WO2015124177A1

WO2015124177A1 - Method and system for calculating a fault indicator indicating a fault in a distribution network

Info

Publication number: WO2015124177A1
Application number: PCT/EP2014/053205
Authority: WO
Inventors: Izudin Dzafic
Original assignee: Siemens Aktiengesellschaft
Priority date: 2014-02-19
Filing date: 2014-02-19
Publication date: 2015-08-27

Abstract

An embodiment of the invention relates to a system (10) for calculating a fault indicator indicating a fault in a distribution network that comprises network sections with different phase numbers, said network sections being separated by at least one phase transition node. The system (10) is characterized by a computer (11) being programmed to carry out the steps of: forming Fortescue-π-Equivalents for every branch element of the distribution network and every phase transition node of the distribution network, determining a system admittance Fortescue matrix ( Y ^F) based on said Fortescue-π- Equivalents, and calculating said fault indicator based on said admittance Fortescue matrix.

Description

Method and system for calculating a fault indicator indicating a fault in a distribution network

The invention relates to a method and a system of calculating a fault indicator which indicates a fault in a distribution network . Background of the invention

U.S. Patent 8,274,294 discloses a method for detecting a ground fault in an electrical supply system. The method in^¬ cludes using a combination of components and their connec^¬ tions within the supply system to form virtual components, allowing presetting of fault indicators for incoming and outgoing supply lines to individual components.

Objective of the present invention

An objective of the present invention is to provide a method and system which allows calculating fault indicators for dis^¬ tribution networks, which comprise network sections with dif^¬ ferent phase numbers, with exceptionally high calculation speed . Brief summary of the invention

An embodiment of the invention relates to a method of calcu^¬ lating a fault indicator indicating a fault in a distribution network, wherein said fault indicator is calculated for a distribution network that comprises network sections with different phase numbers, said network sections being sepa^¬ rated by at least one phase transition node, said method com^¬ prising the steps of:

- forming Fortescue-7i;-Equivalents for every branch element of the distribution network and every phase transition node of the distribution network,

- determining a system admittance Fortescue matrix based on said Fortescue-7i;-Equivalents , and - calculating said fault indicator based on said admittance Fortescue matrix.

An advantage of this embodiment is based on the fact that a Fortescue transformation allows transforming any unbalanced set of n phasors into n -1 balanced n- phase system of dif^¬ ferent phase sequences and one zero phase sequence system. On the basis of a balanced phase system, the calculation of faults indicators is less time-consuming than on the basis of an unbalanced phase system.

According to preferred embodiment, for at least one node of the distribution network, an Fortescue-Equivalent current vector is calculated based on said admittance Fortescue ma- trix, said Fortescue-Equivalent current vector forming said fault indicator.

The method preferably further comprises the steps of deter^¬ mining an impedance Fortescue-Equivalent matrix of the re- spective node, and a Fortescue-Equivalent voltage vector of the respective node based on said system admittance Fortescue matrix, for said at least one node of the distribution net^¬ work. The Fortescue-Equivalent current vector may then be calculated by multiplying the impedance Fortescue-Equivalent matrix and the Fortescue-Equivalent voltage vector.

Further, a Right Hand Side vector may be formed which con^¬ tains zeros for all network nodes of the distribution network except for an injection node of the distribution network, and a voltage vector of the injection node in the Fortescue do^¬ main, as defined by the following equation:

wherein Ι_ζ designates the Right Hand Side vector and vf des^¬ ignates the voltage vector of the injection node in the

Fortescue domain. The system admittance Fortescue matrix is preferably factor- ized into a lower triangular matrix and an upper triangular matrix : v^F ₌ v^F v^F wherein redesignates the system admittance Fortescue matrix, Y_^F _L designates the lower triangular matrix, and designates the upper triangular matrix.

A pre-fault voltage vector of said at least one node and a voltage vector of said at least one node after occurrence of the fault are preferably determined by first solving the equation:

and by assuming

F _yF

—k, fault —'pre wherein Y__pre describes the pre-fault voltage of said at least

F

one node and v_{k ault} describes the voltage of said at least one node after occurrence of the fault.

