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Robust Capacity Assessment of Distributed Generation in Unbalanced Distribution Networks Incorporating ANM Techniques

Xin Chen, Wenchuan Wu, Boming Zhang

arXiv:1709.03746v1eess.SYcs.SY

TL;DR

DG hosting-capacity assessment must address uncertainty in loads and renewable output as well as unbalanced three-phase power flow. The paper proposes a two-stage robust capacity-assessment method incorporating ANM, and reports higher average PV output with zero violation rate across Monte Carlo cases than two comparison methods.

  • Problem

    Uncertain load demands and renewable DG outputs, together with unbalanced three-phase power flow, complicate DG hosting-capacity assessment; earlier approaches may ignore uncertainty or omit relevant network controls.

    Source: §I · p. 2

  • Method

    The two-stage robust RC-CAM combines three-phase modeling with DG installation decisions and ANM schemes, including network reconfiguration, OLTC regulation, and reactive-power control.

    Source: §VI · p. 10

  • Results

    RC-CAM outperformed two other methods with the greatest average PV outputs and zero violation rate in all Monte Carlo cases.

    Source: D. Performance Comparisons via Monte… · p. 10

  • Takeaways & Limitations

    The authors conclude that RC-CAM combines robust capacity evaluation with optimal ANM schemes and can extend to other active distribution network optimization and planning problems.

    Source: D. Performance Comparisons via Monte… · p. 10, §VI · p. 11

Abstract

from arXiv · show

To settle a large-scale integration of renewable distributed generations (DGs), it requires to assess the maximal DG hosting capacity of active distribution networks (ADNs). For fully exploiting the ability of ADNs to accommodate DG, this paper proposes a robust comprehensive DG capacity assessment method considering three-phase power flow modelling and active network management (ANM) techniques. The two-stage adjustable robust optimization is employed to tackle the uncertainties of load demands and DG outputs. With our method, system planners can obtain the maximum penetration level of DGs with their optimal sizing and sitting decisions. Meanwhile, the robust optimal ANM schemes can be generated for each operation time period, including network reconfiguration, on-load-tap-changers regulation, and reactive power compensation. In addition, a three-step optimization algorithm is proposed to enhance the accuracy of DG capacity assessment results. The optimality and robustness of our method are validated via numerical tests on an unbalanced IEEE 33-bus distribution system.

A. Set and Notation

The notation organizes network nodes and branches, model parameters, and decision and operating variables for DG capacity assessment. These definitions cover uncertainty, device limits, power flows, and network status over time.

A. Set and Notation

  • Node sets distinguish all nodes, non-root nodes, DG-eligible nodes, compensator locations, and node neighborhoods; branch sets distinguish all branches, transformer branches, and branches without transformers.The notation also defines time periods, phases A/B/C, and a large positive constant.Source: A. Set and Notation · p. 1

B. Parameter

  • Parameters include DG-output efficiency and load forecasts, uncertainty budgets, device and branch limits, and network impedance and voltage bounds.The parameter list also specifies power-factor limits, compensator limits and increments, and node counts.Source: B. Parameter · p. 1

C. Variable

  • Variables represent installed DG capacity, voltage and output realizations, OLTC tap ratios, branch status and flows, and continuous or discrete VAR injections.A nonnegative integer records the step number for discretely adjustable VAR compensation.Source: C. Variable · pp. 1–2

D. Three-phase Vector and Matrix

The paper frames DG hosting-capacity assessment around uncertainty, unbalanced three-phase power flow, and active network management. It proposes a robust model and a three-step assessment approach that incorporates these elements.

D. Three-phase Vector and Matrix

  • Three-phase vectors and matrices represent node voltages, squared voltage magnitudes, branch power flows, net loads, and branch impedance.The notation distinguishes complex voltage and power quantities from their associated squared-voltage or net-load quantities.Source: D. Three-phase Vector and Matrix · p. 2

I. INTRODUCTION

  • Uncertain loads and renewable outputs, unbalanced phase conditions, and the limits of existing approaches motivate robust three-phase capacity assessment with ANM.The paper identifies network reconfiguration, OLTC regulation, VAR compensation, and DG power-factor control as relevant ANM schemes.Source: §I · p. 2

II. MATHEMATICAL MODEL BUILDING

  • The proposed approach combines exact-OLTC and loss-aware three-phase linear branch-flow models in a three-step optimization algorithm, alongside two-stage robust DG siting and ANM decisions.The first stage chooses DG installations and time-period ANM strategies; the second searches for worst-case scenarios within predefined uncertainty sets.Source: §I · pp. 2–3

A. Three-phase Linear Branch Power Flow Model

The paper develops a three-phase linear branch-flow model for unbalanced networks, including an exact OLTC representation. LBPF-1 simplifies the power-flow equations, with accuracy dependent on network conditions.

