Cooperative Design and Control of Distributed Harmonic and Reactive Compensators

1
Cooperative Design and Control of Distributed
Harmonic and Reactive Compensators
International School on Nonsinusoidal Currents
and Compensation (ISNCC 08) Lagów, (Poland) 10
- 13 June 2008

• Elisabetta Tedeschi, Paolo Tenti
• Department of Information Engineering, University
  of Padova, Italy

2
Presentation Outline

• Compensation goals and methods
• Definition and properties of instantaneous and
  average power terms
• Reactive power under non-sinusoidal conditions
• Orthogonal current decomposition
• Apparent power decomposition
• Cooperative design and control of distributed
  compensators
• quasi-stationary compensators (reactive power
  compensation)
• dynamic compensators (reactive and distortion
  power compensation)
• Control application and results of single-phase
  and three-phase systems
• Conclusions

3
Compensation goals and methods

• Vision
• In addition to nonlinear loads pollution,
  distributed generation is an increasingly
  important issue in view of compensation in
  fact, wind generators, photovoltaic plants, fuel
  cells, micro-turbines are more and more diffused
  in electrical networks
• Optimal utilization of alternative energy sources
  requires matching between generators and grid
  parameters, which is obtained by suitable
  electronic interfaces
• In addition to their adaptation function, such
  interfaces may perform as distributed
  compensation units too
• Coordination of their design and operation may
  improve global and local network utilization
  while avoiding detrimental interactions

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Compensation goals and methods

• Goal of Compensation
• Generally, the goal of compensation is twofold
• On the supply side, compensation aims at reducing
  the energy loss associated to power delivery
• On the load side, compensation aims at improving
  the quality of supply
• The first goal requires proper shaping of the
  current drawn from the grid
• The second goal requires limitation of voltage
  distortion at load terminals

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Compensation goals and methods

• Compensator types
• Stationary (passive) compensators include passive
  filters (low-pass / selective) and fixed
  reactances (inductors / capacitors), designed to
  provide a given amount of reactive power
  absorption and/or harmonic filtering under rated
  conditions.
• These compensators are cheap and quite
  standardized, however they must be optimized for
  each specific load and location, taking into
  account that the behavior of passive elements is
  strongly affected by surrounding network
  parameters.

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Compensation goals and methods

• Compensator types
• Quasi-stationary (reactive) compensators, i.e.,
  Static VAR Compensators (SVC), include
  thyristor-controlled reactors (TCR) and
  thyristor-switched capacitors (TSC). By properly
  gating the thyristors, such compensators provide
  positive or negative reactive power absorption
  from zero to a rated amount.
• These compensators are cheap, however reactive
  power control is normally associated to
  detrimental injection of current harmonics at the
  connecting point.
• Regulation provided by SVC is necessarily slow,
  since thyristors are gated only once per line
  cycle.

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Compensation goals and methods

• Compensator types
• Dynamic (active) compensators include Switching
  Power Compensators (SPC), i.e., Active Power
  Filters (APF), which provide fast control and can
  compensate for transient and/or high-frequency
  current components absorbed by time-varying /
  distorting loads.
• These compensators are more expensive and,
  therefore, less diffused than the other ones.

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Compensation goals and methods

• Distributed Compensation
• Complex networks may include several compensation
  units connected to different network ports
• Such compensating units are normally designed and
  driven independently, neglecting their
  potentiality to coordinate, reinforce or affect
  each others compensation capability
• Cooperative design and control of distributed
  compensation units has the potential to achieve
  full exploitation of the installed compensation
  capability, while avoiding unwanted interactions
  and instabilities

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Cooperative operation of distributed compensators
Compensation goals and methods

• Compensation requirements are expressed in terms
  of supply (input) currents and/or load (output)
  voltages
• Instead, compensation commands can be expressed
  in power terms, since these are conservative
  quantities
• Quasi-stationary compensators are driven to
  absorb a suitable amount of reactive power
• Dynamic compensators are driven to absorb a
  suitable amount of instantaneous complex power

A distributed compensation system may produce the
same effect of an APF acting at the input port
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Goal

