1 Recitation Module E.2 Electricity transmission expansion models Prof. Ignacio J. Pérez-Arriaga Engineering, Economics & Regulation of the Electric Power Sector ESD.934, 6.974 2 Regulation of transmission services INVESTMENT2 3 Annex A model for transmission network planning 4 Nature of the transmission Expansion problem Determine the technical characteristics and installation time of new network facilities, so that:  Total expected cost of supply (including consumer outage costs) is minimized) subject to acceptability criteria  Technical  Reliability  Financial  Environmental  other3 Transmission expansion planning 6 Nature of the transmission expansion problem TIME PERSPECTIVE  LONG-TERM (15-30 YEARS)  Guidelines for network development  Simplified models are acceptable  Synthesis of plans is main priority  MID TERM (6-10 YEARS)  Decisions for network development  Detailed models are required  Analysis of proposed plans is main priority4 7 Mono-attribute optimization of expansion plans MINIMIZE M(p) pє P Subject to Gk, min ≤ Gk (p) ≤ Gk, max, k = 1,..., K Where p: individual plan P: set of all possible plans M: attribute to be minimized (e.g., total cost of supply G: result of each one of the k=1,...K technical/or reliability constraints that the plan has to meet Alternative: Heuristic search model  Same as above, but algorithm (typically computationally efficient) does not guarantee that the optimal plan is obtained5 9 Multi-attribute optimization models 10 Mono-Attribute Optimization Models  STATIC MODELS  Only the final year of the considered time horizon is analyzed  Only models that seem to have been actually used in practical applications  DYNAMIC MODELS  The entire time horizon is simultaneously considered6 11 Methodology. Modeling Aspects (1 of 2)  Main issues  Demand  Generation of scenarios  Expansion alternatives/investment model  Discrete or continuous variables  Financial/economic constraints  Attributes (objectives function)  Reliability: constraint, cost or both  Other attributes (e.g. environmental impact)  Network representation  Transportation, DC, AC, hybrid model  Ohmic losses  Security limits 12 Methodology. Modeling Aspects (2 of 2)  Production cost model  Thermal generation units representation  Hydro units  Security constraints (preventive vs. corrective)  Uncertainty: hydro, load availability  Reliability model  Contingency list vs. Probabilistic approach  Uncertainty: hydro, load availability7 13 Mono-Atribute static & strictly optimization models Main Features  Single attribute: Total supply cost (network investment cost + system operation cost + consumers outage cost)  Optional constraints of the investment subproblem  Maximum number of lines per corridor  Maximum number of lines of a type per corridor  Maximum investment per corridor  Maximum total investment  Maximum non served energy  Several options of network representation (DC has been chosen in the example shown here)  Investment variables  type of line & volume of investment at each corridor Network representation Ohmic losses 14 l corridor identification index λl ohmic losses (nonlinear function) Fl active power flow in line l λl = 2 Gl [ 1 - cos (θi - θj)]8 15 Power System Model  Production cost subproblem subject to MINIMIZEg,r, f ,θZ = cTg +µuTr−Δ − s. f + g + r = d (πd)f −γ.ST.θ= 00 ≤ g ≤ g0 ≤ r ≤ df ≤ f (πq)Δi=12λi, jj∑(losses)16 Power System Model  Reliability subproblem subject to MINIMIZEg,r, f ,θZ = uTr−s. f + g + r = d (πd)f −γ.ST.θ= 00 ≤ g ≤ g0 ≤ r ≤ df ≤ f (πf)9 17 Glosary of terms  g: active power generation at each bus  g: maximum active power generation at each bus  f: active power flow at each line  f: maximum active power flow at each line  r: non served power at each bus  u: unit vector  m: cost of unserved energy  c: variable generation cost  θ: voltage angle at each bus  λl: ohmic losses in line l  S: node-arc incidence matrix  d: active power demand at each node  pd, pf: dual variables of associated constraints  Gl: susceptance of each line l 18 Mono-Attribute static optimization model Solution by heuristic search Case example: CHOPIN  Formulation  Only discrete investment variables are considered in CHOPIN  Production cost & reliability models with DC network formulation  Solution method  The optimization of the investment subproblem is replaced by a heuristic search that consists in a truncated enumeration of the complete solution space (i.e., the set of all possible plans)  Investment restrictions are explicitly accounted for during the search: non feasible solutions are not accepted  The level of network modelling detail is not relevant for the performance of the algorithm  no restrictions to the use of DC (or even AC) models10 19 CHOPIN Solution by heuristic search 20 CHOPIN Algorithm organization11 21 CHOPIN Basic Philosophy of the search algorithm  Start from a user-provided reasonable plan (*)  Local search that is guided by  Sensitivities  heuristic rules  logic  experience from actual use of algorithm  Depth-first search  since truncation here is mostly based on extent of deviations from what locally appears to be the best decision  good solutions in limited time (*) Successful searches have been achieved in all cases even when starting from very poor initial plans 22 CHOPIN Classification of the investment variables  Questioned variables  Lines included by user in initial plan  User considers they may not belong to optimal plan  Initial value = 1  Attractive variables  Lines not included by user in initial plan  User considers they may belong to optimal plan  Initial value = 0  Frozen variables  Cannot change their initial values ( 0 or 1) fixed by user  During the search the questioned & attractive variables become frozen variables12 23 2413 25 CHOPIN Example: Solutions Space 26 CHOPIN Solution Space in a