Dataset for Planning of Active Distribution Systems: Assessing the Impact of PV Inverter Volt-VAR Control on Reinforcement Strategies

Planning of Active Distribution Systems: Assessing the Impact of PV Inverter Volt-VAR Control on Reinforcement Strategies Technical Appendix – Linearization and Model Formulation Jo˜ao Vitor G. de Ara´ujo1, Wandry R. Faria1, and Benvindo R. Pereira Junior1 1Department of Electrical and Computer Engineering, University of S˜ao Paulo (USP), S˜ao Carlos, Brazil [email protected], [email protected], [email protected] 1 Power Flow Formulation for Distribution Systems This subsection summarizes the set of nonlinear equations that represent the steady-state operation of radial distribution systems. The model is based on the power flow formulation for balanced networks, where loads and distributed generators are represented by their active and reactive power injections. Considering Kirchhoff’s current law at each node i, the active and reactive power balance can be expressed as: X ji∈ΩB Pji,t −X ij∈ΩB (Pij,t +RijI2 ij,t)+PS i,t −PD i,t = 0 ∀i∈(ΩS N∪ΩD N),∀t∈ΩT(1) X ji∈ΩB Qji,t −X ij∈ΩB (Qij,t +XijI2 ij,t)+QS i,t −QD i,t = 0 ∀i∈(ΩS N∪ΩD N),∀t∈ΩT(2) where Pij,t and Qij,t denote the active and reactive power flows in branch ij at load level t,Rij and Xij are the branch resistance and reactance, and Iij,t represents the branch current magnitude. The active and reactive demands at node iare PD i,t and QD i,t, respectively, and PS i,t and QS i,t correspond to the power injections from the substation. The magnitude of the branch current can be obtained from the power flow as: I2 ij,t =P2 ij,t +Q2 ij,t V2 j,t ∀ij ∈ΩB,∀t∈ΩT(3) Applying Kirchhoff’s voltage law between nodes iand jyields the voltage drop equation: V2 i,t −V2 j,t = 2(RijPij,t +Xij Qij,t)+Z2 ijI2 ij,t ∀ij ∈ΩB,∀t∈ΩT(4) where Z2 ij =R2 ij +X2 ij. Equations (1)–(4) define the nonlinear steady-state power flow model for radial distribution networks. This formulation is widely adopted in distribution studies as it accurately represents the voltage and power profiles along the feeders. As the proposed planning model is formulated as a Mixed-Integer Linear Programming (MILP) problem, the nonlinear relationships above were linearized to obtain an equivalent linear representation suitable for optimization. The linearization process and the corresponding set of constraints are presented in Section 2. 