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الموضوع: Formation of Zbus

  1. #1
    Junior Engineer الصورة الرمزية سمير الرفاعي
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    Oct 2006
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    Formation of Zbus

    Formation of Zbus


    2.1 Introduction:
    Power systems are, in general, very large. They contain linear, bilateral network of impedances (or admittances), which are interconnected in some specified way at various points called nodes (or buses). The number of these nodes (or buses) may be of the order of several hundreds.
    The modern digital computer has a distinct advantage in that it provides a mean for solving such large networks accurately. Therefore a systematic procedure suitable for digital computer calculation is necessary. System studies can be carried out using [Ybus]or [Zbus].
    [Ybus] is used for load flow computation. For fault calculations (also known as short circuit studies) the use of Zbus is preferable. As compared to the load flow studies, the short circuit studies are simpler because a direct solution is possible and no iterations are required. In order to solve a large power system, it is necessary to organize the system impedance data the network topology data in such a way that this information may be conveniently introduced and stored in the computer memory. The next step is to choose the appropriate method of solving the problem (either Ybus or Zbus, method).
    In the present work a computer program is developed using MATLAB package to perform the fault calculations for Iraqi national grid.
    2.2 Computer Solution Methods Using the Impedance Matrix:
    The conventional admittance matrix method which is generally used to compute fault currents in power system, although simple to implement; it has certain disadvantages in its application to large networks. The impedance matrix, more difficult to derive, has certain advantages for fault computations. This is primarily due to the impedance matrix being an "open circuit" network de******ion, and this coincides with the open circuit approximation usually used in fault studies. The following sections will give the theoretical background of impedance matrix methods for use in fault studies. Also an algorithm for finding the impedance matrix, which is more direct and simpler to implement than performing an inversion of admittance matrix, is given.
    2.3 Impedance Matrix in Shunt Computation:
    Consider the network shown in Figure 3.1 where the network could be the positive, negative or zero sequence network .We arbitrarily define all currents to be entering the network at nodes l, 2,...,i,...n and all voltages to be the voltage drops from each node to the reference. Hence for ith bus, the voltage of the bus (with respect to the reference) is Viand the entering current to the ith bus is Ii.
    From the network theory, this network can be described by the following system of equations:
    Network

    V

    i

    n

    reference











    Figure 2.1 Network Circuit.
    ……………(2.1)

    Or simply Vbus = Zbus Ibus ........................................ ....................(2.2)
    The general entry Zik of Zbus can be obtained from the equation:
    …………………………… (2.3)
    where Ik ≠ 0
    It is possible to calculate Zik from eq. (3.3). Terminate all buses in open circuit except the K th bus that is terminated in a current source of strength Ik. Solve for Vi and calculate Zik. Especially if the current injected in bus k is 1.0 p.u then |Vi|=Zik. However this method is not suitable because calculation of Vi is generally very difficult. The following step-by-step method is very simple and suitable for computer work.
    2.4 Algorithm for Formulating [Zbus]:
    The Algorithm of formulation (Zbus) is described in terms of modifying an existing bus impedance matrix designated as (Zbus)old .The new modified matrix is designated as (Zbus)new. The network consists of a reference bus and a number of other buses. When a new element having self-impedance Zs is added, a new bus may be created (if the new element is a tree branch) or a new bus may not be created (if the new element is a link). Each of these two cases can be subdivided into two cases so that Zs may be added in the following ways:

    1 .Adding Zs from a bus to reference
    2. Adding Zs form a new bus to an old bus
    3. Adding Zs from an old bus to reference
    4. Adding Zs between two old buses.
    In the following discussion I1 and Kk denote old buses and q denoted new bus. The bus impedance matrix is always diagonally symmetric i.e. Zik =Zki .

    2.4.1 Type 1 Modification (Addition of Tree Branch Zs from a New Bus q to Reference): -
    Figure 3.2 shows a branch. Zs connected between reference bus and a new bus q, for this new branch we can write:
    Vq =Zs Iq ........................................ ................................(2.4)
    On adding equation (2.4) to the set of equation (2.1) we get:
    ……………..(2.5)










    Network

    Vq

    i

    n

    reference

    Zs

    q

    Iq








    Figure 2.2 Modification Type 1.


