Power Filters Planning

Abstract

In this chapter, the authors provide a succinct overview of harmonic power filter planning studies, including causes and malicious effects of nonlinear loads and detailed descriptions of passive and active harmonic power filters. Next, different methodologies for solving harmonic power flow problems are precisely classified. Besides these outlines, the chapter develops the formulation of an innovative techno-economic multi-objective framework for the hybrid harmonic power filter (HHPF) planning problem in distribution networks, with consideration of uncertainty in demand and harmonic currents injected by nonlinear loads. The proposed framework is also broken down into a harmonic power flow problem and the HHPF planning problem. The harmonic power flow problem acts as a central core of the HHPF planning problem and is solved via a probabilistic decoupled harmonic power flow (PDHPF) methodology. This chapter widely utilizes an efficient two-point estimate method (two-PEM) in order to handle uncertainty in demand and harmonic currents injected by nonlinear loads in the proposed framework. The proposed PDHPF methodology, according to the efficient two-PEM, is implemented by a deterministic decoupled harmonic power flow (DDHPF) methodology. A loadability-based Newton-Raphson power flow (LBNRPF) methodology is also applied to solve the power flow problem at the principal frequency.

Appendices

Appendix 1: List of Abbreviations and Acronyms

AC	Alternating current
AHPFs	Active harmonic power filters
ANSI	American National Standards Institute
C	Capacitance
DC	Direct current
DDHPF	Deterministic decoupled harmonic power flow
DNO	Distribution network operator
FSM	Fuzzy satisfying method
HA	Harmonic attenuation
HHPFs	Hybrid harmonic power filters
ICS	Index of cost saving
IEEE	Institute of Electrical and Electronic Engineers
L	Inductance
LBNRPF	Loadability-based Newton-Raphson power flow
MLLF	Motor load loss function
MMM-EMSA	Multi-stage computational multi-dimensional multiple-homogeneous enhanced melody search algorithm
MMS-EMSA	Multi-stage computational multi-dimensional single-inhomogeneous enhanced melody search algorithm
MVAr	Megavolt-ampere reactive power
NSGA-II	Non-dominated sorting genetic algorithm-II
PCC	Point of common coupling
PDHPF	Probabilistic decoupled harmonic power flow
PHPFs	Passive harmonic power filters
R	Resistance
SOSA	Symphony orchestra search algorithm
SS-HSA	Single-stage computational single-dimensional harmony search algorithm
SS-IHSA	Single-stage computational single-dimensional improved harmony search algorithm
THDV	Total harmonic distortion of voltage
THDI	Total harmonic distortion of current
TIF	Telephone influence factor
TLLH	Transmission line loss arising from harmonics
TMS-EMSA	Two-stage computational multi-dimensional single-homogeneous enhanced melody search algorithm
TMS-MSA	Two-stage computational multi-dimensional single-homogeneous melody search algorithm
Two-PEM	Two-point estimate method

