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Cited by 15Optimal Power Flow DistributionElectric Power System OptimizationSmart Grid and Power Systems

Optimal power flow in distribution networks

Jordan N. Radosavljević·2018·Metaheuristic Optimization in Power Engineering
DOI 10.1049/pbpo131e_ch9
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Abstract

In this chapter, a generalized approach for the OPF analysis in distribution networks is presented. This implies the inclusion of different types of DG units, taking into account the uncertainties of the input variables, the multiobjective optimization, and application of efficient metaheuristic methods for solving the OPF problem.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proves that a second-order-cone programming (SOCP) relaxation of a modified optimal power flow (OPF) problem in distribution networks is exact under a condition C1 that can be checked before solving the relaxation. Empirical tests on IEEE 13-bus and SCE 47-bus and 56-bus distribution networks show the modification is small and C1 holds, including networks with high distributed-generation penetration.

Key findings

  • Under Condition C1, the SOCP relaxation of the modified OPF problem (SOCP-m) is exact; C1 can be checked in O(n) time and holds for all test distribution networks considered, including IEEE 13-bus and practical SCE networks with high distributed generation.
  • The modification restricts power injections so that only feasible points “close” to voltage upper bounds are eliminated; Table I reports ε = 0.0043 for IEEE 13-bus, 0.0036 for SCE 47-bus, and 0.0106 for SCE 56-bus (the surrounding text gives 0.0031 for SCE 47-bus).
  • If SOCP-m is exact, it has at most a unique solution (Theorem 3), so any convex programming solver gives the same solution.
  • C1 holds for all three test networks; in the worst-case setup the minimum interval covers the range of r/x for the first two networks but not the third, yet the equivalent condition (14) still holds for the third.

Methods

  • The OPF problem minimizes power loss subject to nonlinear power-flow constraints, power-injection constraints, and voltage-regulation constraints on a tree distribution network. The paper transforms OPF to OPF′ using W variables, relaxes rank constraints to positive semidefinite constraints to obtain the SOCP relaxation, and modifies OPF by adding constraints on power injections so that W_lin_ii(s) ≤ |V_i|² always holds.
  • Case studies use the IEEE 13-bus test distribution network (adjusted because it is unbalanced three-phase with devices not modeled by (1)) and two SCE practical networks: a 47-bus network with 6.4 MW nameplate distributed generation over 11.3 MVA peak spot load (>50% penetration), and a 56-bus network with 5 MW nameplate distributed generation over 3.835 MVA peak spot load (>100% penetration).
  • For SCE networks, the study assumes loads draw peak spot apparent power at power factor 0.97, all shunt capacitors are switched on, and distributed generators generate real power at nameplate capacity with zero reactive power; voltage bounds are |V_i| = 1.05 and |V_i| = 0.95, with substation voltage fixed at nominal.

Limitations

  • The paper states that C1 requires p and q to be “small,” that it is a sufficient condition, and that exactness is proved only for the modified OPF problem, not the original OPF; modifying OPF is necessary because the SOCP relaxation is in general not exact.
  • The paper states that the modification eliminates feasible points close to voltage upper bounds, which may be undesirable for robust operation, and that empirical justification is limited to the test networks considered.
  • Evident from the text: several proofs (Theorem 1, Lemma 1, Lemma 2, Theorem 3) are deferred to an extended version [16], and the IEEE 13-bus network required adjustment; the SCE case studies rely on simplifying assumptions about load power factor, capacitor status, and DG output.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1049/pbpo131e_ch9

Cites

23 of 23 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

12 of 19 cited sources checked · 2 supported · 2 flagged · accuracy not yet measured

  • Interpretationcitation
    § A. A sufficient condition · p. 4

    If every optimal solution w to the SOCP relaxation satisfies W_lin_ii(s) ≤ |V_i|^2 for i∈N+, and the condition (C1) holds, then the SOCP relaxation is exact.