Moreover, load admittance matrices may be determined in the Fortescue domain for each load in the distribution network. The load admittance matrices may be incorporated in the sys- tern admittance Fortescue matrix.

Based on the fault indicator, a decision may be taken as to whether and to which extent the distribution network needs to be switched off. The fault indicators may also be used to locate the fault. To this end, fault indicators may be calculated for each node of the distribution network based on the system admittance Fortescue matrix. Then, faulty network nodes may be deter- mined based on the calculated fault indicators.

Alternatively, the fault location and the faulty network node may be determined using a different fault location method. In this case, the fault indicator may be calculated thereafter for the faulty network node based on said admittance Fortes^¬ cue matrix in order to obtain further information about the nature of the fault and in order to facilitate the decision process regarding the decision as to whether and to which extent the distribution network needs to be switched off.

A further embodiment of the invention relates to a system for calculating a fault indicator indicating a fault in a distribution network that comprises network sections with different phase numbers, said network sections being separated by at least one phase transition node, wherein a computer is pro^¬ grammed to carry out the steps of:

- determining a system admittance Fortescue matrix based on said Fortescue-7i;-Equivalents , and

- calculating said fault indicator based on said admittance Fortescue matrix. Regarding the advantages of this embodiment, the explanations above apply mutatis mutandis.

The system is preferably configured to generate switch-off command signals for switching off the distribution network or parts thereof based on said fault indicator.

Brief description of the drawings In order that the manner in which the above-recited and other advantages of the invention are obtained will be readily un^¬ derstood, a more particular description of the invention briefly described above will be rendered by reference to spe^¬ cific embodiments thereof which are illustrated in the ap^¬ pended drawings. Understanding that these drawings depict only typical embodiments of the invention and are therefore not to be considered to be limiting of its scope, the inven^¬ tion will be described and explained with additional speci^¬ ficity and detail by the use of the accompanying drawings in which

Figure 1 shows a three-phase to two-phase lateral phase transition node,

Figure 2 shows a three-phase to single phase lateral

phase transition node,

Figure 3 shows a two-phase to single phase lateral phase transition node,

Figure 4 shows an exemplary embodiment of a Fortescue π equivalent, and

Figure 5 shows an exemplary embodiment of a system capa^¬ ble of calculating fault indicators for distribution networks that comprise network sections with different phase numbers.

Detailed description of the preferred embodiment

The preferred embodiment of the present invention will be best understood by reference to the drawings, wherein identi^¬ cal or comparable parts are designated by the same reference signs throughout.

It will be readily understood that the present invention, as generally described and illustrated in the figures herein, could vary in a wide range. Thus, the following more detailed description of the exemplary embodiments of the present in^¬ vention, as represented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of presently preferred embodiments of the in^¬ vention .

I. Fortescue transformation theorie:

In a seminal paper [1], Fortescue proved that any unbalanced set of n phasors can be transformed into n-1 balanced n-phase system of different phase sequences and one zero phase se^¬ quence system by the following transformation (1) .

where v^o, _l a⁷¹'¹ : voltages in Fortescue coordinates,

v^, : voltages in phase coordinates.

Because of the non-singularity of the transformation it is also proved that the quantities from Fortescue domain to phase domain can be recovered through an inverse n x n trans^¬ formation. In the following paragraphs, the transformation matrices for the special cases n = 3, n = 2 and n = 1 will be briefly reviewed.

__ph3 into

Fortescue coordinates using a₃ = β^^2π/3. The transformed quan- tities (voltage, current) are recovered in phase domain

rppk3

through the inverse transformation

:

Electric current through a three phase element is given by: / - V V

—ph₃ —ph₃—ph₃ (3)

Transforming voltage and current into Fortescue domain, can be rewritten as:

Three phase admittance matrix is transformed into Fortescue domain of order 3 as shown in (4) .