    [Zbus] old is an n x n matrix. The addition of the new branch means the
    addition of an (n+1)th row and (n+l)th column. As seen form equation (2.5)
    Ziq =Zqi =0 for i=l,2,..i,...n.
    Zqq=Zs
    [Zbus]new can therefore, be written as:
    ….....................(2.6)






    2.4.2 Type 2 Modification (Addition of a Tree Brunch Zs from a New Bus q to Old Bus k): -
    Figure 3.3 shows this modification. The equation for the new brands is:
    Vq=Zs Iq + Vk
    = Zs Iq + Zk1 I1 +……+ Zk2 I2+….+ Zkk (Ik+Iq)+….+ Zkn In
    = Zk1 I1+ Zk2 I2+..….+ Zkk Ik+….+ Zkn In+(Zkk+Zs) Iq ………(2.7)

    Network

    i

    n

    reference

    Zs

    q

    Ik

    k Ik+ Iq










    Figure 2.3 Modification Type 2.

    The equation for V1 (i.e. first equation (3.1)) can be written, as under keeping in mind that new current at kth bus is Ik +lq.

    V1=Z11I1+Z12I2+...+Z1k(Ik+Iq)+...+Z1nIn
    =Z11I1+Z12I2+...+Z1kIk+...+Z1nIn+Z1kIq.. ......…………………(2.8)

    The equation for V2,V3,...Vn also get modified in a similar manner. When equation (2.7) is added to the modified equations for V1, V2,...Vn . The complete new set of equations is given by:
    ……………....(2.9)

    ………....(2.10)

    2.4.3 Type 3 Modification (Addition of a Link Zs Between an old Bus k and Reference): -
    Referencing to Figure 3.4. This case can treated as an extension of type (2) modification. Initially let Zs connected between a new bus q and old bus k. and then let the new bus q be connected so that Vq=0 the new set of equitation is:







    Network

    i

    n

    reference

    Zs


    Iq








    Figure 2.4 Modification Type 3.

    ……….(2.11)

    If Iq is eliminated from equation (3.11) i.e. last row and last column of the above impedance matrix are eliminated (using matrix algebra) we have:



    2.4.4 Type 4 Modification (Addition of Link Zs between Two old Buses):
    Referring to Figure3.5. Equation for V1 can be written as:
    V1=Z11 I1+ Z12 I2+…+Z1i (Ii+Iq)+ Z1j Ij+Z1k(Ik-Iq)+…+Z1n In………(2.13a)
    Rearranging:

    Network

    1

    n

    reference

    Zs

    Ik

    Ii


    i

    k










    Figure 2.5 Modification Type 4.

    V1=Z11 I1+ Z12 I2+…+Z1n In+ Z1j Ij+(Z1i-Z1k)Iq……………..…(2.13b)
    Equation similar to equation (3.13b) can be written for all buses. It is to be noted that:
    Vk=Zs Iq +Vi..................................... ....................…….........(2.14a)
    Or Vk=Zk1 I1+ Zk2 I2+…+Zki (Ii+Iq)+ Zkk(Ik-Iq)+….
    =Zs Iq +Zi1 I1+ Zi2 I2+…+Zii (Ii+Iq)+Zij Ij+ Zik(Ik-Iq)…….(2.14b)
    Rearranging:
    0=(Zi1-Zk1)I1 +....+(Zii-Zki)Ii+(Zij-Zkj)Ij+(Zik-Zkk)Ik+....+(Zs+Zii-Zki+Zkk)Iq
    …………………………………………………………………(2.14c)
    Equation (2.13b) along with similar equations for V2,V3,....Vn and equation
    (2.14c) can be written in the following form:


    Elimination Iq from equation (3.15) by matrix algebra, we get:


    2.5 Formation [Zbus] from [Ybus ]: -
    Node equations systematic formulation of equations determined at the nodes of circuit by applying kirchhoff 's current law is the basis for some excellent computer solutions of power-system problem .In order to examine some features of these node equations the circuit of Figure 3.8 has been redrawn with some minor changes. The result of Figure 3.9. The series reactances have been combined for convergence, as was done in drawing the figure for writing the loop equations , and their series impedances have been replaced by equivalent current sources and shunt impedances . Admittance values shown for each branch are the reciprocals of the impedances shown in Figure 3.8 for the same branches. Single-sub****** notation will be used to designate the voltage of each bus with respect to the neutral taken as the
    reference node. Applying kirchhoff 's current law at node (1) with current into the node from the source equated to current a way from the node gives:

    Za

    Zb

    2





    Ze

    Eb

    Zk

    Zd

    Zi

    Zc

    Ec

    Ea

    Zf

    0

    4

    3

    1

    Figure 2.8 The Impedance Circuit for A Simple System.

