Peak Shaving and Price Saving Algorithms for self-generation David Craigie...

Peak Shaving and Price Saving

Algorithms for self-generation

David Craigie

_______________________________________________________Supervised by: Prof. Andy Philpott Dr 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

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Demand Side Management

• Conservation

• Load Shifting

• Contracting

• Self-Generation

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Contracting & Self-Generation

• CFDs don’t alter optimal self-generation strategy:

Charge (in period t) =

where dt = demand realised in period t

pt = spot market price in period t

dc = swap volume

pc = strike price

(based on Mercury Energy’s Swap Contracts)

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)( tcctt ppdpd

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.

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• Objective:

where:

di = demand in period i [MWh]

pi = spot market price in period i [$/MWh]

si = fuel used in period i [L]e = generator efficiency [MWh/L]c = maximum demand charge [$/MWh]N = number of periodsmd = average of m highest load realisations during N [MWh]

By setting e = 1 and removing constant terms:

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mdcpsedN

iiii ..min

1

mdcspN

iii .min

1


• 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 Mi is the set giving the ith way of choosing m periods from a

possible N.

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TsN

ii

1

Niksi ,...,2,1

mN

Mjjj Cisdmd

i

,...,2,1)(

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.

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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:

p1=72 p2=68 p3=60 p4=65 p5=85 [$/MWh]

Initial Cost = + 15(12+11) = $3,317

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5

1iii pd

Initial Load Profile

0

2

4

6

8

10

12

14

1 2 3 4 5

period

Lo

ad (

MW

h)

Peak Shaving ExampleSuppose 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)

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Period Load Price MDSavin

g Range Alloc.

1 6 72 N 72 4 0

2 10 68 N 68 4 0

3 12 60 Y 75 2 0

4 11 65 Y 80 1 0

5 5 85 N 85 4 0

Peak Shaving ExampleIteration 2

Decision: Allocate 1MWh to period 4 (11MWh remaining)Current Load Profile:

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Load Profile after 2 Iterations

0

2

4

6

8

10

12

14

1 2 3 4 5

period

Lo

ad (

MW

h)

Period Load Price MDSavin

g Range Alloc.

1 6 72 N 72 4 0

2 10 68 N 68 4 0

3 12 60 Y 75 2 0

4 11 65 Y 80 1 0

5 1 85 N 0 0 4


*Saving for {2,4} = 1/2 x (68 + 65) + 1/2 x 15Decision: Allocate 2MWh to period 3 (9MWh remaining)

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Load Profile after 3 Iterations

0

2

4

6

8

10

12

1 2 3 4 5

period

Lo

ad (

MW

h)

Period Load Price MD Saving Range Alloc.

1 6 72 N 72 4 0

2 10 68 N 68 4 0

3 12 60 Y 75 2 0

4 10 65 N 65 3 1

5 1 85 N 0 0 4

{2,4} 74* 3


*Saving for {2,3,4} = 1/3 x (68 + 65 + 60) + 2/3 x 15

In general the savings from an n-period MD set tie are:1/n x (p1+ p2 + … + pn) + (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)

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1 6 72 N 72 4 0

2 10 68 N 68 4 0

3 10 60 N 60 2 2

4 10 65 N 65 3 1

5 1 85 N 0 0 4

{2,4} 74 2

{2,3,4} 74.33* 2


Decision: Allocate 1MWh to periods 2 and 4 (1MWh remaining)Iteration 6

Decision: Allocate 1MWh to period 1 - STOP


1 6 72 N 72 3 0

2 8 68 N 68 2 2

3 8 60 N 0 0 4

4 8 65 N 65 1 3

5 1 85 N 0 0 4

{2,4} 74 1

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1 6 72 N 72 1 0

2 7 68 N 68 1 3

3 8 60 Y 0 0 4

4 7 65 N 0 0 4

5 1 85 N 0 0 4

Peak Shaving ExampleBefore:

Cost = $3,317

After:

Cost = $2,081

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Initial Load Profile

0

2

4

6

8

10

12

14

1 2 3 4 5

period

Lo

ad (

MW

h)

Final Load Profile

0

2

4

6

8

10

12

14

1 2 3 4 5

period

Lo

ad (

MW

h)

UoA ExampleUsing 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: $36,164 After: $32,102

Cost = $2,081

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Change in max load periods

520

530

540

550

560

570

580

590

1 2 3 4 5 6 7 8 9 10 11 12 13 14

loa

d (

kW

h)

Before

After

Dual Simplex ApproachWe 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

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Dual Simplex ApproachIteration 1:

Iteration 2:

Iteration 3:

etc…

Cost = $2,081

iMjjj

i

N

ii

N

iii

sdmd

Niks

Ts

mdcsp

)(

,...,2,1

s.t.

.min

1

1

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k

i

Mjjj

Mjjj

i

N

ii

N

iii

sdmd

sdmd

Niks

Ts

mdcsp

)(

)(

,...,2,1

s.t.

.min

1

1

Niks

Ts

mdcsp

i

N

ii

N

iii

,...,2,1

s.t.

.min

1

1

Stochastic Problem Scenario tree for prices (3 periods only)…

Cost = $2,081

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S1

S21

S22

S31

S32

Stochastic Problem N = 3, m = 2. Complete Stochastic LP formulation:

Cost = $2,081

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)()(

)()(

)()(

)()(

)()(

)()(

,

s.t.

min

3232222

323112

222112

3132121

313111

212111

32221

31211

2211,

sdsdmd

sdsdmd

sdsdmd

sdsdmd

sdsdmd

sdsdmd

jiks

Tsss

Tsss

mdcmdcs

ij

jiijij

Stochastic ProblemUse Dual Simplex approach:

Iteration 1:

Iteration 2:

etc…

Cost = $2,081

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jiks

Tsss

Tsss

mdcmdcs

ij

jiijij

,

s.t.

min

32221

31211

2211,

)()(

)()(

,

min

313112

212111

32221

31211

2211,

sdsdmd

sdsdmd

jiks

Tsss

Tsss

mdcmdcs

ij

jiijij

s.t.

Stochastic ProblemDynamic Programming Approach:

Cost = $2,081

))},(,max{,(

.)}(,max{.)(min),,(

1 jsdmdsSV

mdcsdmdcpsdimdSV

ttttt

ttjtt

jij

stt

t

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variable)(decision period in allocated fuel

charge demand maximum

state to state from transition ofy probabilit the

state in pricemarket spot

period in demand

far so realised demand maximum

period ofstart at remaining fuel

:where

ts

c

ji

ip

td

md

tS

t

ij

i

t

t

Stochastic ProblemDynamic 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 1st and 10th highest demands and interpolating between these.

Cost = $2,081

))},(,max{,(

.)}(,max{.)(min),,(

1 jsdmdsSV

mdcsdmdcpsdimdSV

ttttt

ttjtt

jij

stt

t

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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.

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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?

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Thank You For Listening. Questions?

25/25

Peak Shaving and Price Saving Algorithms for self-generation David Craigie...

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