The Fortescue transformation (4) when applied to symmetrical three phase element (stationary or rotating) results in a 3 3 diagonal matrix. The two phase system in Fortescue coordi^¬ nates will only have 0 and 1 sequences. Here, because of n = 2, the operator ₂ = β^^2πΙ2 =—1. The respective transformation matrices are:

Two phase admittance matrix is transformed into Fortescue do^¬ main of order 2 as shown in (6) . ∑_F?= ¾∑_phl7 *² ,/ph₂ e {ab,be,ca} (6)

The admittance matrix in case of a two phase element is:

Substituting the expression for Y_ph₂ from (7) into (6) re^¬ sults in ( 8 ) . y__s + ¾η

(8) o y_s n

In comparison with the phase coordinate system, nothing changes when transforming one phase system admittance into Fortescue coordinate. The transformation matrix for single phase system is a 1 x 1 identity matrix. This is because the value of the operator a_x = e^y'2r/1 , which in this case equals to 1.

Fortescue π Equivalents

A. Three-phase to two-phase lateral:

Let us consider a network segment as shown in Fig. 1. It shows two buses i and / and three lines L_n!, L_!; and. L- .The phase types are shown in Fortescue and in phase coordinates.

The voltages at buses i and j are shown in Fortescue coordi^¬ nates and marked as V_0l2 and V_m , respectively. The bold two- phase line L_;; is connected to the three phase bus i at which line phase and bus phase types do not match. For all remain^¬ ing elements in this example there is no mismatch between the bus and equipment phase type. Bus i is a three-phase bus which requires Fortescue domain of third order F₃. Bus j is a two-phase bus which requires

Fortescue domain of second order F₂. In order to apply

Kirchhoff' s current law, all currents at bus i must be formu^¬ lated in the same Fortescue order, i.e. F₃ in this case. Al- though line L_;; is two-phase line, its current contribution at bus i side needs to be calculated in domain F₃. We will now derive the Fortescue π equivalent described in (9) and already introduced in the previous section.

All four sub-matrices will be derived directly, i.e. bypass^¬ ing the fictitious node introduced in the previous section, from the current equations of the branch π equivalent in the phase domain.

1) Ya sub-matrix:

The Fortescue equivalent sub-matrix is therefore given by:

T~'-^3 r b rplb

—ii — ab—ii —F^ (10)

Equation (10) can be written in general form as:

Xl- =T!l^²T^,_Ph₂^{ab,bc,ca) (11)

The following Table (hereinafter referred to as Table I) shows all possible phase combinations for a two phase line connected between three and two phase buses. Table I shows

_F and L·pL· sub-matrices for different phase types.

1 1 1 1 1 1 1

ab

1 | a₃ 3 1 ₃ £3.

1 3 ₃ 1 1 ₃

be

-matrix :

Equation (12) can be written in general form as:

from (5) .

J-j_j sub-matrix

The Fortescue equivalent sub-matrix is therefore given by: -ab ,ab

∑ji ³=T;¾¾ (1 )

Equation (14) is written in general form as:

m (5) .

4) sub-matrix:

This (self admittance) sub-matrix is not affected by the map- ping at bus i and can therefore be derived directly from (6)

B. Three-phase to single-phase lateral:

This case is illustrated in Fig. 2. Bus i is three phase and bus j is single phase. A special Fortescue transformation is again required for this branch between two different Fortes^¬ cue domains .

Similarly to the previous case, the Fortescue π equivalent for this case can be written as follows:

L 1x3 lxl -1

The sub-matrix terms of this equation are derived in the fol^¬ lowing sub-sections.

1) sub-matrix:

The Fortescue equivalent sub-matrix is therefore given by:

Equation (17) can be written in the general form as:

The following table (hereinafter referred to as Table II) shows all possible combinations for a single phase line con nected between three and single phase buses. Table II shows

-F, and -I sub-matrices for different phase types

2 ) ^ ij sub-matrix:

The Fortescue equivalent submatrix is therefore given by:

= T_a ³Y_ijT_R (19) Equation (19) can be written in the general form as:

^^F' = T^^ T^,ph^{a,b,c } ₍₂₀₎

Where —p^ is obtained from Table II and T_p ^hl = [ljVp/j_j .

3) ¹ β sub-matrix:

The Fortescue equivalent submatrix is therefore given by:

Equation (21) can be written in the general form as:

Where —F₃ is obtained from Table II and T_p^

4) jj sub-matrix:

This (self admittance) sub-matrix is not affected by the map- ping at bus i and can therefore be derived directly from the phase domain.