Appendix 2: List of Mathematical Symbols

Index
a	Index for AHPFs of type m running from 1 to A
$b,{b}^{\prime },\widehat{b}$	Index for buses of the distribution network running from 1 to B
h	Index for harmonic orders running from 2 to H
hc	Index for harmonic order counters running from 1 to HC
i	Index of concentrations for each uncertainty parameter running from 1 to I
k	Index for uncertainty parameters running from 1 to K
m	Index for types of AHPFs running from 1 to M
n	Index for types of PHPFs running from 1 to N
p	Index for PHPFs of type n running from 1 to P
u	Index for iterations of the LBNRPF methodology running from 1 to U
Set
${\Psi}_m^{\mathrm{A}}$	Set of indices of AHPFs of type m
ΨB	Set of indices of buses of the distribution network
${\Psi}_b^{\mathrm{B}}$	Set of indices of buses of the distribution network for buses connected to bus b of the distribution network
ΨH	Set of indices of harmonic orders
ΨI	Set of indices of concentrations
ΨK	Set of indices of uncertainty parameters
ΨM	Set of indices of types of AHPFs
ΨN	Set of indices of types of PHPFs
${\Psi}_n^{\mathrm{P}}$	Set of indices of PHPFs of type n
Parameters
${A}_b^h$	Ratio of the current of a nonlinear load on bus b of the distribution network for harmonic order h to its current in the principal frequency [%]
D	Distribution network bus voltage vector at initial guess
f 1	Principal frequency [Hz]
f h	Harmonic frequency at harmonic order h [Hz]
HOV	Harmonic order vector
${I}_b^{1,\mathrm{NLL}}$	Current of a nonlinear load on bus b of the distribution network for the principal frequency [kA or p.u.]
${I}_b^{h,\mathrm{NLL}}$	Current of a nonlinear load on bus b of the distribution network for harmonic order h [kA or p.u.]
${I}_b^{h,\max }$	Maximum admissible value of the current on bus b of the distribution network for harmonic order h [kA or p.u. or %]
$\left|{I}_l^1\right|$	Magnitude of the current at line l of the distribution network for the principal frequency [kA or p.u.]
Imax	Maximum admissible value of the injected current by each AHPF [kA or p.u.]
P b	Active load—demand—on bus b of the distribution network [MW or p.u.]
${P}_b^1$	Active load—demand—on bus b of the distribution network for the principal frequency [MW or p.u.]
Q b	Reactive load—demand—on bus b of the distribution network [MVAr or p.u.]
${Q}_b^1$	Reactive load—demand—on bus b of the distribution network for the principal frequency [MVAr or p.u.]
Qmax	Maximum admissible value of the principal reactive power compensation by all PHPFs on all buses of the distribution network [MVAr or p.u.]
${Q}_b^{\mathrm{max}}$	Maximum admissible value of the principal reactive power compensation by all PHPFs on bus b of the distribution network [MVAr or p.u.]
${Q}_{n,b}^{\mathrm{max}}$	Maximum admissible value of the principal reactive power compensation by each PHPF of type n on bus b of the distribution network [MVAr or p.u.]
Qmin	Minimum admissible value of the principal reactive power compensation by all PHPFs on all buses of the distribution network [MVAr or p.u.]
${Q}_b^{\mathrm{min}}$	Minimum admissible value of the principal reactive power compensation by all PHPFs on bus b of the distribution network [MVAr or p.u.]
${Q}_{n,b}^{\mathrm{min}}$	Minimum admissible value of the principal reactive power compensation by each PHPF of type n on bus b of the distribution network [MVAr or p.u.]
R b, b'	Resistance between buses b and b′ of the distribution network [Ω or p.u.]
R l	Resistance of line l of the distribution network [Ω or p.u.]
TCHFmax	Maximum allowable value of the TCHF index [M$]
${THDI}_b^{\mathrm{max}}$	Maximum admissible value of THDI on bus b of the distribution network [p.u. or %]
${THDV}_b^{\mathrm{max}}$	Maximum admissible value of THDV on bus b of the distribution network [p.u. or %]
${V}_b^{h,\max }$	Maximum admissible value of voltage on bus b of the distribution network for harmonic order h [p.u. or %]
X b, b'	Inductance between buses b and b′ of the distribution network [Ω or p.u.]
${y}_b^{\mathrm{PA}}$	Parallel-connected admittance connected to bus b of the distribution network that is available in the initial configuration of the distribution network [℧ or p.u.]
${y}_b^{1,\mathrm{PA}}$	Parallel-connected admittance connected to bus b of the distribution network that is available in the initial configuration of the distribution network for the principal frequency [℧or p.u.]
${y}_b^{h,\mathrm{PA}}$	Parallel-connected admittance connected to bus b of the distribution network that is available in the initial configuration of the distribution network for harmonic order h [℧or p.u.]
${y}_{b,{b}^{\prime }}$	Existing admittance between buses b and b′ of the distribution network [℧or p.u.]
${y}_{b,b\hbox{'}}^h$	Existing admittance between buses b and b′ of the distribution network for harmonic order h [℧or p.u.]
${y}_{b,\widehat{b}}$	Existing admittance between buses b and $\widehat{b}$ of the distribution network [℧or p.u.].
${y}_{b,\widehat{b}}^h$	Existing admittance between buses b and $\widehat{b}$ of the distribution network for harmonic order h [℧or p.u.]
${y}_b^h$	Admittance of a linear load on bus b of the distribution network for harmonic order h [℧or p.u.]
${y}_{b,{b}^{\prime}}^h$	Existing admittance between buses b and b′ of the distribution network for harmonic order h [℧or p.u.]
Y b, b	Element (b,b)—main-diagonal element—of the admittance matrix
${Y}_{b,b}^h$	Element (b,b)—main-diagonal element—of the admittance matrix for harmonic order h
${Y}_{b,{b}^{\prime }}$	Element (b,b′)—off-diagonal element—of the admittance matrix
${Y}_{b,{b}^{\prime}}^h$	Element (b,b′)—off-diagonal element—of the admittance matrix for harmonic order h
${Y}_{b,\widehat{b}}$	Element (b,$\widehat{b}$)—off-diagonal element—of the admittance matrix
