Electronic companion — Coordinated Expansion Planning of T&D Systems Using Distributed Optimization Approaches

1 Electronic Companion – Coordinated Expansion Planning of T&D Systems Using Distributed Optimization Approaches Nahid-Ur-Rahman Chowdhury, Graduate Student Member, IEEE, Gregorio Mu˜ noz-Delgado, Senior Member, IEEE, Natalia Alguacil, Senior Member, IEEE, and Javier Contreras, Fellow, IEEE NOMENCLATURE The detailed centralized mathematical problem formulation of the coordinated transmission and distribution systems expansion planning (CTDSEP) problem, corresponding to Section II in the main paper, which is proposed as an instance of mixed-integer linear programming (MILP), is presented in this electronic companion. The proposed model is updated from the model described in [3], where discrete generation investments are considered at both system levels. The notation used throughout the problem formulation is presented in this section. The superscripts “T” and “D” depict the association of sets, parameters, and variables to the transmission and distribution levels, respectively. Indices aIndex for distribution systems. dIndex for demands. gIndex for generators. lIndex for lines. nIndex for nodes. oIndex for operating conditions. pIndex for generator types. Sets AIndex set of distribution systems. AnIndex set of distribution systems connected to interface node n. DD,DTIndex sets of demands. DD n,DT nIndex sets of demands connected to node n. GD pa,GT pIndex sets of existing generators. GD+ pa ,GT+ pIndex sets of candidate generators. GD pn,GT pn Index sets of existing generators connected to node n. GD+ pn ,GT+ pn Index sets of candidate generators connected to node n. LT,LT+Index sets of existing and candidates transmission lines, respectively. LD a,LD+ aIndex sets of existing and candidates distribution lines, respectively. LD,F a,LD,R aIndex sets of existing fixed lines and existing replaceable lines, respectively, where LD,F a∪ LD,R a=LD a. ND a,ND+ a,NT, N∞ Index sets of distribution, new distribution, transmission, and interface nodes, respectively, where ND+ a⊂ ND aand N∞⊂ NT. OIndex set of operating conditions. PD,PTSets of generator types. PD={CD, W D} and PT={CT, WT}, where Cand W stand for conventional and renewable-based power generation, respectively. Parameters fr(l),to(l)Sending and receiving nodes. ICD a, ICTAnnualized investment budgets. ICG,D gp ,ICG,T gp Annualized investment cost coefficients for generators. ICL,D l,ICL,T lAnnualized investment cost coefficients for lines. MD,MTSufficiently large positive constants. OCG,D gp ,OCG,T gp Production cost coefficients for generators. OCU,D d,OCU,T dLoad-shedding cost coefficients. PD,D d,PD,T dPeak consumption levels. PG,T gp ,SG,D gp Power capacities of existing generators. PL,T l,SL,D lLine power flow capacities. RD lDistribution line resistance. VD,VDLower and upper limits for distribution nodal voltages. XG,D gp ,XG,T gp Power capacities of candidate generators. XD l,XT lLine reactances. ∆oNumber of hours for operating conditions o. ζno Nodal power factor. µL