C. Two-phase to single-phase lateral:

This case is illustrated in Fig. 3 where bus i is two phase and bus j is single phase. A special Fortescue transforma^¬ tion is again required for this branch between these two dif ferent Fortescue domains. Similarly to the previous cases, the Fortescue π equivalent for this case can be written as follows:

V ^i<2

1) 1 a sub-matrix: This sub-matrix can be expressed m the general form as :

The following table (hereinafter referred to as Table III) shows all possible combinations for a two-phase branch con^¬ nected between two and single phase nodes. Table III shows ph

F and ¹ ph_l sub-matrices for different phase types:

01(c ) c [1 -l]

01(c ) a [1 1]

2) !y sub-matrix:

This sub-matrix can be expressed in the general form as:

= T% Y* T* ^ {aAc } ,2

Where ^ ph_x is obtained from Table III and p ^hi

"F^F,

3) ¹ jt sub-matrix:

This sub-matrix can be expressed in the general form as:

Where ¹ is obtained from Table III and = [l]V/?/z

4) ¹ jj sub-matrix: This (self admittance) sub-matrix is not affected by the mapping at bus i and can therefore be derived directly from the phase domain.

D. Special transformer connections

Transformer modelling using symmetrical components is ex^¬ plained in [2] . The π equivalent in Fortescue domain for open-wye and open-delta type transformer deserves a careful treatment. Typically, this transformer is used to supply three phase load connected to two phase lateral. Phase domain π equivalent and dimensions of corresponding sub-matrices is shown in {21) .

are calculated from corresponding phase domain sub-matrices using (6) and (4), respectively. The other two sub-matrices are calculated using a similar approach to that used for three to two phase con^¬ nection, as explained in subsection II-A.

III. Generalized Fortescue Equivalent: The generalized nodal current injection equation for A: -phase branch connected between m -phase node i and n -phase node j , where m ≥ n ≥ k as shown in Fig. 4 is:

where all the quantities are in Fortescue coordinates of propriate order. Admittance sub-matrices are calculated

(29) . yF(p)-F(q) _ j,F(p)yph_k , ph

L·pq — L_Pph_hk; ^'—Lp_Pq;—LF_M,p,q { j (29)

Few remarks are appropriate here:

- Dimensions of sub-matrices Y_^^P^^~F^ are defined by the num- ber of phases at nodes p and q .

- Phase domain π branch equivalent is taken as input.

- Dimensions of these sub-matrices are defined by the number of phases of the branch (k) .

- The self and mutual admittance matrices of each node are diagonal except for PTNs .

- When the nodal admittance matrix of the entire system is formed, it requires less storage compared to its phase counterpart and LU factorization is therefore faster.

- For branches connected to PTN nodes or special transformer connections, appropriate transformation matrices as illus^¬ trated throughout sections I and II are necessary.

IV. Fault Analysis: The generalized Fortescue approach can be applied to fault analysis based on any matrix power flow method. In this sec^¬ tion, application of the generalized Fortescue approach for fault analysis based on current mismatches (iterations) method is presented. Generally, fault represents a structural network change that can be modeled as additional fault imped^¬ ance at the faulted node. According to Thevenin' s method, changes in the network caused by addition of fault impedance are equivalent to those caused by added voltage source and fault impedance at fault node with all other voltage sources grounded. Voltage of the added source is equal to the pre- fault voltage of the faulted node. Fault voltages are calcu^¬ lated as sum of prefault voltages and voltage changes ob^¬ tained from Thevenin' s network. Prefault voltages are result of current mismatches power flow method applied to the radial network. An exemplary embodiment of a general power flow algorithm is explained in the following:

Step 1 : Read input data and detect PTNs .

Step 2 : Create Fortescue π equivalents for every branch ele- ment (line, transformer,...). If elements are not connected to PTN use transformations (2) and (5) for three-phase and two phase nodes, respectively. In case of single phase ele^¬ ments phase and Fortescue equivalents are equal. If elements are connected to PTN then Fortescue π equivalents are done according to Section II.

