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Efficient Computation of Sensitivity Coefficients of Node Voltages and Line Currents in Unbalanced Radial Electrical Distribution Networks
Konstantina Christakou, Jean-Yves Le Boudec, Mario Paolone, Dan-Cristian Tomozei
TL;DR
Optimal control in distribution networks requires fast linearized dependencies between controlled quantities and control variables as distributed energy resources increase. This paper derives voltage and current sensitivities analytically using the sparse [Y] compound matrix for generic multiphase radial unbalanced networks, reducing computation time by almost a factor of three versus the traditional Jacobian approach and demonstrating optimal voltage control on the IEEE 34-node feeder.
Problem
Increasing distributed energy resources make optimal voltage and power-flow control important in distribution networks, but real-time operation requires computationally efficient sensitivity coefficients.
Method
The paper analytically derives node-voltage and line-current sensitivities to nodal power injections and transformer tap-changer positions using the sparse [Y] compound matrix for generic unbalanced radial networks.
Results
Computation time is reduced by almost a factor of three versus the traditional Jacobian load-flow matrix, with validation on IEEE 13- and 34-node test feeders and an optimal-voltage-control application on the IEEE 34-node feeder.
Takeaways & Limitations
The analytical sensitivities can support real-time optimal controllers and provide network-response information for closed-loop control or contingency analysis.
Abstract
from arXiv · showhide
The problem of optimal control of power distribution systems is becoming increasingly compelling due to the progressive penetration of distributed energy resources in this specific layer of the electrical infrastructure. Distribution systems are, indeed, experiencing significant changes in terms of operation philosophies that are often based on optimal control strategies relying on the computation of linearized dependencies between controlled (e.g. voltages, frequency in case of islanding operation) and control variables (e.g. power injections, transformers tap positions). As the implementation of these strategies in real-time controllers imposes stringent time constraints, the derivation of analytical dependency between controlled and control variables becomes a non-trivial task to be solved. With reference to optimal voltage and power flow controls, this paper aims at providing an analytical derivation of node voltage and line current flows as a function of the nodal power injections and transformers tap-changers positions. Compared to other approaches presented in the literature, the one proposed here is based on the use of the [Y] compound matrix of a generic multi-phase radial unbalanced network. In order to estimate the computational benefits of the proposed approach, the relevant improvements are also quantified versus traditional methods. The validation of the proposed method is carried out by using both IEEE 13 and 34 node test feeders. The paper finally shows the use of the proposed method for the problem of optimal voltage control applied to the IEEE 34 node test feeder.
I. INTRODUCTION
The paper targets computationally efficient sensitivity coefficients for optimal control in multiphase, unbalanced distribution networks with distributed energy resources. It proposes analytical voltage and current sensitivities using the sparse [Y] compound matrix while extending prior formulations.
- Motivation: Optimal distribution-system controls require linearized links between control variables and controlled quantities, but real-time implementation imposes stringent computational constraints.Relevant controls include voltage and power-flow control in active distribution networks.
- Existing limitations: Traditional sensitivity methods rebuild and invert an updated load-flow Jacobian whenever network operating conditions change.This creates non-trivial computation constraints for centralized or decentralized real-time controllers.
- Proposed approach: The proposed method analytically derives node-voltage and line-current sensitivities from nodal power injections and transformer tap-changer positions using the sparse [Y] compound matrix.The formulation is intended for multiphase, unbalanced distribution networks.
- Contributions: The paper generalizes the formulation to a generic number of slack busses and extends sensitivities to transformer tap-changer positions.Tap-changer changes are represented through changes in slack-bus reference voltages.
- Contributions: The method accounts for multiphase, unbalanced networks and provides a uniqueness proof for the analytical sensitivity solution in radial networks.These properties target the structure of electrical distribution systems.
A. Classical Computation of Sensitivity Coefficients in Power Networks
Classical sensitivity computation relies on perturbed load flows, inverted Newton–Raphson Jacobians, or adjoint networks. These approaches have computational, modeling, or applicability limitations for distribution networks.