• General approach for cooperative design and
  control
• of distributed compensators
• The approach makes use of conservative
  power-like quantities and applies to
  quasi-stationary compensators and dynamic
  compensators single-phase and three-phase
  networks sinusoidal and non-sinusoidal
  conditions
• Steps
• Definition of instantaneous and average power
  terms
• conservative, related to power absorption and
  energy storage, valid under sinusoidal and
  non-sinusoidal conditions
• Decomposition of current
• orthogonal terms, each one related to a physical
  phenomenon (power absorption, energy storage,
  load unbalance,voltage current distortion)
• Decomposition of apparent power
• active, reactive, unbalance and distortion terms
• Control of distributed compensators
• SVC (quasi-stationary control) unbalance and
  reactive power compensation
• APF (dynamic control) instantaneous complex
  power compensation
• Application examples
• Single phase, three-phase (3- and 4-wire),
  asymmetrical voltages, unbalanced load

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Hat operator unbiased integral function

• Given variable x(t), which is real, periodic,
  continuous and alternate, let

Average value
Time integral function
we define the unbiased integral function (hat
operator) as

• Hat quantities are homogeneous to basic variables
• Quantities and are also referred as
  integral voltage and integral current

12
Properties of hat quantities

• From the definitions of internal product and
  norm
• we easily obtain
• Moreover, under sinusoidal operation

13
Instantaneous power definitions(valid for
periodic variables)
Real power
Complex power
Imaginary power

• The three definitions of imaginary power are
  equivalent in average. In instantaneous terms,
  however
• definition (1) is insensitive to resistive
  current components
• definition (2) naturally applies to voltage-fed
  units
• definition (3) naturally applies to current-fed
  units

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A property of imaginary power q1
If instantaneous imaginary power q1 is zero at
any time then current and voltage are proportional
Demonstration
Expressing voltage and current terms in p.u. we
have
which can be derived to obtain
Viceversa, imaginary power q1 is insensitive to
resistive current components
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Conservation of instantaneous power terms

• For any given network p, let u and i be the
  vectors of branch voltages and currents
• Integral branch voltages are consistent with
  network p, i.e. they comply with KLV (Kirchhoffs
  law for voltages)
• Integral branch currents are consistent with
  network p, i.e. they comply with KLC (Kirchhoffs
  law for currents)
• Thus, according to Tellegens Theorem

Real, imaginary and, consequently, complex power
are conservative quantities in every network
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Average power definitions(valid for periodic
quantities)
Active power
Reactive power
Apparent power
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Power terms in passive networks
Resistor
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Power terms in passive networks
Inductor
Inductor energy
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Power terms in passive networks
Capacitor
Capacitor energy
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Power terms in passive networks
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Active and reactive power absorption of a linear
passive network p
Remark Whichever is the origin of reactive
power, including active and nonlinear loads, it
can be compensated by reactive elements with
proper energy storage capability
Total absorbed power
22
Orthogonal current decompositionSingle phase
circuits

• Current terms
• ia active current
• in non-active current
• ir reactive current
• iv void current
• isa scattered active current
• isr scattered reactive current
• ig generated current
• Orthogonality. All terms in the above equations
  are orthogonal, thus

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Orthogonal current decomposition

• Active current (Fryze current)

• Non-active current

• Reactive current

• Void current

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Orthogonal current decomposition

• Active current (Fryze current)

• Non-active current

• Reactive current

The void current reflects the presence of
scattered active, scattered reactive and
generated terms

• Void current

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Scattered active current

• For each co-existing harmonic component of
  voltage and current we define
• Harmonic active current terms

• Total harmonic active current

• Scattered active current

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Scattered reactive current

• For each co-existing harmonic component of
  voltage and current we define
• Harmonic reactive current terms

• Total harmonic reactive current

• Scattered reactive current

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Apparent and distortion power

• Apparent power

Distortion power
Du is voltage distortion power Di is current
distortion power
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Distortion power terms
Voltage distortion power (vanishes for sinusoidal
voltage supply or in absence of reactive power
absorption)
su is voltage distortion factor and vanishes
under sinusoidal operation
Current distortion power (vanishes in absence of
void current)
si is current distortion factor and vanishes if
void current is zero
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Extension to poly-phase systems

• Current decomposition into active, reactive and
  void current can be separately applied to each of
  the N phases

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Extension to poly-phase systems

• When active and reactive components are
  determined globally, balanced active and reactive
  terms are obtained

• Balanced active current

• Balanced reactive current

• By difference unbalanced components are obtained

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Orthogonal current decomposition

• Current terms
• ib balanced currents
• iu unbalanced currents
• iba balanced active currents
• ibr balanced reactive currents
• iua unbalanced active currents
• iur unbalanced reactive currents
• iv void currents
• Orthogonality. All terms in the above equations
  are orthogonal, thus