no,µW no Average factors for demand and renewablebased power production. µLG oDemand growth factor. πoProbability of occurrence of long-term uncertainty scenario for operating conditions o. 2 Variables cI,D a,cI,T Annualized investment costs. cO,D a,cO,T Expected operating costs. pG,D gpo ,pG,T gpo Active power outputs of generators. pL,D lo ,pL,T lo Active power flows across lines. pU,D do ,pU,T do Levels of load shedding. p∞,D nao ,p∞,T no Active power injections at interface nodes. qG,D gpo ,qL,D lo Generation levels and line flows of reactive power. vno,δno Squared magnitude and phase angle of nodal voltages. xG,D gp ,xG,T gp Binary investment variables for generators. xL,D l,xL,T lBinary investment variables for lines. I. CENTRALIZED FORMULATION OF THE CTDSEP PROBLEM In this section, we present the detailed centralized mathematical formulations of the CTDSEP problem under uncertainty. The following constraints have been numbered in such a way that the combination after “EC.” allows identifying the corresponding constraints of Section II in the main manuscript. A. Objective Function and Cost-Related Terms The objective function minimized in (EC.A.1) represents the expected total cost, which includes the annualized investment costs and the anticipated operating costs at the transmission and distribution levels. Expression (EC.A.2) denotes the annualized transmission investment cost, which includes two terms related to installing new generators and new transmission lines, respectively. In (EC.A.3), the transmission operating cost is formulated as the sum of the expected production costs and load shedding costs. Expression (EC.A.4) stands for the distribution system and makes up the same cost components as (EC.A.2). Expression (EC.A.5) corresponds to the distribution system and makes up the same cost components as (EC.A.3). Minimize cI,T ,cI,D a,cO,T ,cO,D a, pG,T gpo ,pG,D gpo ,pL,T lo ,pL,D lo , pU,T do ,pU,D do ,p∞,T no ,p∞,D nao , qG,D gpo ,qL,D lo ,vno,δno, xG,T gp ,xG,D gp ,xL,T l,xL,D l cI,T +cO,T +X a∈A cI,D a+cO,D a (EC.A.1) Mathematically, all these costs are defined as: cI,T =X p∈PTX g∈GT+ p ICG,T gp xG,T gp +X l∈LT+ ICL,T lxL,T l(EC.A.2) cO,T =X o∈O πo∆o X p∈PTX g∈(GT p∪GT+ p) OCG,T gp pG,T gpo +X n∈NTX d∈DT n OCU,T dpU,T do  (EC.A.3) cI,D a=X p∈PDX g∈GD+ pa ICG,D gp xG,D gp +X l∈LD+ a ICL,D lxL,D l;∀a∈A (EC.A.4) cO,D a=X o∈O πo∆o X p∈PDX g∈(GD pa∪GD+ pa ) OCG,D gp pG,D gpo +X n∈ND a X d∈DD n OCU,D dpU,D do  ;∀a∈A.(EC.A.5) B. Constraints on the Investments The constraints corresponding to the generation and network investment decisions related with the transmission system are presented in (EC.B.1)–(EC.B.2). Constraint (EC.B.1) imposes the binary nature of the variables modeling investments in candidate generators at the transmission level. Constraint (EC.B.2) regulates the binary nature of the variables modeling investments in transmission lines. The constraints corresponding to the generation and network investment decisions at the distribution level are presented in (EC.B.3)–(EC.B.4). Constraint (EC.B.3) imposes