    “Lemma 2 If every optimal solution w = ( s, W, s 0 ) to the SOCP relaxation satisfies W lin ii ( s ) ≤ | V i | 2 for i ∈ N + , and the condition”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [16] [16] L. Gan, U. Topcu, N. Li, and S. Low, 'Exact convex relaxation for optimal power flow in tree networks,' arXiv preprint arXiv:1208.4076, 2012. - source

      Lemma 2 in “A. A sufficient condition” (p. 4) gives the stated exactness result using the optimal-solution voltage bound; the proof (p. 18) links its conditions to C1. But Lemma 2 also assumes that f₀ is strictly increasing and that the bounds in (10) exist. The claim omits those prerequisites, making its sufficient condition broader than the paper states. (LLM, conf=0.88)

      Experimental — accuracy not yet measured
  • Premisecitation
    § A. A sufficient condition · p. 4

    The linear approximations S_lin_ij(s) and W_lin_ii(s) are close to S_ij and W_ii in practice.

    “The linear approximations S lin ij ( s ) and W lin ii ( s ) are close to S ij and W ii in practice.”
    Quote in the paperSubstantial claim1 of 3 sources checked
    • cites [19] [19] M. E. Baran and F. F. Wu, 'Optimal capacitor placement on radial distribution networks,' IEEE Transactions on Power Delivery, vol. 4, no. 1, pp. 725-734, 1989. - source
    • cites [21] [21] --, 'Network reconfiguration in distribution systems for loss reduction and load balancing,' IEEE Transactions on Power Delivery, vol. 4, no. 2, 1989. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [22] [22] K. Turitsyn, P. Sulc, S. Backhaus, and M. Chertkov, 'Local control of reactive power by distributed photovoltaic generators,' in IEEE SmartGridComm, 2010, pp. 79-84. - source

      Section III (p. 3) says that, for a broad class of feeder lines, small quadratic terms and voltage deviations allow the power-flow equations to be approximated by linear LinDistFlow equations. It does not identify the claimed S_lin_ij(s) or W_lin_ii(s) quantities or establish that those particular approximations are close to S_ij and W_ii in practice. (LLM, conf=0.83)

      Experimental — accuracy not yet measured
  • Premisecitation
    § D. Related work · p. 4

    The SOCP relaxation is in general not exact.

    “In fact, the SOCP relaxation is in general not exact, and a 2-bus example is provided in the extended version of this paper [16].”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [16] [16] L. Gan, U. Topcu, N. Li, and S. Low, 'Exact convex relaxation for optimal power flow in tree networks,' arXiv preprint arXiv:1208.4076, 2012. - source
  • Interpretationcitation
    § A. A sufficient condition · p. 4

    If (s,S,W,s0) satisfies (7), (10) and (12), then S_ij ⪯ S_lin_ij(s) for i→j and W_ii ≤ W_lin_ii(s) for i∈N+.

    “Lemma 1 If ( s, S, W, s 0 ) satisfies (7) , (10) and (12) , then S ij ⪯ S lin ij ( s ) for i → j and W ii ≤ W lin ii ( s ) for i ∈ N + .”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [16] [16] L. Gan, U. Topcu, N. Li, and S. Low, 'Exact convex relaxation for optimal power flow in tree networks,' arXiv preprint arXiv:1208.4076, 2012. - source

      Lemma 1 (A. A sufficient condition, p. 4) states the claimed upper bounds on line power flow and squared voltage, using the paper’s hatted linear approximations. But its stated premises are (6a), (7), and (11), not the citing sentence’s (7), (10), and (12); the cited copy does not establish the claim under that different set of conditions. (LLM, conf=0.86)

      Experimental — accuracy not yet measured
  • Interpretationcitation
    § C. Uniqueness of solutions · p. 5

    SOCP-m has at most a unique solution if it is exact.

