Comprehensive Modeling of Three-Phase Distribution Systems via the Bus Admittance Matrix

I. INTRODUCTION

• The paper combines distribution-network component models to construct a three-phase Y-Bus, avoiding single-phase simplifications.The framework includes transmission lines, transformers, and step-voltage regulators.Source: §I · p. 1
• Unified Y-Bus models accounting for missing line phases and SVRs, and theoretical analysis of component effects on invertibility, are identified as gaps.The paper addresses these gaps with models and an invertibility analysis.Source: §I · p. 1
• The SVR models derive primary-to-secondary line-to-neutral voltage gain matrices from first principles and combine them with a series transmission line.They do not use sequence network matrices (as per) and can also be used in FB-Sweep.Source: §I · p. 1
• The paper explains that certain transformer self-admittances cause Y-Bus singularities, and proves invertibility for practical component combinations after small shunt admittances are added.Y-Bus invertibility is necessary for computing Z-Bus voltage solutions.Source: §I · pp. 1–2
• Online MATLAB scripts model components and build the Y-Bus for four feeders, with power-flow solutions within 0.75% of benchmark solutions.The feeders are IEEE 37-bus, IEEE 123-bus, 8500-node, and European 906-bus low-voltage systems.Source: §I · p. 2

II. MODELING NOTATIONS

• The network is represented as a graph whose nodes correspond to shunt elements and edges to series elements, with wye and delta nodes distinguished.A slack node represents the substation connection.Source: §II · p. 2

III. THREE-PHASE ZIP LOAD MODELS

• The load models represent wye and delta constant-power, constant-current, and constant-impedance behavior through voltage-dependent nodal current injections.The paper notes that ZIP approximates injection-current voltage dependencies and may not capture high reactive-power voltage sensitivity.Source: §III · pp. 2–3

IV. MODELING OF SERIES ELEMENTS

• Each series element is modeled by two equations relating its endpoint currents and voltages, with self- and mutual-admittance matrices set by the component model.Rows and columns for missing phases are set to zero.Source: §IV · p. 3

A. Transmission lines

• The transmission-line π model uses only the phases present on the line, allowing a line to connect nodes with different available phases.The passage gives a two-phase line between a three-phase node and a two-phase node as an example.Source: A. Transmission lines · p. 3
• Typical line series impedance matrices are symmetric with positive-definite real parts, while shunt admittance matrices are symmetric with positive-semidefinite real parts.The paper states these properties play a role in proving invertibility for networks of three-phase transmission lines.Source: A. Transmission lines · p. 3

B. Step-voltage regulators

• SVRs are adjustable-ratio autotransformer connections installed in series with transmission lines, in wye, closed-delta, or open-delta configurations.Their tap positions determine the turns ratios.Source: B. Step-voltage regulators · pp. 3–4
• The SVR model relates primary and secondary voltages and currents through voltage-gain, current-gain, and impedance matrices.The matrix entries depend on connection type and selected tap positions; the model assumes the primary and secondary have the same available phases.Source: B. Step-voltage regulators · p. 3
• The derived regulator models can include series impedance, although in most cases it depends significantly on tap position and is zero.Under the ideal-autotransformer assumption, the series impedance matrix is zero and the current-gain matrix is the identity.Source: B. Step-voltage regulators · p. 4
• For the three common SVR configurations, the voltage-gain matrix is invertible and the current-gain matrix has the stated inverse relationship.These properties are identified as crucial for guaranteeing Y-Bus invertibility.Source: B. Step-voltage regulators · p. 4

1) Wye-connected SVRs:

• Wye-connected SVR voltage gain and impedance matrices follow from the three phase-specific autotransformer ratios and series impedances.The paper identifies type B regulators as more common and develops these models for type B; it notes that the same procedure can produce type A models.Source: 1) Wye-connected SVRs · p. 4

2) Closed-delta connected SVRs:

• For closed-delta SVRs, the model uses three autotransformers across phases ab, bc, and ca and applies KCL at the intermediate node.The resulting equations determine the regulator matrices.Source: 2) Closed-delta connected SVRs · p. 5

