On the Tightness of the Second-Order Cone Relaxation of the Optimal Power Flow with Angles Recovery in Meshed Networks

1 On the T ightness of the Second-Order Cone Relaxation of the Optimal Po wer Flo w with Angles Reco v ery in Meshed Networks Gine vra Larroux, Matthieu Jacobs, Mario Paolone F ellow , IEEE Abstract —This letter in vestigates properties of the second- order cone relaxation of the optimal po wer flow (OPF) pr oblem, with emphasis on relaxation tightness, nodal voltage angles reco v- ery , and alternating-curr ent-OPF feasibility in meshed networks. The theoretical discussion is supported by numerical experiments on standard IEEE test cases. Implications for power system planning are briefly outlined. Index T erms —Second Order Cone Programming, Optimal Po wer Flow , meshed networks I . I N T RO D U C T I O N Con vex relaxations of the alternating current (A C)-optimal power flow (OPF) problem are widely used in po wer sys- tem optimization, both as computationally tractable surrogates and as tools to assess feasibility and optimality . Among these, second-order-cone (SOC) relaxations based on branch- flow grid models hav e attracted particular attention because they can be solved efficiently with standard off-the-shelf second-order-cone programming (SOCP)/quadratically con- strained quadratic programming (QCQP) solv ers (e.g., interior - point methods). Milestones in the dev elopment of SOC-OPF formulations include, in chronological order , [1]–[6]. A central question for an y con ve x relaxation is tightness , i.e., whether a solution of the relaxed problem corresponds to a feasible solution of the original problem. While SOC branch-flow relaxations can be exact under known conditions in radial networks, cycle-induced consistency constraints on voltage angles make angle recovery and tightness nontrivial in meshed networks. This letter focuses on tw o widely used SOC-OPF for- mulations: the baseline model in [4], which eliminates bus voltage and branch current angles and relaxes the noncon vex power -loss equalities, with angle recovery considered only a posteriori ; and the formulation in [6], which instead retains bus v oltage angles e xplicitly in the network constraints. The main thesis of this letter is that, for meshed networks, the model in [6] is not a relaxation of the AC-OPF but rather an approximation; consequently , a zero relaxation gap does not, in general, guarantee A C-OPF feasibility in meshed networks. The remainder of the paper is or ganized as follows. Sec- tion II introduces the tw o formulations and their properties. Section III dev elops a theoretical argument and provides a counterexample to Theorem 1 in [6]. Section IV discusses implications for planning applications and concludes the paper . The authors are with the ´ Ecole Polytechnique F ´ ed ´ erale de Lausanne (EPFL), Switzerland, email: { ginevra.larroux,mario.paolone } @epfl.ch. I I . P RO B L E M F O R M U L AT I O N A. SOC-OPF formulation by [4] For bre vity , we refer to [4] for the full nomenclature; unless stated otherwise, constraints are understood to hold for all buses in N and branches in E . Given the power injections s i := s g i − s c i for i = 1 , . . . , n , the power flow (PF) variables are x ( s ) := ( S, I , V , s 0 ) , where S := { S ij } ( i,j ) ∈ E , I := { I ij } ( i,j ) ∈ E , V := { V i } i ∈ N \{ 0 } , and s 0 denotes the slack injection. The AC-PF equations are listed in (1). s j = X k : j → k S j k − X i : i → j  S ij − z ij | I ij | 2  + y ∗ j | V j | 2 (1a) V i − V j = z ij I ij (1b) S ij = V i I ∗ ij (1c) In the associated OPF, the injections s are also decision variables and the objective is chosen con vex and independent of phase angles, e.g., f ( ˆ h ( x ) , s ) := P ( i,j ) ∈ E r ij | I ij | 2 . W e refer to [4] for