- Perturbed load-flow method: A perturbation method estimates each sensitivity by repeatedly solving load flows after small changes to individual nodal power injections.The perturbed variables include active and reactive injections.
- Modeling assumptions: Some prior formulations neglect line shunt parameters, which can be non-negligible for coaxial cables typical of urban distribution networks.This modeling choice can limit applicability to such network components.
- Newton–Raphson method: The Newton–Raphson method obtains voltage sensitivities as submatrices of the inverted load-flow Jacobian.Its formulation is based on the voltage-magnitude and voltage-angle variables shown in the Jacobian.
- Newton–Raphson limitations: Newton–Raphson sensitivity computation does not provide sensitivities with respect to transformer tap-changer positions.It also commonly uses a reactive-power approximation that assumes negligible longitudinal resistance-to-reactance ratio.
- Adjoint-network method: Adjoint-network methods apply Tellegen’s theorem but require a base-case load flow and a separately solved adjoint network.The adjoint solution is used to infer the desired sensitivities.
B. Analytical Derivation of Voltage and Current Sensitivity Coefficients
This section develops the paper’s main analytical derivation of voltage sensitivity coefficients. The supplied passage identifies the subsection’s focus but does not provide the derivation details.
- Analytical derivation: The subsection presents the main analytical development for deriving voltage sensitivity coefficients.The available passage does not specify the equations, variables, or derivation steps.
1) Voltage Sensitivity Coefficients:
The paper derives voltage sensitivity coefficients for active and reactive power injections in generic multiphase networks, treating each phase separately so the formulation applies to unbalanced systems. A fixed linear-system matrix enables repeated derivative computations across PQ buses, with uniqueness established for radial networks.
- The formulation considers a K-bus, three-phase network and analyzes each phase separately, allowing application to unbalanced networks.
- The network uses S slack buses and N PQ-injection buses, with constant power injections during each separate perturbation.
- Voltage sensitivity coefficients are obtained by differentiating bus voltages with respect to active and reactive power injections at each PQ bus.
- The derivative systems are linear in rectangular coordinates, reducing the computational effort required to solve for voltage sensitivities.
- The matrix requiring inversion is fixed across the PQ buses; only the equations' left-hand sides change with the selected perturbation bus.
- For every radial electrical network, the active- and reactive-power derivative systems have unique solutions.
2) Current Sensitivity Coefficients:
Branch-current sensitivity coefficients are derived from the previously obtained voltage sensitivities using π-model representations of network lines. The resulting derivatives cover active and reactive power injections.
- Branch currents are represented as functions of the phase-to-ground voltages at the line’s two endpoint nodes using π line models.
- Once voltage sensitivities are known, current sensitivities can be computed directly.
- Because voltages depend on network power injections, branch-current derivatives can be expressed with respect to active and reactive power injections.
C. Sensitivity Coefficients with respect to tap positions of transformers
The paper derives voltage sensitivity coefficients with respect to transformer tap positions by treating tap changes at slack buses as changes in the slack reference voltage. The resulting system shares the voltage-sensitivity matrix and has a unique solution.
- Transformer tap-changers are assumed to be located at slack buses, making tap-position sensitivities equivalent to sensitivities to slack reference voltage.
- The tap-position derivative system is linear in the relevant rectangular-coordinate variables and uses the same associated matrix as the power-injection system.
- The tap-position sensitivity system has a unique solution because its homogeneous system is identical to the one used in Theorem 1.
- After solving the system, voltage sensitivity coefficients with respect to the transformer tap position at slack bus k are obtained.
- Current sensitivities with respect to tap positions can be computed directly once the voltage sensitivities are available.
D. Computational Cost Analysis for Voltage Sensitivities with respect to PQ injections
The computational analysis compares the proposed analytical method with the traditional Jacobian method for voltage sensitivities on IEEE 13- and 34-node feeders. The analytical approach avoids rebuilding an updated Jacobian and improves mean CPU-time performance.