Step 3 : Convert all loads to constant impedance Foretescu π equivalents using the following steps: Step 3a: Calculate phase impedances using given loads per phase and nomival phase-to-ground voltage:

Step 3b: Formulate load admittances for three phase loads a 0 0

0 0 (30) 0 0 1/Z_C\ and two phase loads

0 1/Z_b (31)

Using (2) and (5) convert them to Fortescue domain. Single phase loads are equal in phase and Fortescue domains. Step 4 : Build system admittance Fortescue matrix Y_ using every element in the network. Slack node has index 1.

Factorize Y_ into Y__L and

Step 5 : Take measurements for the slack (injection) node. If there are no measurements at the slack node, assume its node voltage is vf = [1 0 0] , vf = [1 0] , or vf = [1] depending on the number of phases of the slack node.

Step 6 : Factorize Y^F =Y_^F _L

Step 7 : Prepare Right Hand Side (RHS) vector which con^¬ tains zeros for all nodes except for injection (slack) node:

F F F F

Step Calculate pre-fault voltages: Solve Y__L _uY-_pre ⁼ —o for · Form voltage vector for the faulted node k . ¥-k,fault ~ ¥-'pre(Jt) (34)

Note: the dimension of V_{k auh} is number of phases of node k p

Step 9 : Calculate Thevenin' s Fortescue Equivalent z_m for the node on which fault has occurred. a) Three phase fault node:

and form r F F F

00 01 02

F „F F

±JH 10 ll 12 (36)

F F F

L 20 21 22

Two phase fault node

and form

00 ^01

±JH F (38;

10 7l^Fl c) Single phase fault node

Step 10 : Calculate fault currents

(41)

7^F

where: TH ^1S obtained from (36), or (40); ∑_k,fault is obtained from (34) . I k, fault represents a fault indicator that may indicate a fault in the distribution network.

V. Exemplary embodiment of a system for calculating a fault indicator :

Figure 5 shows an embodiment of a system 10 for calculating a

.F

fault indicator k, fault which indicates a fault (e. g. in form of a fault location) in a distribution network. The distribution network that is not shown in Figure 5 for the purpose of clarity, comprises network sections with different phase num^¬ bers. The network sections are separated by phase transition nodes as explained in detail above.

The system 10 comprises a computer 11 and a memory 12. The memory stores a software module SPM which programmes the com^¬ puter 11 to carry out the steps of: - forming Fortescue-7i;-Equivalents for every branch element of the distribution network and every phase transition node of the distribution network,

- calculating said fault indicator based on said admittance Fortescue matrix.

In addition, the computer 11 is preferably programmed to gen- erate switch-off command signals Soff for switching off the distribution network or parts thereof based on the fault in-

.F

dicator i__{k ault} . VI. Conclusion:

The delivery of smart distribution grid operation requires fast computation and time critical automation. Applications of distribution protection demands real-time fault analysis oriented towards network operation rather than planning.

Fault analysis in phase coordinates is a good method, but for systems with many single and two phase laterals a more effec^¬ tive algorithm such the generalized Fortescue method as pre^¬ sented above is advantageous. Using the proposed algorithm in Fortescue coordinates of dimension 1, 2 and 3, the properties of symmetrical components may be extended to power systems with one and two and three phase components. The main advan^¬ tage of the approach presented above is the increase in speed compared to standard phase coordinates method. Proposed tech^¬ nique holds great potential for real time control of distri- bution network. Nomenclature :

Explicit suffix for symmetrical components.

Explicit suffix for phase coordinates.

Voltage and current complex vectors.

Complex element of a vector or matrix.

Complex Fortescue operator a_n = e^j2n:ln .

Nodal admittance sub-matrices in n-Fortescue main for node i

Branch admittance sub-matrices in n-Fortescue do^¬ main for an ij branch.

Fortescue transformation from m-Fortescue domain to n-phase domain.

Fortescue transformation from m-phase domain to n

Fortescue domain.

k^th sequence element or sequence voltage vector.

Current of node i in n-Fortescue domain. Fortescue domain of node i.

PL Voltage magnitude for ph x Voltage angle for phase

Literature : [1] C. L. Fortescue, "Method of symmetrical co-ordinates ap^¬ plied to the solution of polyphase networks," Trans, of the American Institute of Electrical Engineers, vol. 37, no. 2, pp. 1027-1140, June 1918. [2] H. T. Neisius and I. Dzafic, "Three-phase transformer modeling using symmetrical components," Innovative Smart Grid Technologies (ISGT), 2011 IEEE PES, pp. 1-6, Jan. 2011.