A. Three-phase Linear Branch Power Flow Model

  • The three-phase branch model relates voltage drops and branch power flows, represents OLTC turn ratios, and is transformed into a linear branch-flow model for optimization.Its phase-decoupled form neglects mutual impedances and removes quadratic loss terms as negligible relative to branch power flow.Source: A. Three-phase Linear Branch Power… · p. 3

LBPF 1

  • LBPF-1 accuracy depends on network loss rate, with errors increasing under heavy loads or high impedance ratios.The model ignores mutual impedances and power losses; the paper proposes a higher-precision LBPF model and a three-step algorithm combining the two models.Source: LBPF 1 · p. 4
  • The OLTC turn ratio is modeled through discrete tap positions and linearized exactly using a binary expansion and equivalent substitution constraints.The tap position is represented by a nonnegative integer variable, with a specified total tap count and turn-ratio increment.Source: LBPF 1 · p. 4

B. Deterministic Comprehensive Capacity Assessment

The deterministic capacity model maximizes total DG capacity while incorporating ANM strategies across multiple time periods. Its radial-topology constraints use virtual power flow to handle network reconfiguration.

B. Deterministic Comprehensive Capacity Assessment

  • The model optimizes maximum DG penetration and installation strategies alongside the optimal ANM schemes for each time period.It incorporates network reconfiguration, OLTC adjustment, and reactive output control of DGs and VAR compensation.Source: B. Deterministic Comprehensive… · p. 4

1) Objective Function:

2) Radial Topology Constraints:

  • Virtual power balance constraints assign each node a fictitious unit load to ensure radial network topologies.A nonnegative virtual power flow variable is introduced to address transfer buses with zero power injection.Source: 2) Radial Topology Constraints · p. 4

3) DG Output Constraints:

The model constrains DG and VAR-compensator outputs, branch power flows, and line thermal capacity. Linear box constraints approximate the quadratic thermal limit.

3) DG Output Constraints:

  • DG units operate at maximum power point tracking, while their reactive outputs can vary continuously within predetermined ranges.Source: 3) DG Output Constraints · p. 4

4) Reactive Output Constraints for VAR Compensators:

5) Branch Power Flow Equation Constraints:

7) Line Thermal Capacity Constraints:

  • Linear box constraints approximate the quadratic line thermal constraint using the circle constraint linearization method introduced in.Branch apparent-power capacity sets the thermal limit, and branch status limits flow to zero when disconnected.Source: 7) Line Thermal Capacity Constraints · p. 5

8) Voltage Magnitude Limit Constraints:

The deterministic formulation constrains voltage magnitude using bounds, but relies on predicted loads and DG efficiency. The robust model extends it to address load and DG-output uncertainty through two-stage optimization.

8) Voltage Magnitude Limit Constraints:

  • The deterministic model uses predicted load demands and DG efficiency coefficients, so it neglects significant uncertainty factors.Voltage magnitude is bounded by the stated minimum and maximum limits.Source: 8) Voltage Magnitude Limit Constraints · p. 5

C. Robust Comprehensive Capacity Assessment Model

  • The robust model uses two-stage optimization: it seeks worst-case load and DG-output scenarios, then chooses DG installations and ANM strategies to maximize hosting capacity.The first-stage decisions include branch status, VAR injections, OLTC turn ratios, and DG installation capacity.Source: C. Robust Comprehensive Capacity… · pp. 5–6
  • RC-CAM combines ANM with robust decision-making to address uncertainty impacts without being too conservative or too radical.Source: C. Robust Comprehensive Capacity… · p. 6

III. SOLUTION METHOD FOR TWO-STAGE ROBUST MODEL

The solution method casts RC-CAM in compact form and solves it iteratively with column-and-constraint generation. Its master problem approximates uncertainty scenarios and provides an upper bound.

III. SOLUTION METHOD FOR TWO-STAGE ROBUST MODEL

  • The compact formulation groups variables into DG installation, power flow, uncertainty, and ANM strategy vectors, then expresses the constraints in a unified form.Source: §III · p. 6
  • Column-and-constraint generation decomposes RC-CAM into a master problem and subproblem, solved through a master–subproblem iteration.Source: §III · p. 6

A. Master Problem of RC-CEM

  • The master problem uses finite partial-enumeration uncertainty scenarios to approximate the uncertainty set, and its objective gives an upper bound on the original problem.Source: A. Master Problem of RC-CEM · p. 6

B. Sub-Problem of RC-CEM

The RC-CAM sub-problem is a minimax bi-level optimization whose objective provides a lower bound; strong duality yields a dual form, and column-and-constraint generation iterates between sub-problem and master bounds. The accuracy method adds a loss-aware three-phase LBPF model and a three-step algorithm.