Ybus	Admittance matrix of the distribution network
${Y}_{\mathrm{bus}}^h$	Admittance matrix of the distribution network for harmonic order h
Z bus	Impedance matrix of the distribution network
${Z}_{\mathrm{bus}}^h$	Impedance matrix of the distribution network for harmonic order h
${Z}_{b,{b}^{\prime}}^h$	Impedance between buses b and b′ of the distribution network for harmonic order h [Ω or p.u.]
θ b, b	Phase angle of the element (b,b)—main-diagonal element—of the admittance matrix [rad]
${\theta}_{b,{b}^{\prime }}$	Phase angle of the element (b,b′)—off-diagonal element—of the admittance matrix [rad]
${\theta}_{b,\widehat{b}}$	Phase angle of the element (b,$\widehat{b}$)—off-diagonal element—of the admittance matrix [rad]
ω 0	The rated angular frequency of the distribution network [rad/sec]
${\tau}_{n,b,p}^{\mathrm{PHPF}}$	Characteristic coefficient corresponding to the fixed cost of PHPF p of type n on bus b of the distribution network [M$]
${\tau}_{m,b,a}^{\mathrm{AHPF}}$	Characteristic coefficient corresponding to the fixed cost of AHPF a of type m on bus b of the distribution network [M$]
${\upsilon}_{n,b,p}^{\mathrm{PHPF}}$	Characteristic coefficient corresponding to the variable cost of PHPF p of type n on bus b of the distribution network [M$/p.u.]
${\upsilon}_{m,b,a}^{\mathrm{AHPF}}$	Characteristic coefficient corresponding to the variable cost of AHPF a of type m on bus b of the distribution network [M$/p.u.]
${\kappa}_n^{\mathrm{PHPF}}$	Certain portion that connects the cost of the inductance and resistance of the PHPFs of type n to the cost of capacitance of corresponding type of PHPFs [%]
Variables
C n, b, p	Capacitance of PHPF p of type n on bus b of the distribution network [μF or p.u.]
D ( u)	Distribution network bus voltage vector at iteration u of the LBNRPF methodology
E(Y)	First moment of Y
E(Y2)	Second moment of Y
F i, b	Reactive power in power transmission lines connected to bus b of the distribution network [MVAr or p.u.]
F r, b	Active power in power transmission lines connected to bus b of the distribution network [MW or p.u.]
I h	Current for harmonic order h [kA or p.u.]
I h, DN	Current of the distribution network’s source for harmonic order h [kA or p.u.]
${I}_b^h$	Current on bus b of the distribution network for harmonic order h [kA or p.u.]
$\left|{I}_b^h\right|$	Magnitude of the current on bus b of the distribution network for harmonic order h [kA or p.u.]
${I}_l^h$	Current at line l of the distribution network for harmonic order h [kA or p.u.]
I m, b, a	RMS injected current by AHPF a of type m on bus b of the distribution network [kA or p.u.]
${I}_{m,b,a}^h$	Injected current by AHPF a of type m on bus b of the distribution network for harmonic order h [kA or p.u.]
${I}_{m,b,a}^{h,\mathrm{r}}$	Real part of the injected current by AHPF a of type m on bus b of the distribution network for harmonic order h [kA or p.u.]
${I}_{m,b,a}^{h,\mathrm{i}}$	Imaginary part of the injected current by AHPF a of type m on bus b of the distribution network for harmonic order h [kA or p.u.]
${I}_{n,b,p}^{h,\mathrm{cap}}$	Harmonic current passing through the capacitance relevant to PHPF p of type n on bus b of the distribution network for harmonic order h [kA or p.u.]
$\left|{I}_l^h\right|$	Magnitude of the current at line l of the distribution network for harmonic order h [kA or p.u.]
J	Jacobian matrix of the distribution network
L n, b, p	Inductance of PHPF p of type n on bus b of distribution network [mH or p.u.]
${N}_{m,b,a}^{\mathrm{AHPFs}}$	Number of installed AHPFs on each candidate bus of the distribution network
${N}_{n,b,p}^{\mathrm{PHPFs}}$	Number of installed PHPFs on each candidate bus of the distribution network
${\mathrm{OF}}_1^{\mathrm{HHPFs}}$	First objective function of the HHPF planning problem [p.u.]
${\mathrm{OF}}_2^{\mathrm{HHPFs}}$	Second objective function of the HHPF planning problem [p.u.]
${\mathrm{OF}}_3^{\mathrm{HHPFs}}$	Third objective function of the HHPF planning problem [p.u.]
${\mathrm{OF}}_4^{\mathrm{HHPFs}}$	Fourth objective function of the HHPF planning problem [M$]
Q n, b, p	Principal reactive power compensation by PHPF p of type n on bus b of the distribution network [MVAr or p.u.]
R n, b, p	Resistance of PHPF p of type n on bus b of the distribution network [Ω or p.u.]
TCHF	Total cost of harmonic power filters [M$]
THDI	Total harmonic distortion of the current [p.u.]
THDI l	Total harmonic distortion of the current at line l of the distribution network [p.u.]
THDV	Total harmonic distortion of the voltage [p.u.]
THDV b	Total harmonic distortion of the voltage on bus b of the distribution network [p.u.]
TLLH	Total line loss arising from harmonics [p.u.]
TLLH h	Total line loss arising from harmonic order h [p.u.]
V b	Magnitude of the RMS voltage on bus b of the distribution network [kV or p.u.]
${V}_{\widehat{b}}$	Magnitude of the RMS voltage on bus $\widehat{b}$ of the distribution network [kV or p.u.]
${V}_{b^{\prime }}$	Magnitude of the RMS voltage on bus b′ of the distribution network [kV or p.u.]
${V}_b^h$	RMS voltage on bus b of the distribution network for harmonic order h [kV or p.u.]
${V}_{b^{\prime}}^h$	RMS voltage on bus b′ of the distribution network for harmonic order h [kV or p.u.]
V h	Voltage at harmonic order h [kV or p.u.]
${V}_{n,b,p}^{h,\mathrm{cap}}$	Harmonic voltage drop on the capacitance associated with PHPF p of type n on bus b of the distribution network for harmonic order h [kV or p.u.]
|Vb|	Magnitude of the RMS voltage on bus b of the distribution network [kV or p.u.]
$\left|{V}_b^1\right|$	Magnitude of the RMS voltage on bus b of the distribution network for the principal frequency [kV or p.u.]
$\left|{V}_b^h\right|$	Magnitude of the voltage on bus b of the distribution network for harmonic order h [kV or p.u.]