3) Open-delta connected SVRs:

• The open-delta SVR model derives its gain and impedance matrices from the two autotransformers and the connection of phase b across the regulator nodes.The derivation also applies KCL at node K to obtain the current-gain matrix.Source: 3) Open-delta connected SVRs · pp. 5–6

C. Three-phase transformers

• Common distribution transformers are represented by series leakage-admittance and shunt core-loss blocks, with nodal admittances specified for common connections.The listed connections include delta–wye grounded, wye–delta, wye–wye, open-wye–open-delta, delta–delta, and open-delta–open-delta.Source: C. Three-phase transformers · p. 6, Table III · p. 6
• Rank deficiency in transformer matrices other than Y1 is identified as the only source of Y-Bus singularity, and the paper presents a remedy.The passage identifies the remedy but does not specify it here.Source: C. Three-phase transformers · p. 6

V. Y-BUS CONSTRUCTION AND THE Z-BUS METHOD

• The Y-Bus is assembled from block matrices representing transmission lines, SVRs, and transformers, with slack-bus voltage and current specified.The paper defines the resulting network matrix as the bus admittance matrix.Source: §V · p. 6, Y YNS YSN YSS · p. 6
• The Z-Bus method decomposes current into ZIP-load components and iterates a fixed-point voltage equation when Y + YL is invertible.The iteration uses Z = (Y + YL)^-1; in practice, the paper says an LU decomposition is computed instead of explicitly inverting the matrix.Source: Y YNS YSN YSS · pp. 6–7

Y YNS YSN YSS

• Transformer connections other than wye-g–wye-g can produce rank-deficient matrices and a non-invertible Y, which may also make Y + YL non-invertible.The passage says this occurs for all test feeders investigated in Section VIII.Source: Y YNS YSN YSS · p. 7
• The described remedy adds small shunt admittances to selected self-admittance matrices, producing a modified Y for the invertibility proof.The paper then uses Z = (Ŷ + YL)^-1 to compute voltages.Source: Y YNS YSN YSS · p. 7

VI. INVERTIBILITY OF THE BUS ADMITTANCE MATRIX Y

• The small-shunt modification targets Y's numerical properties for steady-state power-flow solutions, not standard grounding practice.The paper distinguishes this modification from grounding through a grounding transformer.Source: §VI · p. 7
• A1–A4 impose properties on transmission lines, transformer and SVR models, connectivity, and available phases.The assumptions include symmetric line matrices, non-ideal transformers, and a connected graph; the line condition allows untransposed lines.Source: §VI · pp. 7–8
• The proof establishes that Y is symmetric and Re[Y] is positive definite, yielding invertibility.Theorem 1 proves symmetry under A1; Theorem 2 proves positive definiteness under A1–A4.Source: §VI · p. 8
• Under A1–A4, the Y-Bus is invertible.The proof combines symmetry of Y with positive definiteness of Re[Y].Source: §VI · p. 8
• The proof is modular: invertibility holds regardless of element count, order, or radial versus meshed topology when series elements satisfy A1–A4.The argument depends on the properties of the series-element block admittances.Source: §VI · p. 8

VII. EXTENSIONS

• The extensions cover practical feeders with single-, two-, and three-phase laterals and open-wye–open-delta transformers.The paper also considers invertibility of Y + YL, reactive modifications for zero-ohmic-loss transformer models, and alternative practical transformer connections.Source: §VII · pp. 8–9

A. Invertibility under transmission lines with missing phases and open-wye–open-delta transformers

• Under A1′, A2, A3, and A4′, Re[Y] remains positive definite with missing-phase lines and open-wye–open-delta transformers.A4′ requires a path from the slack bus to each node and phase using edges that include that phase; the paper says typical feeders with one-, two-, and three-phase laterals satisfy it.Source: A. Invertibility under transmission… · p. 9