1 2 Linearization of the AC Power Flow for Distribution Systems Some nonlinearities in the power flow equations can be removed through variable substitutions. In particular, replace I2 ij,t and V2 i,t by ISQ ij,t ≥0 and VSQ i,t ≥0, respectively. The set of equations (5)–(8) is an exact equivalent of the power flow relations presented in Section 1: ISQ ij,t VSQ j,t =P2 ij,t +Q2 ij,t ∀ij ∈ΩB,∀t∈ΩT(5) VSQ i,t −VSQ j,t = 2(RijPij,t +Xij Qij,t)+Z2 ijISQ ij,t ∀ij ∈ΩB,∀t∈ΩT(6) X ji∈ΩB Pji,t −X ij∈ΩB (Pij,t +RijISQ ij,t )+PS i,t −PD i,t = 0 ∀i∈ΩN,∀t∈ΩT(7) X ji∈ΩB Qji,t −X ij∈ΩB (Qij,t +XijISQ ij,t )+QS i,t −QD i,t = 0 ∀i∈ΩN,∀t∈ΩT(8) The substitution does not remove the nonlinearity of the current equation. Expression (5) remains nonlinear due to the product of variables ISQ ij,t VSQ j,t on the left-hand side and the quadratic terms P2 ij,t +Q2 ij,t on the right-hand side. A direct linear approximation of the product ISQ ij,t VSQ j,t is obtained by fixing the voltage magnitude at its nominal value, that is, VSQ j,t =V2 nom. Since nodal voltages Vj,t lie in a narrow interval [V,¯ V] and Vnom ∈[V , ¯ V], the approximation in (9) is accurate and computationally efficient: ISQ ij,t VSQ j,t ≈ISQ ij,t (Vnom)2∀ij ∈ΩB,∀t∈ΩT(9) To linearize the quadratic terms P2 ij,t and Q2 ij,t in (5), we adopt a piecewise linear approximation with Λ equal-width segments. Let ¯ ∆S ij denote the segment size. The linearization of P2 ij,t +Q2 ij,t is given by (10)–(17): P2 ij,t +Q2 ij,t ≈ Λ X l=1 (2l−1) ¯ ∆S ij (∆P ij,t,l + ∆Q ij,t,l)∀ij ∈ΩB,∀t∈ΩT(10) Pij,t =P+ ij,t −P− ij,t ∀ij ∈ΩB,∀t∈ΩT(11) Qij,t =Q+ ij,t −Q− ij,t ∀ij ∈ΩB,∀t∈ΩT(12) P+ ij,t +P− ij,t = Λ X l=1 ∆P ij,l,t ∀ij ∈ΩB,∀t∈ΩT(13) Q+ ij,t +Q− ij,t = Λ X l=1 ∆Q ij,l,t ∀ij ∈ΩB,∀t∈ΩT(14) 0≤∆P ij,t,l ≤¯ ∆S ij/(Vnom)2∀ij ∈ΩB,∀t∈ΩT,∀l∈ {1,...,Λ}(15) 0≤∆Q ij,t,l ≤¯ ∆S ij/(Vnom)2∀ij ∈ΩB,∀t∈ΩT,∀l∈ {1,...,Λ}(16) P+ ij,t, P− ij,t, Q+ ij,t, Q− ij,t ≥0∀ij ∈ΩB,∀t∈ΩT(17) The segment size is ¯ ∆S ij =Vnom ¯ Iij Λ∀ij ∈ΩB,(18) 2 where (2l−1) ¯ ∆S ij is the slope of the l-th linear piece for branch ij, Λ is the number of segments, and ∆P ij,t,l and ∆Q ij,t,l are the l-th segment variables for active and reactive power at load level t. The auxiliary variables P+ ij,t, P− ij,t, Q+ ij,t and Q− ij,t model absolute values |Pij,t|and |Qij,t|through (12)–(14), while (15)–(16) set segment bounds. To improve the fit near the origin, the first segment slope can be set to 5/6. In this case, P2 ij,t +Q2 ij,t ≈5 6¯ ∆S ij (∆P ij,t,1+ ∆Q ij,t,1) + Λ X l=2 (2l−1) ¯ ∆S ij (∆P ij,t,l + ∆Q ij,t,l)∀ij ∈ΩB,∀t∈ΩT. (19) Finally, the following linear constraints compose the MILP power flow surrogate used in the planning model: X ji∈ΩB Pji,t −X ij∈ΩB (Pij,t +RijISQ ij,t )+PS i,t −PD i,t = 0 ∀i∈ΩN,∀t∈ΩT(20) X ji∈ΩB Qji,t −X ij∈ΩB (Qij,t +XijISQ ij,t )+QS i,t −QD i,t = 0 ∀i∈ΩN,∀t∈ΩT(21) VSQ i,t −VSQ j,t = 2(RijPij,t +Xij Qij,t)+Z2 ijISQ ij,t ∀ij ∈ΩB,∀t∈ΩT(22) ISQ ij,t (Vnom)2=5 6¯ ∆S ij Mij,t,1+ Λ X l=2 (2l−1) ¯ ∆S ij Mij,t,l ∀ij ∈ΩB,∀t∈ΩT(23) Mij,t,l = ∆P ij,t,l + ∆Q ij,t,l ∀ij ∈ΩB,∀t∈ΩT(24) Pij,t =P+ ij,t −P− ij,t ∀ij ∈ΩB,∀t∈ΩT(25) Qij,t =Q+ ij,t −Q− ij,t ∀ij ∈ΩB,∀t∈ΩT(26) P+ ij,t +P− ij,t = Λ X l=1 ∆P ij,l,t ∀ij ∈ΩB,∀t∈ΩT(27) Q+ ij,t +Q− ij,t = Λ X l=1 ∆Q ij,l,t ∀ij ∈ΩB,∀t∈ΩT(28) 0≤∆P ij,t,l ≤¯ ∆S ij/(Vnom)2∀ij ∈ΩB,∀t∈ΩT,∀l∈ {1,...,Λ} (29) 0≤∆Q ij,t,l ≤¯ ∆S ij/(Vnom)2∀ij ∈ΩB,∀t∈ΩT,∀l∈ {1,...,Λ} (30) P+ ij,t, P− ij,t, Q+ ij,t, Q− ij,t ≥0∀ij ∈ΩB,∀t∈ΩT(31) V2≤VSQ i,t ≤¯ V2∀i∈ΩN,∀t∈ΩT(32) 0≤ISQ ij,t ≤¯ I2 ij ∀ij ∈ΩB,∀t∈ΩT(33) 2.1 Load Model An essential input for power flow calculations is the nodal demand. To capture voltage sensitivity, we adopt the ZIP model (voltage-dependent load), which represents the active and reactive demands as a second-order polynomial combining constant-impedance (Z), constant-current (I), and constant-power (P) components. Let αZ i, αI i, αP ibe the activepower participation factors and βZ i, βI i, βP ithe reactive-power counterparts at node i. 3 Then, for load level t: ˆ PD i,t =PD i,t "αZ i VSQ i,t (Vnom)2+αI i Vi,t Vnom +αP i#∀i∈ΩN,∀t∈ΩT(34) ˆ QD i,t =QD i,t "βZ i VSQ i,t (Vnom)2+βI i Vi,t Vnom +βP i#∀i∈ΩN,∀t∈ΩT(35) where ˆ PD i,t and ˆ QD i,t are the voltage-dependent active/reactive demands; PD i,t and QD i,t are the nominal (at Vnom) active/reactive demands. Because the ZIP model depends on both VSQ i,t and Vi,t, we introduce a linear approximation to retrieve Vi,t =qVSQ i,t from the square-voltage variable. Let V0:= (¯ V+V)/2 be the midpoint of the admissible voltage range, and note that √xlinearized at x0=V2 0 yields: Vi,t ≈V0+1 2V0VSQ i,t −V2 0∀i∈ΩN,∀t∈ΩT(36) This first-order expansion around V0provides a tight linear surrogate on the typical operating band [V , ¯ V]. Finally, the voltage-dependent demands in (34)–(35) replace PD i,t and QD i,t in the nodal balance constraints (20)–(21). Constraint (36) must be added to the linear set (20)–(33) to compute Vi,t from VSQ i,t . 2.2 Model of Dispatchable Distributed Generators Synchronous distributed generators are modeled through a polygonal approximation of the P–Qcapability curve, suitable for MILP implementation. The same formulation is applied to the generator located at the substation, which is also a synchronous unit, using its own active and reactive power variables (PS i,t,QS i,t) and rated capacity ¯ SS i,t. 0≤PDG i,t ≤¯ SDG i,t (37) −√2¯ SDG i,t −PDG i,t ≤QDG i,t ≤√2¯ SDG i,t −PDG i,t (38) − ¯ SDG i,t sin(π 8)−tan3π 8PDG i,t !