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Apparent power decomposition
From current decomposition
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Apparent power decomposition
S is useful power P is active power Q is
reactive power Su is unbalance power Sua
active unbalance power Sur reactive unbalance
power D is distortion power Du is voltage
distortion power Di is current distortion power
34
Distributed compensatorsDesign criteria

• Passive filters are designed to provide local
  reactive/harmonic compensation under rated
  stationary conditions
• Static VAR Compensators (TCR and TSC) are
  cooperatively designed together with passive
  filters so as to provide, as a whole, the energy
  storage capability required to compensate for
  worst-case unbalance and reactive power
  absorption
• Active power filters are designed to provide, as
  a whole, a proper dynamic compensation
  capability, so as to face distortion power,
  supply and load transients, delays of other
  compensators etc.

35
Distributed compensatorsUse of power command

• If the compensation command is given in power
  terms, it is not affected by turn ratio and phase
  rotation of transformers
• Moreover, compared to current command, the power
  command is less affected by network dynamics,
  i.e., by cross-impedances between input port and
  compensation ports (parasitic phenomena due to
  resonances, voltage drops on line impedances,
  transfer function of passive filters, nonlinear
  loads)
• Finally, a power command generated at the input
  port can be shared among various compensation
  units acting in the same network according to
  their power ratings and distance from the input
  port

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Distributed compensatorsControl criteria

• SVC units (TCR, TSC) are driven by an average
  complex power command and adjust the inductive or
  capacitive energy storage according to unbalance
  and reactive power compensation need at the input
  port. SVCs execute the power command within a
  line cycle, providing quasi-stationary control of
  unbalance and reactive power.
• APF units are driven by an instantaneous complex
  power command, which is converted into a suitable
  current command to provide dynamic compensation
  of (unbalance reactive ) void currents at the
  input port.

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Distributed compensatorsDynamic control

• For the purpose of dynamic control of distributed
  switching compensators, assuming that they are
  voltage-fed, we select the following definition

real power
imaginary power
and consequently
complex power

• For control purposes, all definitions of
  imaginary power are viable
• However, the above choice makes control simpler
  for voltage-fed shunt APF
• In fact, no reference is made to integral
  current, which would introduce an additional
  state variable in control

38
Structure of dynamic control
The Central Control Unit determines the power
command as a function of the desired behavior at
the input port The Local Control Units transform
the power command in a current command for each
APF
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Structure of dynamic control

• Central Control Unit
• CRG (Current Reference Generator) determines
  input current references as a function of input
  port variables and requirements
• ii f(pi, ui, PF, THD)

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CRG - Current Reference Generator
CRG generates input current references according
to suitable criteria, e.g.
Unity power factor (active currents)
Constant input power
Minimum currents conveying instantaneous real
power
Only positive sequence, sinusoidal currents etc.
41
Structure of dynamic control
42
EA - Error Amplifier

• The error amplifier is designed according to
  usual criteria taking into account the transfer
  function between APF current and input current.
• For this purpose, reference is made to the
  equivalent single-phase p.u. circuit.
• In fact, phase rotation and voltage shift caused
  by transformers do not affect the power command.

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Structure of dynamic control

• Central Control Unit
• PS (Power Sharing unit) distributes complex power
  reference among the various compensators

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PS - Power Sharing Unit

• The Power Sharing unit distributes the power
  command to the various APF considering
• Power rating of each APF
• Distance of each APF from the input port (i.e.,
  response time and attenuation factor of the
  transfer function between APF current and input
  current)
• Residual compensation capability of each APF,
  which can perform local compensation too
• While parameters 1 and 2 can be assigned at
  system set up, parameter 3 requires real-time
  interaction between APF and PSU

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Structure of dynamic control

• Local Control Unit (LCU)
• This unit transforms the complex power command
  generated by PS unit for each APF into a suitable
  current reference

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Local Control Unit

• The function of the LCU is to convert the complex
  power command given to the APF into a current
  command.
• The power-current conversion algorithm performs
  differently depending on network type, i.e.
• Three-phase three-wire
• Single-phase
• Three-phase four-wire

47
Three-phase three-wire systems Power-current
conversion algorithm

• In a three-phase three-wire system there is a
  univocal correspondence between the complex power
  absorbed at a specified port and the
  corresponding currents. Let

the conversion equations are
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Three-phase three-wire systems Application
example