the definition of the projection ˆ h ( · ) . Operational constraints include bounds on generations and loads, voltage magnitudes, and branch currents or po wers. For bre vity , we explicitly report only the ampacity constraint and use (2) to refer to all operational constraints. | I ij | ≤ I ij (2) The resulting non-approximated OPF problem is (OPF) min x,s f ( ˆ h ( x ) , s ) s.t. (1) , (2) . (3) The relaxation in [4] is obtained in two stages. First, voltage and current angles are eliminated by replacing (1) with (4), yielding the (OPF-ar) model, in the lifted v ariables l ij := | I ij | 2 and v i := | V i | 2 . Second, the nonconv ex quadratic equality (4d) is relaxed into a second-order cone inequality , resulting in the con vex model (OPF-cr). p j = X k : j → k P j k − X i : i → j ( P ij − r ij ℓ ij ) + g j v j (4a) q j = X k : j → k Q j k − X i : i → j ( Q ij − x ij ℓ ij ) + b j v j (4b) v j = v i − 2 ( r ij P ij + x ij Q ij ) +  r 2 ij + x 2 ij  ℓ ij (4c) l ij = P 2 ij + Q 2 ij v i (4d) If the hypotheses 1 hold, the authors show that any optimal solution of (OPF-cr) is also optimal for (OPF-ar), ev en on 1 (I) The network graph G is connected; (II) the cost function is con ve x; (III) the cost function is strictly increasing in l , non-increasing in s c , and independent of S ; (IV) the optimal A C-OPF is feasible. 2 meshed networks, provided there are no upper bounds on ac- tiv e and reacti ve loads. Crucially , they also note that recov ering angles from an (OPF-ar) solution is alw ays possible in radial networks, but not necessarily in meshed networks. Theorem 2 in [4] giv es a condition to determine whether a branch- flow solution can be recovered from an (OPF-ar) solution, together with the corresponding recov ery computation. As this is nonconv ex, two angle-recov ery algorithms (“centralized” and “distrib uted”) are proposed, b ut they are not guaranteed to succeed in meshed networks. In the related work [5], a con vexification approach for meshed networks based on phase shifters is proposed when the angle-recovery condition fails. On a final note, [7] discusses alternati ve suf ficient conditions for e xactness in radial networks that do not rely on allo wing load oversatisf action (unrealistic in practice). B. SOC-OPF formulation by [6] Reference [6] proposes a formulation that explicitly includes bus voltage angles and distinguishes between measurable branch quantities (denoted with ˜ · ) and branch-flow OPF vari- ables that exclude shunt contributions, enabling the correct treatment of branch current ampacity limits. Using the notation of [6], the A C-PF equations are in (5). p n − p d n = X l  A + nl p s l − A − nl p o l  + G n V n (5a) q n − q d n = X l  A + nl q s l − A − nl q o l  − B n V n (5b) V s l − V r l = 2 R l p s l + 2 X l q s l − R l p o l − X l q o l (5c) v s l v r l sin θ l = X l p s l − R l q s l (5d) V n = v 2 n (5e) θ l = θ s l − θ r l (5f) q o l = p 2 s l + q 2 s l V sl X l (5g) p o l X l = q o l R l (5h) Define A s l n , A r l n ∈ R | L |×| N | the branch-to-node incidence matrices with A s ( r ) l n = 1 if node n is the sending (recei ving) end of branch l . Then (5f) can be equiv alently written as (6). θ l = ( A s l n − A r l n ) θ n (6) As above, for brevity we e xplicitly report only the ampacity constraint among the operational limits. Note that, in this formulation, bounds are also imposed on bus voltage angles and on branch angle differences. ˜ i 2 s ( r ) l = ˜ p 2 s ( r ) l + ˜ q 2 s ( r ) l V s ( r ) l ≤ e K l (7) The corresponding non-approximated A C-OPF is written as ( o-A COPF ) min Ω o-ACOPF f (Ω o-A COPF ) s.t. (5) , (7) (8) where f ( · ) is a con ve x objectiv e function. The proposed SOC-OPF formulation then modifies two constraints: the non-con ve x relation (5d) is replaced by the