X	Input uncertainty parameter vector
Y	Output uncertainty parameter vector, which is directly affected by input uncertainty parameters
$\tilde{\mathrm{Y}}$	Output uncertainty parameter vector, which is indirectly affected by input uncertainty parameters
${Y}_{n,b,p}^h$	Equivalent admittance associated with PHPF p of type n on bus b of the distribution network for harmonic order h [℧or p.u.]
α n, b, p	Binary variable, which is 0 if there is no PHPF p of type n on bus b of the distribution network; 1 if there is a PHPF p of type n on bus b of the distribution network
β m, b, a	Binary variable, which is 0 if there is no AHPF a of type m on bus b of the distribution network; 1 if there is an AHPF a of type m on bus b of the distribution network
${\varpi}_{n,b,p}^{\mathrm{PHPF}}$	Nominal capacity of the capacitance of PHPF p of type n on bus b of the distribution network [μF or p.u.]
${\varpi}_{m,b,a}^{\mathrm{AHPF}}$	Nominal capacity of AHPF a of type m on bus b of the distribution network [kA or p.u.]
δ k, 1	Location of the first concentration of probabilistic parameter k
δ k, 2	Location of the second concentration of probabilistic parameter k
λ k, 3	Skewness of probabilistic parameter k
ρ k, 1	Probability of the first concentration of probabilistic parameter k
ρ k, 2	Probability of the second concentration of probabilistic parameter k
x k, 1	First concentration point of probabilistic parameter k
x k, 2	Second concentration point of probabilistic parameter k
μ X, k	Mean value of vector Xk
σ X, k	Standard deviation of vector Xk
η b, a	Correction coefficient related to AHPF a on bus b of the distribution network
ζ h	Harmonic attenuation coefficient for harmonic order h
ϕ b	Phase shift of the voltage on bus b of the distribution network with respect to the voltage phase angle of the slack bus [rad]
ϕ b	Phase angle of the RMS voltage on bus b of the distribution network [rad]
${\phi}_{\widehat{b}}$	Phase angle of the RMS voltage on bus $\widehat{b}$ of the distribution network [rad]
${\phi}_{b^{\prime }}$	Phase angle of the RMS voltage on bus b′ of the distribution network [rad]
χ n, b, p	Tuned frequency of PHPF p of type n on bus b of the distribution network
γ n, b, p	Quality coefficient of PHPF p of type n on bus b of the distribution network
ω	The angular frequency [rad/s]
ΔCn, b, p	Variation of the capacitance of PHPF p of type n on bus b of the distribution network due to perturbation problems [μF or p.u.]
ΔD(u)	Corrective vector at iteration u of the LBNRPF methodology
Δf1	Variation of the principal frequency of the distribution network due to perturbation problems [Hz]
ΔIh	Additional harmonic current coefficient vector arising from the installation of different types of AHPFs on different buses of the distribution network for harmonic order h
$\Delta {I}_m^h$	Additional harmonic current coefficient vector arising from the installation of AHPFs of type m on different buses of the distribution network for harmonic order h
ΔLn, b, p	Variation of the inductance of PHPF p of type n on bus b of the distribution network due to perturbation problems [mH or p.u.]
ΔRn, b, p	Variation of the resistance of PHPF p of type n on bus b of the distribution network due to perturbation problems [Ω or p.u.]
ΔPb	Active mismatch power on bus b of the distribution network [MW]
ΔQb	Reactive mismatch power on bus b of the distribution network [MVAr]
ΔW(u)(D(u))	Mismatch power vector at iteration u of the LBNRPF methodology
ΔWb	Element b of the mismatch power vector or mismatch power on bus b of the distribution network
ΔYh	Additional admittance matrix resulting from the installation of various types of PHPFs on different buses of the distribution network for harmonic order h
$\Delta {Y}_n^h$	Additional admittance matrix resulting from the installation of PHPFs of type n on different buses of the distribution network for harmonic order h
$\Delta {Y}_1^h$	Additional admittance matrix resulting from the installation of the fifth second-order series resonant band-pass PHPFs on different buses of the distribution network for harmonic order h
$\Delta {Y}_2^h$	Additional admittance matrix resulting from the installation of the seventh second-order series resonant band-pass PHPFs on different buses of the distribution network for harmonic order h
$\Delta {Y}_3^h$	Additional admittance matrix resulting from the installation of the second-order damped high-pass PHPFs on different buses of the distribution network for harmonic order h
ΔZh	Additional impedance matrix resulting from the installation of various types of PHPFs on different buses of the distribution network for harmonic order h
$\Delta {Z}_n^h$	Additional impedance matrix resulting from the installation of PHPFs of type n on different buses of the distribution network for harmonic order h
$\Delta {Z}_1^h$	Additional impedance matrix resulting from the installation of the fifth second-order series resonant band-pass PHPFs on different buses of the distribution network for harmonic order h
$\Delta {Z}_2^h$	Additional impedance matrix resulting from the installation of the seventh second-order series resonant band-pass PHPFs on different buses of the distribution network for harmonic order h
$\Delta {Z}_3^h$	Additional impedance matrix resulting from the installation of the second-order damped high-pass PHPFs on different buses of the distribution network for harmonic order h
ΔZbus	Variation of the impedance of the distribution network due to perturbation problems [Ω]
$\Upsilon \left({\vartheta}_{n,b,p}^{\mathrm{PHPF}}\right)$	Cost function of PHPF p of type n on bus b of the distribution network [M$]
$\Upsilon \left({\vartheta}_{m,b,a}^{\mathrm{AHPF}}\right)$	Cost function of AHPF a of type m on bus b of the distribution network [M$]