the binary nature of the variables modeling investments in candidate generators at the distribution level. Constraint (EC.B.4) regulates the binary nature of the variables modeling investments in distribution lines. Additionally, expression (EC.B.5) sets a limit on the annual investments budget at the transmission level, and expression (EC.B.6) restricts a distinct budget for each distribution system. xG,T gp ∈{0,1};∀g∈GT+ p,∀p∈PT(EC.B.1) xL,T l∈{0,1};∀l∈LT+(EC.B.2) xG,D gp ∈{0,1};∀g∈GD+ pa ,∀p∈PD,∀a∈A (EC.B.3) xL,D l∈{0,1};∀l∈LD+ a,∀a∈A (EC.B.4) cI,T ≤ICT(EC.B.5) cI,D a≤ICD a;∀a∈A.(EC.B.6) C. Constraints on the Transmission System The operational aspects of transmission system components are modeled in (EC.C.1)–(EC.C.11) excluding the interface nodes where the distribution systems are connected via the corresponding substations. The interface nodes expressions are detailed later. For the remaining components of the transmission system, the operational model is defined by the set of constraints (EC.C.1)–(EC.C.11) for the set of operating conditions ∀o∈O. At first, the impact of the transmission network is characterized by a linearized dc network model, as outlined in (EC.C.1), which is standard in transmission network planning [28], [29]. Specifically, expression (EC.C.1) addresses power balance for all transmission nodes, excluding interface nodes. Power flows for existing transmission lines are described in (EC.C.2). Then, voltage angles at the reference node are defined in (EC.C.3). Later, constraints (EC.C.4)–(EC.C.6) establish the upper and lower bounds on flows and injections 3 for existing assets within the transmission system, including lines, as well as conventional and renewable generators. Furthermore, power flows for candidate transmission lines are described in (EC.C.7), with the linearized approach by using the disjunctive-constraint-based transformation technique outlined in [28]. Later, constraints (EC.C.8)–(EC.C.10) establish the upper and lower bounds on flows and injections for candidate assets within the transmission system, including lines, as well as conventional and renewable generators. Lastly, nodal load shedding is constrained in (EC.C.11). X p∈PTX g∈(GT pn ∪GT+ pn ) pG,T gpo −X l∈(LT∪LT+) |fr(l)=n pL,T lo +X l∈(LT∪LT+) |to(l)=n pL,T lo =X d∈DT n µLG oµL noPD,T d−pU,T do ;∀n∈(NT\N ∞),∀o∈O (EC.C.1) pL,T lo =1 XT lδfr(l)o−δto(l)o;∀l∈LT,∀o∈O (EC.C.2) δno = 0; ∀n:ref.,∀o∈O (EC.C.3) −PL,T l≤pL,T lo ≤PL,T l;∀l∈LT,∀o∈O (EC.C.4) 0≤pG,T gpo ≤PG,T gp ;∀g∈GT p,∀p∈CT,∀o∈O (EC.C.5) 0≤pG,T gpo ≤µW no PG,T gp ; ∀n∈N T,∀g∈GT pn,∀p∈WT,∀o∈O (EC.C.6) −MT(1 −xL,T l)≤pL,T lo −1 XT lδfr(l)o−δto(l)o ≤MT(1 −xL,T l); ∀l∈LT+ ,∀o∈O (EC.C.7) −PL,T lxL,T l≤pL,T lo ≤PL,T lxL,T l;∀l∈LT+,∀o∈O (EC.C.8) 0≤pG,T gpo ≤xG,T gp XG,T gp ;∀g∈GT+ p,∀p∈CT,∀o∈O (EC.C.9) 0≤pG,T gpo ≤µW no xG,T gp XG,T gp ; ∀n∈N T,∀g∈ GT+ pn ,∀p∈WT,∀o∈O (EC.C.10) 0≤pU,T do ≤µLG oµL no PD,T d;∀n∈NT ,∀d∈DT n,∀o∈O (EC.C.11) D. Constraints on the Distribution Systems The operational model for the distribution system is defined by the set of constraints (EC.D.1)–(EC.D.22) for