≤QDG i,t ≤ ¯ SDG i,t sin(π 8)−tan3π 8PDG i,t (39) − ¯ SDG i,t sin(3π 8)−tanπ 8PDG i,t !≤QDG i,t ≤ ¯ SDG i,t sin(3π 8)−tanπ 8PDG i,t (40) −PDG i,t tanarccos(¯τcap i)≤QDG i,t ≤PDG i,t tanarccos(¯τind i)(41) ∀i∈ΩDG N, t ∈ΩT.(42) 2.3 Model of Capacitor Banks Capacitor banks (CBs) are incorporated to provide reactive power support, reduce losses, and improve voltage profiles. The proposed formulation models two types of devices: fixed (CBF) and switched (CBSW ) banks, whose reactive injection depends on the nodal 4 voltage magnitude. The total reactive power injected by all CBs connected to node iis expressed as: QCB i,t =X bf∈ΩF CB qF i,bf,t +X bsw∈ΩSW CB qSW i,bsw,t,∀i∈ΩN, t ∈ΩT.(43) Reactive injections are defined as functions of the squared voltage VSQ i,t and the susceptance of each installed unit: qF i,bf,t −BF bf VSQ i,t ≥ − ¯ V2BF bf (1 −xCB i,bf ),(44) qF i,bf,t −BF bf VSQ i,t ≤ −V2BF bf (1 −xCB i,bf ),(45) qF i,bf,t ≤¯ V2BF bf xCB i,bf ,∀i∈ΩN, bf ∈ΩBCF ix, t ∈ΩT.(46) qSW i,bsw,t −BSW bsw VSQ i,t ≥ − ¯ V2BSW bsw (1 −xSWop i,bsw,t),(47) qSW i,bsw,t −BSW bsw VSQ i,t ≤ −V2BSW bsw (1 −xSWop i,bsw,t),(48) qSW i,bsw,t ≤¯ V2BSW bsw xSWop i,bsw,t, xSWop i,bsw,t ≤xSWinst i,bsw ,∀i∈ΩN, bsw ∈ΩBCSw, t ∈ΩT.(49) Finally, the number of installed fixed and switched units is limited by: X i,bf xCB i,bf ≤¯ NCBF,X i,bsw xSWinst i,bsw ≤¯ NCBSW .(50) The nonlinear dependency between voltage and reactive injection is represented through adisjunctive Big-M formulation, where binary variables xCB i,bf and xSWinst i,bsw indicate the installation of fixed and switched CBs, while xSWop i,bsw,t controls the operational status of switched units across representative load levels. This approach ensures a linear and tractable representation of the voltage-dependent behavior of CBs in the MILP model. 2.4 Voltage Regulator Model Voltage regulators (VRs) are devices installed in distribution feeders to maintain adequate voltage levels. Their allocation and operation are represented by a linearized model using disjunctive Big-M constraints, as defined below: VSQ j,t =a2 ij,tVSQ i,t ,∀ij ∈ΩB, t ∈ΩT,(51) δV R ij,t =VSQ j,t −VSQ i,t =VSQ i,t (a2 ij,t −1),∀ij ∈ΩB, t ∈ΩT,(52) δV R ij,t +∆V R ij (∆V R ij + 2)VSQ i,t ≥M(1 −XV R ij ),∀ij ∈ΩB, t ∈ΩT,(53) δV R ij,t +∆V R ij (∆V R ij + 2)VSQ i,t ≤M(1 −XV R ij ),∀ij ∈ΩB, t ∈ΩT,(54) −MXV R ij ≤δV R ij,t ≤MXV R ij ,∀ij ∈ΩB, t ∈ΩT,(55) X g∈ΩV R XV R ij,g =XV R ij ,∀ij ∈ΩB,(56) X ij∈ΩB XV R ij ≤¯ NV R,(57) XV R ij,g ¯ I2 g≥X a∈ΩL ISQ ij,a,t −¯ I2 a(1 −XV R ij,g ),∀ij ∈ΩB, g ∈ΩV R, t ∈ΩT.(58) 5