_{\mathrm{r}\mathrm{e}\mathrm{f}}\hfill \\ \varepsilon =\frac{\mathrm{\Delta }V}{0.05V_{\mathrm{r}\mathrm{e}\mathrm{f}}}\hfill & \iff \hfill & -0.05V_{\mathrm{r}\mathrm{e}\mathrm{f}}\le \mathrm{\Delta }V\le 0.05V_{\mathrm{r}\mathrm{e}\mathrm{f}}\hfill \\ \varepsilon =1\hfill & \iff \hfill & \mathrm{\Delta }V\ge 0.05V_{\mathrm{r}\mathrm{e}\mathrm{f}}\hfill \end{array}\right.$

  (9-60)

where V_ref is the DC bus voltage reference.

• Signal ɛ and the reference compensation currents ($i_C^{*}\text{≡}\left[i_{ca}^{*},i_{cb}^{*},i_{\mathit{cc}}^{*}\right]\text{,}$ Fig. 9.35) are employed to generate gating signals for the shunt converter switches (Fig. 9.31). For example, ɛ can act as a dynamic offset level that is added to the hysteresis-band limits in the PWM current control of Fig. 9.31:

$\left\{\begin{array}{l}\mathrm{upper}\mathrm{hysteresis}\mathrm{band}\mathrm{limit}=i_C^{*}+\mathrm{\Delta }\left(1+\varepsilon \right)\hfill \\ \mathrm{lower}\mathrm{hysteresis}\mathrm{band}\mathrm{limit}=i_C^{*}-\mathrm{\Delta }\left(1-\varepsilon \right)\hfill \end{array}\right.$

  (9-61)

where $i_C^{*}\text{≡}\left[i_{ca}^{*},i_{cb}^{*},i_{\mathit{cc}}^{*}\right]$ are the instantaneous current references (computed by the shunt converter control system) and Δ is a fixed semi-bandwidth of the hysteresis control. Therefore, signal ɛ shifts the hysteresis band to change the switching times of the shunt converter such that

$\left\{\begin{array}{ll}\varepsilon >0\hfill & \mathrm{increases}{V}_{C1}\mathrm{and}\mathrm{decreases}{V}_{C2}\hfill \\ \varepsilon <0\hfill & \mathrm{increases}{V}_{C2}\mathrm{and}\mathrm{decreases}{V}_{C1}\text{.}\hfill \end{array}\right.$

9.12.2.5 Control of Series Converter using Instantaneous Power Theory

Inputs of the series converter control system (Fig. 9.37) are the three-phase supply voltages (v_sa, v_sb, v_sc), load voltages (v_a, v_b, v_c), and the line currents (i_a, i_b, i_c). Its outputs are the reference-compensating voltages v_ca*, v_cb* and v_cc* (e.g., signal v_C* in Fig. 9.31). The compensating voltages consist of two components:

• One is the voltage-conditioning component $v_s^{\prime }-v_s$ (e.g., the difference between the supply voltage v_s ≡ [v_sa, v_sb, v_sc] and its fundamental positive-sequence component v′_s ≡ [v′_sa, v′_sb, v′_sc]) that compensates harmonics, fundamental negative-, and zero-sequence components of voltage, and regulates the fundamental component magnitude.

• The other voltage component (v_h ≡ [v_ha, v_hb, v_hc]) performs harmonic isolation and zero-sequence current damping, as well as damping of harmonic oscillations caused by the series resonance due to the series inductances and the shunt capacitor banks. v_h can also compensate for the oscillating instantaneous active and reactive powers (${\tilde{p}}_h$ and ${\tilde{q}}_h$).

The series controller operates as follows (Fig. 9.37):

• PSVD, αβ0, and power calculation blocks are relied on to compute oscillating instantaneous powers ${\tilde{p}}_h$ and ${\tilde{q}}_h$. This procedure is similar to the shunt converter control system.

• ${\tilde{p}}_h$ and ${\tilde{q}}_h$ are compared with their reference values (p_ref and q_ref) to generate the desired compensation powers. The reference active power (p_ref) forces the converter to inject a fundamental voltage component orthogonal to the load voltage. This will cause change in the power angle δ (phase angle between the voltages at both ends of transmission line) and mostly affects the active power flow. The reference reactive power (q_ref) will cause a compensation voltage in phase with the load voltage. This will change the magnitudes of the voltages and strongly affects the reactive power flow of the system.

• A current reference calculation block is used to generate the reference compensation currents in the αβ0 space:

$\left[\begin{array}{c}\hfill i_{h\alpha }\hfill \\ \hfill i_{h\beta }\hfill \end{array}\right]=\frac{1}{{v_{\alpha }^{\prime }}^2+{v_{\beta }^{\prime }}^2}\left[\begin{array}{cc}\hfill v_{\alpha }^{\prime }\hfill & \hfill -v_{\beta }^{\prime }\hfill \\ \hfill v_{\beta }^{\prime }\hfill & \hfill v_{\alpha }^{\prime }\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\tilde{p}}_h+{\tilde{p}}_C\hfill \\ \hfill {\tilde{q}}_h+{\tilde{q}}_C\hfill \end{array}\right]\text{.}$

  (9-62)

• Signal i_h0, which provides harmonic isolation and damping for zero-sequence currents flowing through the series converter, is separated from the signal i₀ by a low-pass filter.

• The αβ0 transformation block is employed to generate the reference compensation voltages for harmonic isolation and harmonic oscillation damping ($v_h\text{≡}\left[v_{\mathit{ha}}^{*},v_{hb}^{*},v_{hc}^{*}\right]$). These voltages are added with $v_s^{\prime }\text{≡}\left[v_{sa}^{\prime },v_{sb}^{\prime },v_{sc}^{\prime }\right]$ and subtracted from the supply voltages $v_s\text{≡}\left[v_{sa},v_{sb},v_{sc}\right]$ to form the reference compensation voltage for the series converter:

$\left\{\begin{array}{l}{v}_{ca}^{*}=v_{\mathit{ha}}+\left(v_{sa}^{\prime }-v_{sa}\right)\hfill \\ v_{cb}^{*}=v_{hb}+\left(v_{sb}^{\prime }-v_{sb}\right)\hfill \\ v_{\mathit{cc}}^{*}=v_{hc}+\left(v_{sc}^{\prime }-v_{sc}\right)\text{.}\hfill \end{array}\right.$

  (9-63)

• The reference compensation voltages and a sinusoidal PWM control approach are relied on to produce the gating signals of the series converter (Fig. 9.31).

The UPQC series control system (Fig. 9.37) may be simplified by eliminating the reference compensation currents as shown in Fig. 9.38. Voltage regulation is controlled by V_ref. The converter is designed to behave like a resistor at harmonic frequencies to perform damping of harmonic oscillations. To do this, harmonic components of the line current are detected and multiplied by a damping coefficient.