    “Theorem 3 SOCP-m has at most a unique solution if it is exact.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [16] [16] L. Gan, U. Topcu, N. Li, and S. Low, 'Exact convex relaxation for optimal power flow in tree networks,' arXiv preprint arXiv:1208.4076, 2012. - source

      Theorem 3 (C. Uniqueness of solutions, p. 5) says that exact SOCP-m has at most one solution, but only if the specified objective functions and feasible sets are convex. The citing statement omits those conditions. (LLM, conf=0.91)

      Experimental — accuracy not yet measured
  • May combine two claims. 8
  • Classified as probably not a substantial claim. 6
  • The quote only partly states this claim. 2

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    Prior sufficient conditions for exactness of the SOCP relaxation require load over-satisfaction on some or all buses and do not hold in practice.

    “these conditions require some bus to be able to draw infinite power, which is referred to as load over-satisfaction in the literature. For example, the conditions given in [9], [13], [14] require load over-satisfaction on some/all of the buses.”
    Quote in the paperSubstantial claim3 of 3 sources checked
    • cites [9] [9] M. Farivar, C. R. Clarke, S. H. Low, and K. M. Chandy, 'Inverter var control for distribution systems with renewables,' in IEEE SmartGridComm, 2011, pp. 457-462. - source

      The Introduction (page not supplied) says the paper’s convex relaxation is exact provided load over-satisfaction is allowed. It does not specify that a bus must be able to draw infinite power, identify which buses require over-satisfaction, or say the condition does not hold in practice. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
    • cites [13] [13] B. Zhang and D. Tse, 'Geometry of injection regions of power networks,' IEEE Transactions on Power Systems, to appear, 2012. - source

      Theorem 2 (section IV.A; page not supplied) relates the paper’s lower-bound conditions to load over-satisfaction, supporting that part of the attribution. But the Introduction explicitly says simulations with practical distribution networks show these requirements are “not stringent in actual operations,” contrary to the claim that they do not hold in practice. (LLM, conf=0.87)

      Checked by a reasoning model — may be wrong.

      Experimental — accuracy not yet measured
    • cites [14] [14] S. Bose, D. Gayme, S. Low, and K. Chandy, 'Quadratically constrained quadratic programs on acyclic graphs with application to power flow,' arXiv:1203.5599, 2012. - source

      Section III-C (p. 18) describes exact-relaxation conditions that remove load bounds and calls this “load oversatisfaction”; its examples remove bounds at some or all buses. The paper’s own result concerns a semidefinite relaxation, while Prior work (p. 13) reports SOCP exactness under either no upper bounds on loads or no upper bounds on voltage magnitudes. Our copy does not say these conditions do not hold in practice. (LLM, conf=0.86)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Premisecitation
    § D. Related work · p. 4

    All sufficient conditions in the literature that guarantee exactness of the SOCP relaxation require removing some of the constraints in OPF, and do not hold in practice.

    “To summarize, all sufficient conditions in the literature that guarantee the exactness of the SOCP relaxation require removing some of the constraints in OPF, and do not hold in practice.”
    Quote in the paperSubstantial claim2 of 3 sources checked
    • cites [9] [9] M. Farivar, C. R. Clarke, S. H. Low, and K. M. Chandy, 'Inverter var control for distribution systems with renewables,' in IEEE SmartGridComm, 2011, pp. 457-462. - source

      The Introduction says this paper's convex relaxation is exact provided oversatisfaction of load is allowed, supporting the narrower point that its exactness guarantee depends on relaxing a load requirement. It does not say that all sufficient conditions in the literature require removing OPF constraints or that those conditions fail in practice. (LLM, conf=0.83)

      Experimental — accuracy not yet measured
    • cites [13] [13] B. Zhang and D. Tse, 'Geometry of injection regions of power networks,' IEEE Transactions on Power Systems, to appear, 2012. - source

      The Introduction (page unavailable) says the paper's conditions retain voltage, line-loss, and line-flow constraints, restrict certain bus-power bounds, and are 'not stringent in actual operations'—contrary to the claim that such conditions do not hold in practice. The paper discusses a convex relaxation; these passages do not identify it specifically as an SOCP relaxation or establish a claim about every sufficient condition in the literature. (LLM, conf=0.82)

      Checked by a reasoning model — may be wrong.

