Optimization Based Power Generation Scheduling

Optimization Based Power Generation Scheduling Xiaohong Guan Tsinghua / Xian Jiaotong University

In this talk • Introduction to Power Generation Scheduling • Motivations and background • Difficulties • Current approaches • Problem Formulation • Solution Based on Lagrangian Relaxation • Numerical Testing Results • Issues of Homogenous Units and Resolution

Power Generation Scheduling(Unit commitment) • Background and Motivations • Many generating units in a power system connected through transmission network to supply demand • Scheduling unit on/off and generation levels to meet the system demand, reserve and individual constraints • Minimizing the total generation cost • Potential for significant cost savings: over 10 millions of US dollars per year for a large generation company • Hot research topics for decades

Power Generation Scheduling(Unit commitment) • Difficulties • Complicated discrete and continuous and dynamic constraints of individual units  hybrid dynamics and constraints • System wide constraints coupling the operation of individual units • NP-hard mixed integer programming problem: extremely difficult to obtain the optimal schedule • Integrated with bidding problems in the market environment

Problem of computational complexity Power Generation Scheduling(Unit commitment) • Current approaches • Priority list and other heuristics  little control on schedule quality • Enumeration such as branch and bound • Dynamic programming • Bender’s decomposition • Lagrangian relaxation  our approach

Lagrangian Relaxation Based Scheduling Algorithms • Relax system wide constraints to form a two level optimization problems • Solve low level individual subproblems with much less efforts • Update Lagrange multipliers at the high level • Modify dual solution into a feasible schedule • Quantitative estimate of solution quality since the dual cost is the lower bound of the primal cost

Primal Cost d’ d Dual Cost l Duality Gap and Solution Quality

C, with C = Problem Formulation Objective function Subject to • System wide constraints: • System demand

reserve contribution generation level minimum generation generation capacity generation range • Reserve requirements

if if discrete decision variable of unit i at time t, “1” for up, “-1” for down • Individual unit constraints (thermal units): • Discrete state transitions

if if if if • Operating regions of thermal units • Minimum up/down time

maximum generation feasible region of minimum generation t t+1 • Ramping constraint  continuous dynamics

or • Individual unit constraints (hydro units): • Reservoir dynamics • Operating regions of hydro units • Reservoir level limit • Initial and terminal reservoir levels

Hydraulic coupling

Lagrangian Relaxation • Lagrangian function

Multipliers Generation levels Subproblems for other special units Subproblems for thermal units Subproblems for hydro units Lagrangian relaxation framework Update Multipliers Obtain feasible schedule 18

Min Li with Solve Thermal Subproblems • Objective function • Individual constraints of thermal units • Discrete dynamic state transactions • Minimum down/up times • Discontinuous operating regions • Major method: Dynamic Programming

Stage-wise cost generation cost cti(pti(t)) total stage-wise cost pti(t) p*ti(t) shadow cost -l(t) pti(t)-m(t) pti(t) • Optimal generation at a particular hour (ignoring ramping constraints) Obtain optimal stage-wise cost and generation by optimizing a single variable function !

Start up cost Si(xti(t)) exponential Saturate linear xti(t) Cold start time Minimum down time • Typical start up cost functions

Up min time generation costs cti(pti(t)) Up two hour Up one hour Down one hour Down two hour Start up costs Sti(xti(t)) Down min down time Down cold start time • Optimal states across time obtained with efficiently with only a few states and transitions t+1 t States

Dealing with ramping constraints • Difficulties of ramping constraints • Ramping couples generation levels across time Continuous dynamics • The optimal generation of an “Up State” can no longer be a single point • Dynamic programming on discrete states can no longer be applied straightforwardly

Approach to resolve the issue • Heuristics • Discretizing generation levels  greatly increasing the number of states and computational efforts • Relaxation of ramping constraints  three level optimization structure • Constructive dynamic programming for continuous optimal generation level and regular dynamic programming for optimal discrete states • Double dynamic programming method for solving subproblems with ramping constraints  best algorithm so far

Ideas of constructive dynamic programming • Optimal generation level can only be at the corner points of the cost function or the active points of the ramping constraints • The optimal generation levels of the previous or next stage w.r.t the above points can be mapped across time systematically • The possible optimal generation levels are constructed backwardly without discretization

mapping of optimal generation levels t t+1 • The number of states would increase but not significantly • The method is efficient

