Peak Shaving and Price Saving Algorithms for self-generation David Craigie _______________________________________________________ Supervised by: Prof. Andy PhilpottDr Golbon Zakeri

Today Demand side management Contracting & Self-Generation The Peak Shaving Problem The Peak Shaving Algorithm Example using UoA demand data The stochastic problem Conclusions Future Work Questions 1/25

Demand Side Management Conservation Load Shifting Contracting Self-Generation 2/25

Contracting & Self-Generation CFDs don’t alter optimal self-generation strategy: Charge (in period t) = where d t = demand realised in period t p t = spot market price in period t d c = swap volume p c = strike price (based on Mercury Energy’s Swap Contracts) 3/25

The Peak Shaving Problem The deterministic peak shaving problem can be stated: Given a load profile, price profile and a capacity of generation, find the optimal allocation of a limited quantity of generator fuel with a sunk cost in order to minimize the cost of electricity consumption. Or: Allow for an unlimited amount of fuel but at a given unit cost. However, the algorithm is the same in either case, only the stopping criterion changes. 4/25

The Peak Shaving Problem Objective: where: d i = demand in period i[MWh] p i = spot market price in period i [$/MWh] s i = fuel used in period i[L] e = generator efficiency [MWh/L] c = maximum demand charge [$/MWh] N = number of periods md = average of m highest load realisations during N[MWh] By setting e = 1 and removing constant terms: 5/25

The Peak Shaving Problem Constraints: (i)Total fuel allocation cannot exceed available quantity: (ii)Fuel allocation in any period cannot exceed generator capacity: (iii)The maximum demand quantity must be equal to the greatest sum of m demands after generation: where M i is the set giving the i th way of choosing m periods from a possible N. 6/25

The Peak Shaving Algorithm The combinatorial number of maximum demand constraints makes the Peak Shaving Problem intractable for the RSM. However, we can use a greedy algorithm that will give the same solution as the RSM but without the computational cost. At every iteration the algorithm will choose a period to allocate fuel to that will give the maximum savings. It will cease allocation to that period when either capacity of the generator is reached, fuel supply is exhausted or savings need to be recalculated. 7/25

Peak Shaving Example Suppose we will be charged for the average of the highest 2 load realizations in the following five period demand profile, at a rate of $30/MWh: p 1 =72 p 2 =68 p 3 =60 p 4 =65 p 5 =85 [$/MWh] Initial Cost = + 15(12+11) = $3,317 8/25

Peak Shaving Example Suppose we have 16MWh worth of fuel, but the capacity in any one period is 4MWh. We seek the allocation of fuel among the 5 periods that will obtain the greatest savings: Iteration 1 Decision: Allocate 4MWh to period 5 (12MWh remaining) 9/25 PeriodLoadPriceMDSavingRangeAlloc. 1672N N Y Y N 40

Peak Shaving Example Iteration 2 Decision: Allocate 1MWh to period 4 (11MWh remaining) Current Load Profile: 10/25 PeriodLoadPriceMDSavingRangeAlloc. 1672N N Y Y N004

Peak Shaving Example Iteration 3 *Saving for {2,4} = 1/2 x ( ) + 1/2 x 15 Decision: Allocate 2MWh to period 3 (9MWh remaining) 11/25 PeriodLoadPriceMDSavingRangeAlloc. 1672N N Y N N004 {2,4} 74*3

Peak Shaving Example Iteration 4 *Saving for {2,3,4} = 1/3 x ( ) + 2/3 x 15 In general the savings from an n-period MD set tie are: 1/n x (p 1 + p 2 + … + p n ) + (n-1/n) x c Thus the best tie to consider will be the highest priced n periods where n maximizes the above expression Decision: Allocate 2MWh to periods 2,3 and 4 (3MWh remaining) 12/25 PeriodLoadPriceMDSavingRangeAlloc. 1672N N N N N004 {2,4} 742 {2,3,4} 74.33*2

Peak Shaving Example Iteration 5 Decision: Allocate 1MWh to periods 2 and 4 (1MWh remaining) Iteration 6 Decision: Allocate 1MWh to period 1 - STOP PeriodLoadPriceMDSavingRangeAlloc. 1672N N N N N004 {2,4} /25 PeriodLoadPriceMDSavingRangeAlloc. 1672N N Y N N004

Peak Shaving Example Before: Cost = $3,317 After: Cost = $2,081 14/25

UoA Example Using demand data from 18 Symonds St (Engineering Building) for the month of August (1488 periods) and price data from OTA reference node. Using a 100kW generator at 3.6kWh/L with 14000L of fuel. Maximum demand charge of $8/kWh multiplied by the average of the 10 highest load realisations. Before: $36,164After: $32,102 Cost = $2,081 15/25

Dual Simplex Approach We can solve a relaxed version of the LP by removing all the maximum demand constraints. This solution will (most likely) be infeasible in the original problem, but we know at least one of the constraints it violates, so we can add that constraint and resolve using the Dual Simplex Algorithm. We continue to add constraints until they cease to change the solution. By doing this we will obtain all the binding constraints from the optimal solution of the original problem. Depending on how many of the maximum demand constraints are binding in the optimal solution of the original LP, this may be a faster approach. Cost = $2,081 16/25

Dual Simplex Approach Iteration 1: Iteration 2: Iteration 3: etc… Cost = $2,081 17/25

Stochastic Problem Scenario tree for prices (3 periods only)… Cost = $2,081 18/25 S 1 S 21 S 22 S 31 S 32

Stochastic Problem N = 3, m = 2. Complete Stochastic LP formulation: Cost = $2,081 19/25

Stochastic Problem Use Dual Simplex approach: Iteration 1: Iteration 2: etc… Cost = $2,081 20/25

Stochastic Problem Dynamic Programming Approach: Cost = $2,081 21/25

Stochastic Problem Dynamic Programming Approach: This approach works when the size of the maximum demand set is 1 (as in the recursion above) or close to 1. However when m=10, we need to store 10 demands and consequently the state space “explodes” (curse of dimensionality). To overcome this problem, we might consider storing the 1 st and 10 th highest demands and interpolating between these. Cost = $2,081 22/25

Conclusions Under normal market conditions, self generation using diesel generators is not economical. However, if fuel is needed as a backup supply and is approaching expiration there is an optimal way to use it. Formulated as an LP the Peak Shaving problem is intractable for the RSM. A relatively simply greedy algorithm exists that will determine the optimal allocation. Alternatively, a dual simplex approach can be employed. The stochastic problem is significantly larger and requires a large number of constraints if formulated using an SLP or an enormous state space if formulated as a DP. 23/25

Future Work Is the stochastic problem more sensitive to demand or price uncertainty? Are we better to use the SLP and trim the scenario tree, or the DP and interpolate the state space? Is back-up generation really worth it? Given that peak shaving can reduce the cost of this security of supply, how risk averse does one need to be for back-up generation to be a sensible strategy. The level of contracting does not alter the optimal self generation plan but the converse is not necessarily true. Does the added security of back-up generation alter the optimal contract – spot mix. Other demand side questions: –Would a liquid hedge market enable large consumers to more effectively manage price risk and at less cost? –Can better contracts be negotiated by consumers acting as a group? –What about demand side bids? –Every little bit counts: Are there optimal conservation strategies? How valuable are they? 24/25

Thank You For Listening. Questions? 25/25