Appendix 3: Input Data

Table 7.24 Data for the transmission lines of the modified IEEE 18-bus distorted test network

Table 7.25 Bus data of the modified IEEE 18-bus distorted test network

Table 7.26 Predicted harmonic contents of the nonlinear loads in the modified IEEE 18-bus distorted test network

Table 7.27 The characteristic coefficients associated with the cost of the PHPFs in the modified IEEE 18-bus distorted test network

Table 7.28 The characteristic coefficients related to the cost of the AHPFs in the modified IEEE 18-bus distorted test network

Table 7.29 The characteristic coefficients related to the cost of the AHPFs in the modified IEEE 18-bus distorted test network

Table 7.30 Maximum permissible values of the individual harmonic distortion of voltage and current along with the maximum admissible values of the THDV and THDI in accordance with the ANSI and IEEE 519-1992 standards

Table 7.31 Other required information in the modified IEEE 18-bus distorted test network

Table 7.32 Parameter adjustments of the multi-objective SOSA

Table 7.33 Data for the transmission lines of the modified 34-bus distribution test network

Table 7.34 Bus data for the modified 34-bus distribution test network

Table 7.35 Other required information in the modified 34-bus distribution test network

Table 7.36 Parameter adjustments of the multi-objective TMS-EMSA

Table 7.37 Parameter adjustments of the multi-objective continuous/discrete TMS-MSA

Table 7.38 Parameter adjustments of the multi-objective SS-HSA

Table 7.39 Parameter adjustments of the multi-objective SS-IHSA

Table 7.40 Parameter adjustments of the NSGA-II