Redefine the discrete state as an “up” or “running” cycle • Apply constructive dynamic programming to obtain optimal general levels and cost for all running cycles • Apply dynamic programming to obtain the optimal cycle

Ui: the number of hours before the unit committed (up) for the ith time, Di: as the number of hours before the unit decommitted (down) for the ith time.

min Lj, with Solve Hydro Subproblems (including Pumped Storage) • Objective function • Individual constraints of hydro units • Water balance • Reservoir levels • Discontinuous operating regions • Discrete operating constraints such as minimum down times • Major difficulties • Hydraulic coupling integrated with discontinuous operating regions and discrete dynamic constraints

t t+1 v1(t) -w1(t) v1(·) w1(t) w1(t) v2(t) v2(·) -w2(t) w2(t) • Method 1: Network flow optimization • Ignore discontinuous operating regions and discrete operating constraints • Apply minimum cost flow optimization to schedule generation levels with water balance and reservoir level constraints • Meet other constraint by heuristics • Two reservoir example:

Method 2: Relaxation of reservoir level constraints • Substitute continuous hydro dynamics and relax limits on reservoir levels • Solve subproblems w.r.t. individual hydro units using dynamic programming similar to thermal subproblems • Apply minimum cost flow optimization with fixed discrete states to schedule generation levels to meet water balance and reservoir level constraints

Method 3: General mixed integer programming • Solve the hydro subproblems as a mixed integer programming problem using solver such as CPLEX • Efficiency closely related to the problem formulation

with F(l(t), m(t)) = Solve High Level Dual Problem Subgradient

Updating Multipliers • The multipliers are updated using an efficient subgradient algorithm • Adaptive step sizing • Good initial multipliers using priority list scheduling

Total generation Over generationPrice down System demand Under generation  price up Time Updating Multipliers

Obtaining Feasible Schedules • Goal: to satisfy once relaxed system demand and reserve requirement constraints • Heuristics should be applied • If possible, satisfy these constraints by adjusting generation levels only  economic dispatch • For piece-wise linear cost function, sorting all power blocks of all scheduled “up” units and piling these blocks up till the the system wide constraints satisfied

Adjust discrete operating (commitment) states. • Calculate the “opportunity cost” of state change based on the state transition and cost-to-go information in the dual solution for all units • Adjust the commitment state of a unit with the smallest cost increase • Repeat if sufficient and necessary feasibility conditions not satisfied

Numerical results • Based on Northeast Utilities system with 70 thermal units, 7 hydro units and 1 large pumped storage unit

Consistent convergence and near optimal schedules obtained • Only a few seconds on P-IV computer • Significant cost saving in comparison with the schedules by NU engineers • Production use for many years

Inherent Issues of Lagrangian Relaxation Based Scheduling Algorithms • Homogeneous subproblem solutions to the subproblems of identical units • May deviate far away from the optimal schedule • Difficult to obtain feasible solution since not much information on solution structure • Long been recognized and considered as a major obstacle for applying Lagrangian approach • Existing approaches to solving these issue  parameter perturbation (heuristics)

Subject to with Optimal solutions: Homogenous solution: 2-Unit Example

Subject to if if Dual solutionwith Lagrangian Relaxation Optimal dual solution:

or Dual solution patterns: Primal optimal solution: or Dual solutionwith Lagrangian Relaxation

Dual solutionwith Lagrangian Relaxation • Solutions oscillation around as the multiplier being updated • Subproblem solutions far away from primal optimum • Primal optimal solution never obtained

Key Idea of the New Algorithm: Differentiate homogenous subproblems • Add quadratic or piece-wise linear penalty terms to Larangian • Solve individual subproblem successively with each high level iteration to keep decomposability • Update Lagrange multipliers using surrogate subgradient at the high level

Step 1 Update multipliers Initialization Let w = 0, solve standard LR Solve only one subproblems Step 3 Step 2 Check Convergence New Algorithm: Successive Subproblem Solving Method (SSS) Step 0

Features of the New Algorithm • Surrogate subgradient = proper search direction • Dual cost still the lower bound of the primal cost • Larger penalty weight  smaller constraint violation in the dual problem • Rigorous convergence proof

Numerical Testing for SSS Algorithm • Testing results of the simple problem with two identical units • Testing results of generation scheduling problem of 10-units with two groups of identical units • Excellent results observed

Optimization Based Power Generation Scheduling