• Distorted input voltage (5th harmonic 10, 7th
  harmonic 5)
• Wye/delta transformer
• Ohmic-inductive load RL-LL ZL1 pu, cos f 0.9
• Ls 0.05 pu at fundamental frequency
• SVC is driven to compensate load reactive power
• APF performs remaining compensation

49
Three-phase three-wire systems Application
example

• As a whole the network appears as a resistive
  load
• APF and SVC perform cooperatively though
  connected to different points

50
Single-phase systems Power-current conversion
algorithm

• For a given APF supply voltage u, the algorithm
  must determine current reference i which better
  suits complex power reference s

.

• Since p and q set two independent conditions on
  variable i, in general only an approximate
  solution can be found

• Current reference i is selected to minimize the
  error function

resulting in
51
Single-phase systems Power-current conversion
algorithm

• Coefficients a and b can be dynamically adjusted
  to provide good accuracy, in spite of the
  variation of power and voltage terms

• Smooth operation is ensured also around critical
  times, when and vanish

• An effective solution is

while keeping
52
Single-phase systemsApplication example
APF

• Ohmic-inductive load RL-LL ZL1 p.u., cos f 0.9
• Very high line inductance (ZLs 0.2 p.u.)
• Capacitor bank (Zc 2.3 p.u.)
• Distorted input voltage (5th harmonic 10, 7th
  harmonic 5)

53
Single-phase systemsApplication example

• Input current and voltage

Input and APF voltage

• The input current tracks the input voltage with
  good accuracy, in spite of the inherent error
  introduced by the algorithm

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Extension to three-phase four-wire systems

• The APF can control the homopolar current, which
  can compensate for zero-sequence power absorption

• If u0 and i0 are voltage and current of the
  neutral wire, instantaneous power terms absorbed
  by the APF are

• Selecting the voltage reference so that
• and considering that

where
we have
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Extension to three-phase four-wire systems

• According to these definitions, zero-sequence
  components can be treated separately from the
  other ones

• Given total power reference
• and zero-current power reference
• non-zero current references are obtained by

56
Three-phase unbalanced system Application example

• Distorted input voltage (5th harmonic 10, 7th
  harmonic 5)
• Wye/delta transformer
• Single-phase ohmic-inductive load RL-LL ZL1 pu,
  cos f 0.9
• Ls 0.05 p.u. at fundamental frequency
• APF performs the entire compensation

57
Three-phase unbalanced system Application example

• Input currents

APF currents

• The proposed control technique inherently
  compensates also for load unbalance

58
Compensation of 3-phase unbalanced systems by
SVCs and APFs

• Steinmetz method for unbalance compensation can
  be extended to asymmetrical and distorted networks

An exact solution for Steinmetz network
parameters is possible only in case of
symmetrical voltages. Otherwise an approximate
solution can be found.
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Distributed compensation of load unbalance and
reactive power

• Distributed SVCs can be used to compensate for
  unbalance and reactive current components
• APFs allow compensation of void currents too
• The power commands are generated as follows

• Based on this approach, every unwanted current
  component (reactive, unbalanced and void) can be
  eliminated by cooperative operation of the
  compensators. Almost balanced active currents are
  absorbed at PCC.

60
Three-phase unbalanced system Application example

• Distorted and unbalanced input voltage
• Single-phase ohmic-inductive load RL-LL ZL1 pu,
  cos f 0.9
• APF and SVC share the compensation duty

61
Three-phase unbalanced system Application example
t0 all compensators off t1 fixed capacitors CF
are inserted t2 TCR is connected t3 APF is
switched on

• Voltage and currents at PCC

• Unbalance, reactive and harmonic compensation can
  effectively be achieved by cooperative operation
  of the various kinds of compensators

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Conclusions

• A general approach to cooperative design and
  operation of distributed unbalance, reactive and
  harmonic compensators has been presented.
• It is based on power-like quantities, which are
  conservative and have physical meaning, being
  related to power absorption and energy storage.
• The approach requires averaging of control
  quantities over a line period and applies both to
  quasi-stationary control of SVCs and dynamic
  control of APFs. In the latter case, however,
  fast communication between Central Control Unit
  and Local Control Units is required, which is a
  problem in practical applications.
• According to the proposed control, the system of
  distributed compensators performs as an APF
  connected at the input port.