linear (9a), and the losses equality (5g) is relaxed to the conic inequality (9b). In addition, the ampacity constraint (7) is re written in terms of measurable quantities (9c). Finally , a tightness-promoting constraint (9d) is introduced under the hypothesis ( θ min l , θ max l ) ⊆  − π 2 , π 2  , θ min l = − θ max l . θ l = X l p s l − R l q s l (9a) q o l ≥ p 2 s ( r ) l + q 2 s ( r ) l V s ( r ) l X l (9b) K o l =  e K l − V s ( r ) l B 2 s ( r ) l + 2 q s ( r ) l B s ( r ) l  X l ≥ q o l (9c) V s l V r l sin 2 ( θ max l ) ≥ θ 2 l (9d) The resulting SOC-OPF model is ( SOC-A COPF ) min Ω SOC-ACOPF f (Ω SOC-A COPF ) s.t. (5a) , (5b) , (5c) , (5f) , (5h) , (9) . (10) Reference [6] provides sev eral theoretical results on tight- ness and on the recovery of A C-OPF-feasible operating points. The theorems most rele v ant to this letter are recalled below . Theorem 1 (Theorem 1 in [6]) . Assume ( θ min l , θ max l ) =  − π 2 , π 2  and ( v min n , v max n ) = (0 . 9 , 1 . 1) , r eplacing (5d) with (9a) relaxes the (o-A COPF) pr oblem. Theorem 2 (Theorem 4 in [6]) . If the optimal solution Ω ∗ of the (SOC-A COPF) model with the additional constr aint (9d) gives a tight solution q ∗ o l = p ∗ 2 s l + q ∗ 2 s l V ∗ s l X l , ∀ l ∈ L , the global optimum solution Ω ∗ 0 of the (o-A COPF) model is Ω ∗ 0 := { Ω ∗ \ ( θ ∗ l , V ∗ n ) } ∪ n v ∗ 0 ,n := p V ∗ n , θ ∗ 0 ,l := arcsin  θ ∗ l v ∗ s l v ∗ r l o . I I I . D I S C U S S I O N This section provides a counterargument to Theorem 1 in [6]. W e show that there exist operating points that are feasible for the A C-PF (and hence belong to the feasible re gion of the corresponding AC-OPF, the (o-A COPF) model in [6]), but that do not belong to the feasible set of the SOC-OPF model proposed in [6] (SOC-ACOPF) once the linearized angle relation (9a) is imposed together with angle consistency in meshed netw orks. Consequently , in meshed networks, (SOC- A COPF) is an appr oximation rather than a r elaxation of (o- A COPF). This also undermines Theorems 2–4 in [6], whose proofs build upon Theorem 1. A. Theoretical considerations Constraint (9a) is obtained by linearizing the nonlinear relation (5d) around the operating point θ l ≈ 0 and v n ≈ 1 . The proof strategy in Theorem 1 of [6] argues that, under the bounds ( θ min l , θ max l ) =  − π 2 , π 2  and ( v min n , v max n ) = (0 . 9 , 1 . 1) , the feasible set induced by the linear constraint (9a) contains that of the nonlinear term v s l v r l sin( θ l ) , hence suggesting that the modified model is a relaxation. Ho we ver , this ar gument does not account for c ycle consistenc y of v oltage angles (11) in meshed networks. X l ∈ C θ l = 0 mo d 2 π, ∀ cycles C (11) In [6] it is stated that, introducing θ n as decision v ariables and defining θ l through (5f), “implicitly” enforces the cyclic 3 condition (11). Lemma 1 highlights where the relaxation claim fails for meshed grids. Lemma 1. Consider a meshed network and nodal voltage angle differ ences across br anches defined by (6) . If the non- appr oximated branc h Kir chhoff voltage law (KVL) equation (5d) is replaced by the linearized constraint (9a) in the power flow equations, then the resulting branch angle differ ences ar e not guaranteed to satisfy the cycle constraint (11) . Pr oof. In radial networks, for any θ l , the nodal angles θ n are determined uniquely up to the slack reference, since A s l n − A r l n is full rank and the Rouch ´ e–Capelli theorem holds by construction. In meshed networks, by contrast, θ l must sat- isfy the additional c ycle constraints (11). Ev en when the conic inequality (9b) is tight (i.e., the relaxation gap is zero), if the (ov erdetermined) system ( A s l n − A r l n , θ l ) satisfies a solu- tion θ n , the system  A s l n − A r l n , arcsin  θ l ( A s l n − A r l n ) √ V n  may not be satisfied, unless (9a) and (5d) are simultaneously satisfied. This can occur only in degenerate cases, i.e., when v s l v r l = 1 and θ l ≈ 0 . B. Numerical counter example W e consider two standard IEEE test systems in pandapower : the IEEE 33-bus system (radial) and the IEEE 39-bus system (meshed). F or each case, we start from the pandapower A C-PF solution, denoted by ( p 0 n , q 0 n , v 0 n , θ 0 n ) . a) Step 1: verify feasibility in the non-r elaxed (o- A COPF) model: W e substitute ( p 0 n , q 0 n , v 0 n , θ 0 n ) into the non- relaxed formulation of [6] in (5) by re writing it as the linear system Ax = b in (12).     A + nl Z n − A − nl Z n Z n A + nl Z n − A − nl 2 R 2 X − R − X X − R Z l Z l Z l Z l X − R       p s l q s l p o l q o l   =     p 0 n − G n V 0 n q 0 n + B n V 0 n A l V 0 n A l v 0 n sin( θ 0 l ) 0 | L |     (12) Here Z n := 0 | N |×| L | and Z l := 0 | L |×| L | denote zero matri- ces, R := diag( R l ) , X := diag ( X l ) , and A l := A s l n − A r l n . The branch angle differences are obtained from b us angles using their definition, θ 0 l = A l θ 0 n . W e solv e (12) numerically using a least-squares method and verify that the residuals are (up to numerical tolerances) zero. b) Step 2: test cycle consistency of the voltag e angle differ ences acr oss branc hes implied by the (SOC-ACOPF) model: W e now construct the voltage angle differences across branches under the (SOC-ACOPF) model, i.e. θ 1 l := v 0 s l v 0 r l sin( θ 0 l ) . W e then test whether it is compatible with a set of bus voltage angles by checking if there exists θ 1 n such that ( A sl − A rl ) θ 1 n = θ 1 l . W e solve this system in least squares after removing the slack-b us column (gauge in v ariance), yielding an exact square system for the radial case and an o verdetermined system for the meshed network. The residual norm is zero if and only if the cycle constraints are satisfied. Residual distrib utions for both networks are reported in Fig. 1. For the system ( A, b ) , residuals are numerically zero in both cases, confirming (o-ACOPF) feasibility of the pandapower operating points. For the system ( C, d ) with -1e-14 -1e-10 -1e-06 1e-06 1e-10 1e-14 [p.u.]/[rad] (A,b) (C,d) radial meshed Fig. 1. Residuals when solving the linear systems ( Ax ≈ b ) and ( C θ n ≈ d ) , for the radial IEEE 33-bus (red) and meshed IEEE 39-bus (blue) networks. C = A s l n − A r l n and d = θ 1 l , residuals are numerically zero for the radial network (as expected), whereas they are non- negligible for the meshed network, with a maximum residual of 0 . 0245 degrees. This de viation, when propagated through the AC-PF equations, yields branch po wer -flo w errors on the order of 0 . 5 p.u., indicating that θ 1 l violates cycle consistency . I V . C O N C L U S I O N S This letter sho ws that, in meshed networks, the SOC-OPF formulation in [6] cannot guarantee A C-OPF feasibility , ev en when the conic relaxation of the quadratic loss equality is tight. At the same time, [6] provides a practically relev ant way to incorporate v oltage angles directly into the OPF constraints. Although this yields an approximation of the AC-OPF, the resulting operating point is often close to an A C-PF-feasible one. By contrast, formulations such as [4] rely on a posteriori cycle-consistenc y checks and angle recovery , which may fail and thus yield a solution that is infeasible or arbitrarily inaccurate. The approach in [5] instead restores exactness in meshed networks via phase shifters, but at a cost that may be of questionable economic viability . In both cases, an a posteriori procedure is needed to obtain an A C-OPF feasible solution e ven when the relaxation is tight. R E F E R E N C E S [1] M. Baran and F . 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