      Experimental — accuracy not yet measured
    • cites [18] [18] L. Gan, N. Li, U. Topcu, and S. H. Low, 'On the exactness of convex relaxation for optimal power flow in tree networks,' in IEEE CDC, 2012, pp. 465-471. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationcitation
    § C. The OPF problem · p. 3

    For any feasible point x of OPF, φ(x) is feasible for OPF' and L(x)=L'(φ(x)); the map φ is bijective for tree networks.

    “Theorem 1 For any x ∈ F OPF, the point φ ( x ) ∈ F OPF' and satisfies L ( x ) = L ′ ( φ ( x )) . Furthermore, the map φ : F OPF →F OPF' is bijective for tree networks.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [16] [16] L. Gan, U. Topcu, N. Li, and S. Low, 'Exact convex relaxation for optimal power flow in tree networks,' arXiv preprint arXiv:1208.4076, 2012. - source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § D. Related work · p. 3

    The SOCP relaxation is exact if p_i = -∞ and q_i = -∞ for i ∈ N+.

    “Proposition 1 ( [9]) The SOCP relaxation is exact if p i = -∞ and q i = -∞ for i ∈ N + .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [9] [9] M. Farivar, C. R. Clarke, S. H. Low, and K. M. Chandy, 'Inverter var control for distribution systems with renewables,' in IEEE SmartGridComm, 2011, pp. 457-462. - source

      The introduction (p. 457) says the paper’s convex relaxation is exact when oversatisfaction of load is allowed. The available passage does not state the specific conditions p_i = q_i = −∞ for i ∈ N+, so that formulation cannot be verified from our copy. (LLM, conf=0.78)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Findingown data
    § A. Test distribution networks · p. 5

    The SCE 56-bus network has 5 MW nameplate distributed generation capacity, over 100% penetration in comparison with 3.835 MVA peak spot load.

    “the 56-bus network has 5 MWnameplate distributed generation capacity (over 100% penetration in comparison with 3 . 835 MVA peak spot load).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § A. Test distribution networks · p. 5

    The SCE 47-bus network has 6.4 MW nameplate distributed generation capacity, over 50% penetration in comparison with 11.3 MVA peak spot load.

    “the 47bus network has 6 . 4 MW nameplate distributed generation capacity (over 50% penetration in comparison with 11 . 3 MVA peak spot load)”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingnone given
    § TABLE II · p. 7

    (14), which is equivalent to C1, still holds for the third network in the worst case.

    “However, (14), which is equivalent to C1, still holds for the third network.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § TABLE II · p. 7

    In the worst case, the minimum interval covers the range of r/x for the first two networks, but not the third.

    “In the worst case, the minimum interval covers the range of r/x for the first two networks, but not the third one.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S1 (asserted).

    Premisecitation
    § B. Feasible sets of OPF-m and OPF are close. · p. 6

    Given s = p + i q, there exists a unique voltage V(s) near the nominal value that satisfies the power flow constraint (1).

    “It is claimed in [25] that given s = p + i q , there exists a unique voltage V ( s ) near the nominal value that satisfies the power flow constraint (1).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [25] [25] H.-D. Chiang and M. E. Baran, 'On the existence and uniqueness of load flow solution for radial distribution power networks,' IEEE Transactions on Circuits and Systems, vol. 37, no. 3, pp. 410-416, 1990. source

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § Theorem 2 The SOCP-m relaxation is exact if C1 holds. · p. 5

    C1 requires p and q to be small.

    “C1 requires p and q be 'small'.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Interpretationreasoning
    § Summary of contributions · p. 2

    If SOCP-m has a unique solution, any convex programming solver gives the same solution.

    “In this case, any convex programming solver gives the same solution.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § Theorem 2 The SOCP-m relaxation is exact if C1 holds. · p. 5

    The power injection constraints in (2) can be generalized to s ∈ S where S is an arbitrary set.