- The comparison evaluates voltage-sensitivity computation on IEEE 13- and 34-node test feeders using 1000 iterations.
- The traditional method builds and inverts an updated Newton–Raphson Jacobian matrix of size 2N × 2N.
- The analytical method inverts a 2N × 2N matrix and performs N multiplications with vectors of size 2N × 1.
- 2.34 improvement is reported for the IEEE 13 node test feeder, while 2.52 improvement is reported for the IEEE 34 node test feeder.
- The performance advantage increases with network size and depends on both the number of buses and the sparsity of the admittance matrix.
- Traditional Jacobian-based sensitivity computations do not account for tap-changer variations.
III. NUMERICAL VALIDATION
The proposed sensitivity calculations are validated on IEEE 13- and 34-node feeders against Jacobian-based and numerical approaches. The analytical results closely match the reference calculations while revealing phase coupling, distance effects, and the stronger influence of reactive power.
- Validation approach: The validation compares voltage sensitivities with inverse-Jacobian calculations and current sensitivities with numerical load-flow perturbations.The inverse Jacobian does not provide current sensitivity coefficients, so current accuracy is evaluated numerically.
- Power-injection sensitivities: Voltage-sensitivity comparisons for bus 8 and bus 9 produce relative errors on the order of 10^-6.The cases cover active and reactive power absorption or generation, including same-phase and different-phase dependencies.
- Power-injection sensitivities: Current-sensitivity coefficients for phase a of branch 10-13 are evaluated against active and reactive power generation or absorption at phase a of node 13, with extremely low errors.The figure reports relative errors between the analytical and numerical values.
- Tap-changer sensitivities: Tap-position voltage sensitivities on the IEEE 13-node feeder differ from numerical calculations by approximately 10^-4.The test varies the slack-bus voltage by ±6% over 72 tap positions and evaluates bus 7 phase a against tap positions on phases a, b, and c.
- Sensitivity structure: Same-phase voltage and perturbation coefficients show the largest coupling, while non-negligible cross-phase dependencies remain.The reported voltage-sensitivity behavior is observed in the multiphase feeder validation.
- Sensitivity structure: Sensitivity magnitudes generally increase with distance from the slack bus, and reactive-power absorption has a larger influence on voltage variations than active-power absorption.The distance-dependent behavior is illustrated for power absorption at phase a of bus 13; the resulting views may support closed-loop control or contingency analysis.
IV. APPLICATION OF THE PROPOSED PROCEDURE TO THE PROBLEM OF OPTIMAL VOLTAGE CONTROL
The IEEE 34-node feeder application formulates optimal voltage control as a linear optimization using voltage sensitivities to DER injections and transformer tap positions. It compares three-phase DER control with independent phase control.
- The DNO controls active and reactive DER injections at buses 18, 23, 24, and 33, along with transformer tap positions.
- The optimization uses bus voltages as controlled variables and DER active/reactive injections and tap positions as control variables.
- Transformer taps make the formulation formally mixed-integer, but tap positions are treated as pseudo-continuous and rounded after optimization.
- The linearized problem is solved by linear least squares under rectangular DER capability curves in the P-Q plane.
- Two scenarios compare equal three-phase DER set points with independent phase control, while tap-changer positions remain shared.
- Independent phase control produces a better optimal voltage profile than controlling only the three-phase DER output.
V. CONCLUSION
The paper concludes that its analytical sensitivity method supports unbalanced radial-network control and reduces computation time relative to Jacobian-based methods. IEEE 13- and 34-node feeders validate the method and demonstrate voltage-control use.
- The method computes voltage and current sensitivities from nodal power injections, supports multiple slack buses and tap-changer positions, and has a unique radial-network solution.
- Nearly a factor-of-three computation-time reduction versus the traditional Jacobian load-flow matrix supports potential real-time optimal-controller implementation.
- The analytical sensitivities can reduce computation time for real-time centralized control, contingency analysis, and optimal planning.