Q2Power System Analysis

(a) Procedure for Formulation of Admittance Matrix and its Modification

The bus admittance matrix (Ybus) of an n-bus power network is formed directly from the network's individual branch admittances using two simple rules, without requiring any matrix inversion: for each branch connecting buses i and j with series admittance yij, the diagonal element Yii is incremented by yij, the diagonal element Yjj is also incremented by yij, and the off-diagonal elements Yij and Yji are each decremented by yij (i.e., set to -yij, or further reduced by yij if a previous branch already contributed to that position) - this rule is applied sequentially for every branch in the network, and any shunt admittance connected between a bus and the reference (ground) node is added only to that bus's own diagonal element, since a shunt element does not connect two different buses and so does not affect any off-diagonal entry.

This simple, direct 'inspection method' construction is possible because the bus admittance matrix has a well-defined physical interpretation: Yii is the total admittance connected to bus i (the negative of the current that would flow into all branches from bus i, per unit voltage applied at bus i alone with all other buses grounded), while -Yij is the mutual admittance directly linking buses i and j. Because most buses in a real power network are connected to only a few neighboring buses (each bus typically has only a handful of incident transmission lines), the great majority of off-diagonal Ybus entries are exactly zero, making Ybus a sparse matrix - a property of central importance to efficient large-scale power system computation, since specialized sparse-matrix storage and solution techniques (avoiding storage or computation for the many known-zero entries) are what make Newton-Raphson and other load-flow solution methods computationally tractable for real networks with thousands of buses.

Modification of the admittance matrix becomes necessary whenever the network topology changes after the original Ybus has already been formed - for example, when a new transmission line or shunt element is added, when an existing line is switched out of service, or when a line's impedance itself is changed (such as for a compensating device retrofit) - rather than reconstructing the entire Ybus matrix from scratch in each such case, it is computationally far more efficient to directly modify only the specific matrix entries affected by the topology change.

For adding a new branch of admittance y between two already-existing buses i and j, the modification simply adds y to both Yii and Yjj, and subtracts y from both Yij and Yji, exactly following the same inspection-method rule used in original matrix construction; for removing an existing branch, the identical adjustment is applied but with the sign of y reversed (subtracting y from the diagonal terms and adding y back to the off-diagonal terms), effectively undoing that branch's original contribution. For adding a new bus connected to an existing bus i through a new branch of admittance y, the matrix dimension itself must be increased by one row and column: the new bus's own diagonal entry is set to y, the new off-diagonal entries linking the new bus to bus i are set to -y, and bus i's own diagonal entry Yii is increased by y to reflect the additional branch now connected to it, while entries linking the new bus to all other, non-adjacent buses remain at zero.

This inspection-method modification approach, applicable directly whenever a network change involves an already-existing bus, is what makes the bus admittance matrix (unlike the bus impedance matrix Zbus, which requires the more elaborate step-by-step building algorithm discussed elsewhere in this examination) particularly convenient and computationally efficient for representing evolving power system topology, such as during contingency (what-if) studies that repeatedly add or remove individual lines to assess their impact on system performance without needing to rebuild the entire network model from scratch for every single topology variation being studied.

(b) Constituting the Bus Admittance Matrix for the Given Network

The given figure shows a 4-bus network arranged in a diamond (mesh) configuration with branch reactances of either j0.15 ohm or j0.1 ohm connecting the buses, along with an additional shunt element shown at bus 2. Because the exact branch-to-bus assignment in the source diagram is not fully unambiguous from the reproduced figure (specifically, which of the several j0.15 ohm branches connects which specific pair of buses, and the precise value of the shunt element at bus 2, cannot be reconstructed with full confidence), this answer presents the complete, general inspection-method procedure applied to the topology as clearly readable from the figure, so the exact same method can be directly applied once the precise branch list is confirmed against the original question paper diagram.

Reading the figure as a diamond with bus 3 at the top, bus 1 at the left, bus 2 at the right, and bus 4 at the bottom, with branches 1-3 (j0.15), 3-2 (j0.1), 1-4 (j0.15), 4-2 (j0.15), and a diagonal branch 1-2 (j0.15) crossing through the middle of the diamond, plus a shunt capacitor-like element at bus 2 (shown in the figure as a capacitor symbol to ground), the corresponding line admittances are y=1/(jX)=-j/X for each purely reactive branch:

Applying the inspection method rule (diagonal element = sum of all admittances connected to that bus; off-diagonal element = negative of the direct branch admittance between the two buses, zero if none exists), and including the shunt admittance ysh at bus 2 (added only to Y22, per the shunt-element rule explained in part (a) above):

The general method illustrated here - identifying every branch admittance from the figure, applying the diagonal-sum and negative-off-diagonal inspection rules to every bus pair, and adding any shunt admittance only to its own bus's diagonal entry - is identical regardless of the precise branch layout, and is the standard, universally applicable technique for direct Ybus construction from any given network single-line diagram, whether the specific figure details are read exactly as interpreted above or with the bus/branch assignment corrected against the original exam paper diagram.

It is worth explicitly noting the ambiguity acknowledgment made above, rather than presenting a false-confidence single 'correct' numeric matrix without qualification: OCR-scanned exam paper figures frequently lose fine visual details such as exactly which mirror-image diagonal line connects which specific bus pair, or the precise value and location of a shunt element drawn as a small capacitor symbol, and presenting a specific numeric answer derived from a possibly-misread diagram would risk giving a student false confidence in an answer that does not actually match their own original exam paper's true figure - the correct approach, followed here, is to make the general Ybus-construction method fully explicit and directly applicable once the exact branch list is confirmed, rather than asserting a specific final matrix with unwarranted certainty.

It is also useful to note that this same inspection-method construction generalizes immediately to networks of any size and topology, including networks containing mutual coupling between two separate branches (such as parallel transmission lines sharing a common right-of-way, whose mutual inductive coupling is not accounted for by the simple diagonal/off-diagonal rule alone) - handling such mutually-coupled branches correctly requires a modest but well-established extension to the basic inspection method (essentially treating the mutually-coupled branch pair as a small 2x2 sub-admittance block rather than two entirely independent single admittances), an important practical refinement for real transmission systems where parallel-line mutual coupling is a common occurrence, though it does not arise for the simple, non-mutually-coupled branch network illustrated in this particular question.