    I1 = V1 Yf + (V1 – V2) Ya + (V1 – V4) Yc ……………….(2.17)
    And for node 2:
    0 = (V2 – V1)Ya + (V2 – V3)Yb + (V2 – V4)Ye ……….…….. (2.18)
    Rearranging these equations yields:
    I1 = V1 (Yf + Ya + Yc)- V2 Ya + V4 Yc ………………(2.19)
    0 = – V1 Ya + V2 (Ya + Yb + Ye) – V3 Yb – V4 Ye .….....….……..(2.20)
    Ya

    Yb

    2




    Ye

    Yd

    Yk

    Yi

    I4

    Yc

    Yf

    I1

    I3

    3

    0

    4

    1

    Figure 2.9 The Admittance Circuit for System of Figure 2.8

    Similar equations can be formed for nodes 3 and 4, and the four equations can be solved simultaneously for the voltages V1, V2, V3 and, V4. All branch currents can be found when these voltages are known, and so the required number of node equations is one less than number of nodes is the network.
    A node equation formed for the fifth node would yield no further information.
    At any node one product is the voltage of that node times the sum of all the admittances, which terminate on the node. This product accounts for the current that flows a way from the node if the voltage is zero at each other node . Each other product equals the negative of the voltage at another node times the admittance connected directly between the other node and the node at which the equation is formulated. For instance, at node 1 a product is - V2 Ya, for - V2 is the negative of the voltage at the node which is connected to node 1 by Ya . This product accounts for the current a way from node 1 when all node voltage are zero except that at node 2 .The standard form for the four independent node equations is:
    I1 = Y11V1 + Y12V2 + Y13V3 + Y14V4

    I2 = Y21V1 + Y22V2 + Y23V3 + Y24V4
    I3 = Y31V1 + Y32V2 + Y33V3 + Y34V4 ….……..(2.21)
    I4 = Y41V1 + Y42V2 + Y43V3 + Y44V4

    The symmetry of the equations in this form makes them easy to remember, and their extension to any number of nodes is apparent .The order of the sub******s is effect-cause, as in the standard form for loop equations. The admittances Y11 , Y22 , Y33 , and Y44 are called the self-admittances at the nodes , and each equals the sum of all the admittances terminating on the node identified by the repeated sub******s. The other admittances are the mutual admittances of the nodes, and each equals the negative of the sum of all admittances connected directly between the nodes identified by the double sub******s. For the network of Figure 2.9 the mutual admittance Y1 equals -Ya. Some authors call the self and mutual admittances of the driving-point and transfer admittances of the nodes.
    The general expression for the source current toward node K of a network having N independent nodes , that is, N buses other than the neutral, is:
    …………………………………………..…………..(2.22)

    For a network of four independent nodes:
    Y11 Y12 Y13 Y14

    Ybus = Y21 Y22 Y23 Y24 ……………(2.23)
    Y31 Y32 Y33 Y34
    Y41 Y42 Y43 Y44

    The inverse of the bus admittance matrix is called the bus impedance matrix and identified by Zbus by definition:
    …………………………………(2.24)

    And for a network of four independent nodes:

    Z11 Z12 Z13 Z14
    Zbus = Z21 Z22 Z23 Z24 ……………(2.25)
    Z31 Z32 Z33 Z34
    Z41 Z42 Z43 Z44

  2. #2
    Junior Engineer الصورة الرمزية eng.shahenaz
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    Apr 2009
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    رد: Formation of Zbus

    السلام عليكم


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  3. #3
    Senior Engineer الصورة الرمزية Ibrahim Nasrallah
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    Dec 2009
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    رد: Formation of Zbus

    شكرا جزيلا مهندس سمير
    في الواقع موضوع Zbus and Ybus من المواضيع المهمة جدا والتي يقوم عليها power system
    جزاك الله خيرا
    مشكوووووووووور
    أبو محمود
    لا إله إلا أنت سبحانك إني كنت من الظالمين

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