the set of distribution systems ∀a∈ A, and the set of operating conditions ∀o∈ O. Here, constraints (EC.D.1)–(EC.D.3), and (EC.D.11)–(EC.D.12) depict a linearized ac network model that is appropriate for a radial distribution network [30], [31]. Active and reactive power balances at distribution nodes are conformed in (EC.D.1) and (EC.D.2), respectively. Constraints (EC.D.3), (EC.D.11)–(EC.D.12) correlate nodal voltages to the active and reactive power flows across existing fixed lines, existing replaceable lines, and candidate distribution lines, respectively. Analogous to (EC.C.7), constraint (EC.D.11) and (EC.D.12) are linearized formulations using the big M method with the replacement of the bilinear terms [28]. Furthermore, constraint (EC.D.4) restricts voltage magnitudes at distribution nodes. Then, constraints (EC.D.5)– (EC.D.10), and (EC.D.13)–(EC.D.20) define the upper and lower limits on active and reactive power flows and injections for both existing and potential assets in the distribution network, including existing fixed lines, existing replaceable lines, candidate lines, and both conventional and renewable-based generators. Following [30], and [31], expressions (EC.D.5)– (EC.D.10), and (EC.D.13)–(EC.D.20) offer linear approximations of the apparent power limits that couple active and reactive power flows and injections. It should be noted that consistent with [30] and [31], the active and reactive power flows in distribution lines are constrained by their maximum apparent power flow levels in (EC.D.5)–(EC.D.6), and (EC.D.13)–(EC.D.16). Similarly, an extension of the model for distribution lines applies to the active and reactive power outputs of distributed generators, which are also limited by their respective maximum apparent power production levels, as depicted in (EC.D.7)–(EC.D.10), and (EC.D.17)–(EC.D.20). Lastly, expression (EC.D.21) limits load shedding, and expression (EC.D.22) ensures the radial topology is maintained in each expanded distribution network. X p∈PDX g∈(GD pn ∪GD+ pn ) pG,D gpo −X l∈(LD a∪LD+ a) |fr(l)=n pL,D lo +X l∈(LD a∪LD+ a) |to(l)=n pL,D lo =X d∈DD n µLG oµL noPD,D d−pU,D do ;∀n∈N D a,∀a∈A,∀o∈O (EC.D.1) X p∈PDX g∈(GD pn ∪GD+ pn ) qG,D gpo −X l∈(LD a∪LD+ a) |fr(l)=n qL,D lo +X l∈(LD a∪LD+ a) |to(l)=n qL,D lo =tan(cos−1(ζno)) X d∈DD n µLG oµL noPD,D d−pU,D do ; ∀n∈N D a,∀a∈A,∀o∈O (EC.D.2) vto(l)o−vfr(l)o+ 2RD lpL,D lo +XD lqL,D lo =0; ∀l∈LD,F a,∀a∈A,∀o∈O (EC.D.3) (VD)2≤vno ≤(VD)2;∀n∈N D a,∀a∈A,∀o∈O (EC.D.4) −SL,D l≤pL,D lo ≤SL,D l;∀l∈LD,F a,∀a∈A,∀o∈O (EC.D.5) −SL,D l≤qL,D lo ≤SL,D l;∀l∈LD,F a,∀a∈A,∀o∈O (EC.D.6) 0≤pG,D gpo ≤SG,D gp ;∀g∈GD pa,∀p∈CD,∀a∈A,∀o∈O (EC.D.7) −SG,D gp ≤qG,D gpo ≤SG,D gp ; ∀g∈GD pa,∀p∈CD,∀a∈A,∀o∈O (EC.D.8) 0≤pG,D gpo ≤µW noSG,D gp ; ∀n∈ND a,∀g∈GD pa,∀p∈WD,∀a∈A,∀o∈O (EC.D.9) 4 −µW noSG,D gp ≤qG,D gpo ≤µW noSG,D gp ; ∀n∈ND a,∀g∈GD pa,∀p∈WD,∀a∈A,∀o∈O (EC.D.10) −MD[1 −(1 −xL,D l′)] ≤vto(l)o−vfr(l)o+2 RD lpL,D lo +XD lqL,D lo  ≤MD[1 −(1 −xL,D l′)]; ∀l∈LD,R a,∀l′∈LD+ a|(fr(l′)=fr(l)∧to(l′)=to(l)), ∀a∈A,∀o∈O (EC.D.11) −MD(1 −xL,D l) ≤vto(l)o−vfr(l)o+2 RD lpL,D lo +XD lqL,D