B. Invertibility of Y + YL

• Y + YL is invertible under A1′, A3, A2, and A4′; alternatively, unmodified Y2 or Y4 cases qualify when the affected transformer side has a constant-impedance load with positive definite real part.The extension establishes symmetry and positive semidefiniteness of Re[YL] before proving invertibility of the sum.Source: B. Invertibility of Y + YL · p. 9

C. Invertibility using the imaginary parts of ǫ′ and ǫ′′

• Y is invertible for transformers represented only by series reactances when A1′′, A2, A3′′, and A4′ hold.The assumptions require nonzero leakage inductance and use imaginary-part conditions, including reactive self-admittance modifications.Source: C. Invertibility using the imaginary… · pp. 9–10

D. Alternative transformer models

• If only transformer series admittances are available, the Table III models are presented as reasonable for unbalanced studies and were used for the tested feeders.For common-core or shell-type transformers, measured primitive admittances can be converted to nodal admittances; the invertibility proof can be adapted to the resulting model.Source: D. Alternative transformer models · p. 10

VIII. NUMERICAL RESULTS

• For the 37-bus and 123-bus delta–delta models, successive-run voltage solutions converge below ǫ/|yt| thresholds of 10^-5 and 10^-3; the 8500-node and ELV feeders converge below 10^-3.The paper attributes the smaller 37-bus threshold to its two delta–delta transformers, versus one in the 123-bus feeder.Source: §VIII · p. 10, Figure 4 · p. 10, Figure 5 · p. 11
• Computed voltage magnitudes are below 0.75% of benchmark solutions across the reported comparisons.Table V uses a fixed ǫ = 10^-6|yt|.Source: Table V · p. 11, §VIII · p. 11

A. IEEE 37-bus feeder

• When SVR series impedance is relatively high, the non-ideal regulator models provide more accurate results than ideal models against OpenDSS.This comparison is for the IEEE 37-bus feeder.Source: A. IEEE 37-bus feeder · p. 11, Table VI · p. 11

B. IEEE 123-bus test feeder

• The IEEE 123-bus test includes three-, two-, and single-phase laterals, four wye-connected SVRs, and a delta–delta transformer.Its voltage profile is compared with the benchmark in Fig. 7.Source: B. IEEE 123-bus test feeder · p. 11, Figure 7 · p. 12

C. 8500-node test feeder

• The 8500-node feeder includes three-, two-, and single-phase laterals, a delta–wye substation transformer, and four individually operated three-phase SVRs.The reported voltage profile uses ideal regulator models and includes magnitudes and phases.Source: C. 8500-node test feeder · p. 11

D. The European 906-bus low voltage feeder

• The ELV feeder has three-phase laterals only and a delta–wye substation transformer; its voltage profile is compared with the benchmark.The profile’s intensity reflects the feeder’s three-phase-only laterals.Source: D. The European 906-bus low voltage… · p. 12

IX. CONCLUDING REMARKS

• The paper lays out invertibility conditions for practical radial or meshed networks with transformers, SVRs, and missing phases, and provides scripts to build Y-Bus matrices and compute power-flow solutions.The paper also reviews line and transformer models and derives SVR models that account for tap positions and series admittance.Source: §IX · p. 12

• For Y = G + jB with G ≻ O and B = B^T, the lemma states that Y^-1 exists and Re[Y^-1] ≻ O.Source: Appendix A · p. 12
• The proof begins by contradiction, assuming a nonzero complex vector lies in the nullspace of Y.Source: Appendix A · p. 12
• The proof proceeds by using the real and imaginary parts of the assumed nullspace vector in its subsequent equalities.The supplied passage introduces x = x_R + jx_I and begins the nullspace argument.Source: Appendix A · p. 12

• The proof of the linear-algebra lemma uses the real-part equality to show that both terms vanish, forcing x_R and then x_I to be zero.Source: I GxI+ xT · p. 12
• The proof states that, since G ≻ O, the expression G + BG^-1B is positive definite and therefore concludes that R ≻ O.Source: I GxI+ xT · p. 12
• Under Assumption A1(a) and the Table II choices for A_i and Z_R, Lemma 4 states that the SVR matrix F_R is invertible.Source: I GxI+ xT · pp. 12–13