B. Sub-Problem of RC-CEM

  • The sub-problem is a minimax bi-level optimization whose objective provides a lower bound for RC-CAM, and strong duality reformulates it as a monolithic dual sub-problem.Integer uncertainty budget makes deviation variables binary, allowing the bilinear terms to be linearized equivalently.Source: B. Sub-Problem of RC-CEM · p. 6

C. Column-and-Constraint Generation Algorithm

1) STEP 1 Initialization:

  • Initialization sets the bounds and iteration counter, then solves DC-CAM to obtain initial ANM strategies and an objective value.The initial objective updates the upper bound.Source: 1) STEP 1 Initialization · pp. 6–7

2) STEP 2 Solving Sub-problem:

  • With the current ANM strategy fixed, the sub-problem finds the optimal uncertainty deviation and objective, then adds a constraint to the master problem and updates the lower bound.The added constraint is expression (38).Source: 2) STEP 2 Solving Sub-problem · p. 7

3) STEP 3 Solving master problem:

  • The master problem returns updated optimal ANM strategies and an objective value, which updates the upper bound.The upper bound is the minimum of its current value and the master objective.Source: 3) STEP 3 Solving master problem · p. 7

4) STEP 4 Checking for convergence:

IV. METHOD TO IMPROVE MODEL ACCURACY

  • A three-phase LBPF model that considers power losses and a three-step optimization algorithm are proposed to improve capacity-assessment accuracy.The algorithm is intended to carry out DG capacity evaluation precisely and efficiently.Source: §IV · p. 7

A. Three-phase LBPF Model Considering Power Losses

  • LBPF-2 is derived by linearizing power-flow equations around anticipative operating points and approximating quadratic loss terms with a first-order Taylor expansion.The model substitutes given operating voltages for voltage variables before approximating the loss terms.Source: A. Three-phase LBPF Model Considering… · p. 7

LBPF 2

In worst-case scenarios, RC-CAM maintained voltage security and achieved higher real PV output than the deterministic and robust-without-reconfiguration alternatives. The study also reports improved assessment accuracy from the three-step process and tests the models on a modified unbalanced IEEE 33-bus system.

LBPF 2

  • LBPF-2 accounts for power losses through linearization around operating points, but its accuracy depends heavily on those points.It is state-dependent and cannot be applied directly to long-term assessment and planning, particularly with network reconfiguration and OLTC regulation.Source: LBPF 2 · p. 7

A -CAM LBPF - 2

  • The three-step algorithm uses RC-CAM for ANM decisions and worst-case scenarios, exact power flow for those decisions, and A-CAM for final capacity assessment.The process is described as improving precision and efficiency.Source: A -CAM LBPF - 2 · p. 7, Figure 2 · p. 7

A. Test System and Consideration

  • The numerical tests use a modified three-phase IEEE 33-bus system with four candidate PV sites and 24 daily assessment periods.The tests compare RC-CAM with a deterministic model using all ANM techniques and a robust model without network reconfiguration.Source: A. Test System and Consideration · p. 8

B. Accuracy Analysis of Capacity Assessment Models

D. Performance Comparisons via Monte Carlo Simulation

Monte Carlo comparisons favor RC-CAM over M-1 and M-2, while uncertainty-budget tests show how conservativeness and PV bottlenecks vary with budget. The paper also reports computational timings and concludes that the method is robust and extensible to other distribution-network planning problems.

D. Performance Comparisons via Monte Carlo Simulation

E. Impact of Uncertainty Budget

F. Computational Platform and Efficiency

  • RC-CAM took about 2.6 minutes per process on the modified IEEE 33-bus system and usually converged in two to three iterations.The authors describe this off-line assessment time as acceptable for practical applications.Source: F. Computational Platform and Efficiency · p. 10

VI. CONCLUSION

  • Worst-case tests on an unbalanced IEEE 33-bus system verified robustness, and the authors state that the model can extend to other ADN optimization and planning issues.Monte Carlo comparisons against two other state-of-the-art approaches are reported as demonstrating RC-CAM’s optimality.Source: §VI · pp. 10–11

APPENDIX

The appendix derives exact three-phase branch power-flow relationships for radial distribution networks. It uses Kirchhoff’s current law and Ohm’s law, then removes voltage phase angles by multiplying an equation by its conjugate.

A. Derivation of Exact Three-phase Branch Power Flow

B. Derivation of LBPF-2 Model

The LBPF-2 derivation linearizes nonlinear power-flow terms using first-order approximations around an operating point. It defines intermediate terms and functions before presenting the resulting linear expression.

B. Derivation of LBPF-2 Model

  • For equation (2), the derivation uses a diagonal-matrix notation and the all-ones vector to express a nonlinear term and define functions h^p_ij and h^q_ij.The functions are specified in terms of branch power and impedance quantities.Source: B. Derivation of LBPF-2 Model · p. 11
  • Equation (46) is linearized by a first-order approximation around operating-point values of p_ij,t and q_ij,t.The resulting expressions define coefficients for the linearized terms.Source: B. Derivation of LBPF-2 Model · p. 11
  • The derivation defines intermediate terms for equation (1), then rewrites its nonlinear terms using those definitions.The intermediate definitions include β, γ, and α.Source: B. Derivation of LBPF-2 Model · p. 11
  • The functions h^u_ij(p,q) are linearized around the operating point using a first-order Taylor expansion to obtain equation (53).The derivation defines the associated partial-derivative terms.Source: B. Derivation of LBPF-2 Model · p. 11
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