lo  ≤MD(1 −xL,D l); ∀l∈LD+ a,∀a∈A,∀o∈O (EC.D.12) −SL,D l(1 −xL,D l′)≤pL,D lo ≤SL,D l(1 −xL,D l′); ∀l∈LD,R a,∀l′∈LD+ a|(fr(l′)=fr(l)∧to(l′)=to(l)), ∀a∈A,∀o∈O (EC.D.13) −SL,D l(1 −xL,D l′)≤qL,D lo ≤SL,D l(1 −xL,D l′); ∀l∈LD,R a,∀l′∈LD+ a|(fr(l′)=fr(l)∧to(l′)=to(l)), ∀a∈A,∀o∈O (EC.D.14) −SL,D lxL,D l≤pL,D lo ≤SL,D lxL,D l;∀l∈LD+ a,∀a∈A,∀o∈O (EC.D.15) −SL,D lxL,D l≤qL,D lo ≤SL,D lxL,D l;∀l∈LD+ a,∀a∈A,∀o∈O (EC.D.16) 0≤pG,D gpo ≤xG,D gp XG,D gp ;∀g∈GD+ pa ,∀p∈CD,∀a∈A,∀o∈O (EC.D.17) −xG,D gp XG,D gp ≤qG,D gpo ≤xG,D gp XG,D gp ; ∀g∈GD+ pa ,∀p∈CD,∀a∈A,∀o∈O (EC.D.18) 0≤pG,D gpo ≤µW noxG,D gp XG,D gp ; ∀n∈ND a,∀g∈GD+ pa ,∀p∈WD,∀a∈A,∀o∈O (EC.D.19) −µW noxG,D gp XG,D gp ≤qG,D gpo ≤µW noxG,D gp XG,D gp ; ∀n∈ND a,∀g∈GD+ pa ,∀p∈WD,∀a∈A,∀o∈O (EC.D.20) 0≤pU,D do ≤µLG oµL no PD,D d; ∀n∈ND a,∀d∈DD n,∀a∈A,∀o∈O (EC.D.21) X n∈ND+ a X l∈LD+ a |(fr(l)=n∨to(l)=n) xL,D l=card(ND+ a), ∀a∈A,∀o∈O (EC.D.22) E. Coupling Constraints on the Interface Nodes As mentioned earlier, the interface nodes serve as transmission nodes that function as root nodes for distribution systems. These nodes facilitate the connection of distribution systems to the transmission system. Given that the transmission network’s impact is represented through a dc power flow model, as commonly employed in transmission network expansion planning [28], [29], reactive power at interface nodes is typically neglected. The operational balance constraints at the interface nodes are formulated in (EC.E.1)–(EC.E.4) for the set of interface nodes ∀n∈N∞, and the set of operating conditions ∀o∈O. Constraint (EC.E.1) specifies the active power injected into or extracted from interface nodes that are exclusively connected to the transmission system. Concurrently, constraint (EC.E.2) details the active power injection or extraction at interface nodes solely linked to the distribution lines associated with these nodes. It can be mentioned here that in this study, active power injections at interface nodes, p∞,T no and p∞,D nao are regarded as the coupling variables between the transmission and distribution systems, and these are the only exchange parameters with the central coordinator while solving the CTDSEP problem using the distributed approach. Notably, the coupling constraint (EC.E.3) is characterized as linear which ensures the power flows across both transmission and distribution lines are considered, thus explicitly modeling the interconnection between these two levels. Here, λ∞ no represents the dual variable associated to constraint (EC.E.3). Lastly, consistent with the linearized dc network model utilized for transmission system operations, constraint (EC.E.4) imposes the voltage magnitude to be equal to 1 p.u. at the interface nodes. p∞,T no =X p∈PTX g∈(GT pn ∪GT+ pn ) pG,T gpo −X l∈(LT∪LT+) |fr(l)=n pL,T lo +X l∈(LT∪LT+) |to(l)=n pL,T lo −X d∈DT n µLG oµL noPD,T d−pU,T do ; ∀n∈N ∞,∀o∈O (EC.E.1) p∞,D nao =−X l∈(LD a∪LD+ a) |fr(l)=n pL,D lo +X l∈(LD a∪LD+ a) |to(l)=n pL,D lo ; ∀n∈N ∞,∀a∈An,∀o∈O (EC.E.2) p∞,T no +X a∈An p∞,D nao =0:λ∞ no;∀n∈N ∞,∀o∈O (EC.E.3) vno = 1; ∀n∈N ∞,∀a∈An,∀o∈O.(EC.E.4)