Claims

1. Method of calculating a fault indicator indicating a fault in a distribution network,

characterized in that

said fault indicator is calculated for a distribution network that comprises network sections with different phase numbers, said network sections being separated by at least one phase transition node, said method comprising the steps of:

- determining a system admittance Fortescue matrix ( Y_^F )

based on said Fortescue-7i;-Equivalents , and

- calculating said fault indicator based on said admittance Fortescue matrix.

2. Method of claim 1,

further characterized by

for at least one node of the distribution network, calculat-

.F

ing an Fortescue-Equivalent current vector i lk, fault ) based on said admittance Fortescue matrix ( Y_^F ) , said Fortescue-

.F

Equivalent current vector i hk, fault ) forming said fault indica^¬ tor .

3. Method of claim 2,

further characterized by

- for said at least one node of the distribution network, determining an impedance Fortescue-Equivalent matrix ( z^F _TH ) of the respective node, and a Fortescue-Equivalent voltage

F

vector (∑k, fault ) °f the respective node based on said system admittance Fortescue matrix, and

- calculating said Fortescue-Equivalent current vector (

.F

Ik, fault ) by multiplying the impedance Fortescue-Equivalent

F

matrix ( z_m ) and the Fortescue-Equivalent voltage vector (

F

—k, fault ) ·

4. Method according to any of the preceding claims,

characterized in that

a Right Hand Side vector is formed which contains

- zeros for all network nodes of the distribution network except for an injection node of the distribution network, and

- a voltage vector of the injection node in the Fortescue domain,

as defined by the following equation:

wherein I_^F designates the Right Hand Side vector and vfdes- ignates the voltage vector of the injection node in the Fortescue domain.

5. Method according to any of the preceding claims,

characterized in that

said system admittance Fortescue matrix ( Y_^F ) is factorized into a lower triangular matrix and an upper triangular matrix :

Y^F = Y^F Y^F wherein redesignates the system admittance Fortescue matrix, Y_^F designates the lower triangular matrix, and redesignates the upper triangular matrix.

6. Method of claim 5,

characterized in that

a pre-fault voltage vector of said at least one node and a voltage vector of said at least one node after occurrence of the fault are determined by

- first solving the equation: i Y-^FLiY-^FU!V—^Fpre =—I^Fo f^ro^or^r —V—^Fpre - and by assuming

— ^Fk, fault -—V^F'pre - wherein Y__pre describes the pre-fault voltage of said at

F

least one node and Y_kjault describes the voltage of said at least one node after occurrence of the fault.

7. Method according to any of the preceding claims,

further characterized by

- determining load admittance matrices in the Fortescue do^¬ main for each load in the distribution network and

- incorporating said load admittance matrices in the system admittance Fortescue matrix.

8. Method according to any of the preceding claims,

further characterized by

based on said fault indicator, making a decision as to whether and to which extent the distribution network needs to be switched off.

9. Method according to any of the preceding claims,

characterized in that

- in case of a fault, the fault location and the faulty net- work node is determined using a fault location method, and

- said fault indicator is calculated thereafter for the

faulty network node based on said admittance Fortescue ma^¬ trix .

10. Method according to any of the preceding claims,

characterized in that

- fault indicators are calculated for each node of the dis^¬ tribution network based on said system admittance Fortes^¬ cue matrix, and

- faulty network nodes are determined based on the calcu^¬ lated fault indicators.

11. System (10) for calculating a fault indicator indicating a fault in a distribution network that comprises network sections with different phase numbers, said network sections be^¬ ing separated by at least one phase transition node,

characterized by

a computer (11) being programmed to carry out the steps of:

- determining a system admittance Fortescue matrix ( Y_^F )

based on said Fortescue-7i;-Equivalents , and

- calculating said fault indicator based on said admittance Fortescue matrix.

12. System of claim 11,

characterized in that

said system (10) is configured to generate switch-off command signals for switching off the distribution network or parts thereof based on said fault indicator.