    “The power injection constraints in (2) can be generalized to s ∈ S where S is an arbitrary set.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § B. Feasible sets of OPF-m and OPF are close. · p. 6

    OPF-m is sandwiched between OPF and OPFε, which are ε apart.

    “To summarize, OPF-m is 'sandwiched' between OPF and OPFε , which are ' ε ' apart”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § D. Related work · p. 4

    The SOCP relaxation is exact if |V_i| = ∞ for i ∈ N+, and any one of conditions (i)-(iv) hold.

    “Proposition 2 ( [18]) The SOCP relaxation is exact if | V i | = ∞ for i ∈ N + , and any one of the following conditions hold:”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [18] [18] L. Gan, N. Li, U. Topcu, and S. H. Low, 'On the exactness of convex relaxation for optimal power flow in tree networks,' in IEEE CDC, 2012, pp. 465-471. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § C. Condition C1 holds in all test networks. · p. 6

    (14) is implied by (15).

    “note that (14) is implied by”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § A. Test distribution networks · p. 5

    The IEEE 13-bus test network is an unbalanced three-phase network with some circuit devices that are not modeled by (1).

    “The IEEE 13-bus test network [23] is an unbalanced threephase network with some circuit devices that are not modeled by (1).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [23] [23] IEEE distribution test feeders, available at http://ewh.ieee.org/soc/pes/ dsacom/testfeeders/. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

16 of 16

  • The paper relies on this without support.

    Distribution networks have a tree topology.

    “A distribution network is composed of buses and lines connecting these buses, and has a tree topology.”

  • The paper relies on this without support.

    The substation has fixed voltage and flexible power injection for power balance.

    “There is a substation in the network, which has fixed voltage and flexible power injection for power balance.”

  • The paper relies on this without support.

    The substation voltage V0 is given and fixed.

    “the substation voltage, V 0 , is given and fixed.”

  • The paper relies on this without support.

    The power flow model (1) accurately describes the physical laws.

    “In this paper, we use (1) to model the power flow.”

  • The paper relies on this without support.

    In practice, p_i and q_i are finite for every i ∈ N+.

    “In practice, p i and q i are finite for every i ∈ N + .”

  • The paper relies on this without support.

    In distribution networks, the constraints |V_i| ≤ |V_i| cannot be ignored, especially with distributed generators making the voltages likely to exceed |V|.

    “In distribution networks, the constraints | V i | ≤ | V i | cannot be ignored, especially with distributed generators making the voltages likely to exceed | V | .”
6 of 6

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Conflicting ε values for the SCE 47-bus network

    Minor

    Table I gives ε as 0.0036 for the SCE 47-bus network, while the text gives 0.0031. The stated OPFε upper bound of 1.0994 is consistent with 0.0031, not the table value.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. [23][23] IEEE distribution test feeders, available at http://ewh.ieee.org/soc/pes/ dsacom/testfeeders/. -

Cited by

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Related

Authors
Radosavljević, J. N.
Corresponding author
Jordan N. Radosavljević
ORCID
Jordan N. Radosavljević
Venue
Metaheuristic Optimization in Power Engineering
Publisher
Institution of Engineering and Technology
Published
2018-05-04
Pages
297-335
DOI
10.1049/pbpo131e_ch9
OpenAlex
W2944753221
Semantic Scholar
05eee822b11d0d349e26cc4e1888f9909024ea7e
Cited by
15
References
10
Citations by year
2020: 2 · 2018: 1 · 2017: 2 · 2016: 1 · 2015: 1 · 2014: 6 · 2013: 2
FWCI
2.53
Type
book chapter
Language
en
Open access
closed
Locations
Institution of Engineering and Technology eBooks (publishedVersion)
Topics
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Optimal Power Flow Distribution
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Electric Power System Optimization
  • Physical Sciences › Engineering › Electrical and Electronic Engineering › Smart Grid and Power Systems
Keywords
optimal power flow, distribution networks, distributed generation, uncertainty, multi-objective optimization, metaheuristic optimization, power distribution planning

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