Workshop COOPS 2017 in Milano

Bidding System Design for Multiperiod
Electricity Markets

Takayuki Ishizaki (Tokyo Institute of Technology)
M. Koike (Tokyo Uni of MST) N. Yamaguchi (Tokyo Uni of Sci) J. Imura (Tokyo Tech)

Contents
Part I: What is bidding system design?

Problem formulation

relation between market clearing problem and convex optimization
relation between bidding curves and cost functionsdifficulty in bidding system design for multiperiod markets

Part II: How to design a bidding system for multiperiod markets?
basis transformation compatible with energy shift market
sequential market clearing schemeAn approximate
numerical examples
solution method

2/28

Day-Ahead Energy Markets
Energy Market (ISO)Agg 1energybiddingcurves
Agg 2

total bidding curves
Agg 3

price

multiple spots

Example 2 time spots, 3 aggregators
Market Results

Aggregator 1(producer)

Aggregator 2(consumer)

Aggregator 3(prosumer)

Clearing Price

Spot 1 (AM)

150 [kWh]-250 [kWh]100 [kWh]

10 [yen/kWh]Spot 2 (PM)100 [kWh]-50 [kWh]-50 [kWh]

5 [yen/kWh]

Decision variables:Balanced

Market clearing: Find “desirable”&such that.

Market Clearing as Optimization
Market Results

Aggregator 1(producer)

Aggregator 2(consumer)

Aggregator 3(prosumer)

Clearing Price

Spot 1 (AM)

150 [kWh]-250 [kWh]100 [kWh]

10 [yen/kWh]Spot 2 (PM)100 [kWh]-50 [kWh]-50 [kWh]

5 [yen/kWh]Profit of Agg α:(selfish objective function)

incomecost

Social profit:(social objective function)

Social profit maximization = Social cost minimization

See later how todetermine(especially for prosumer!)social costprimal:dual:
4/28

Bidding Curves for Market Clearing
Suppose that bidding curves for each spot are submitted to ISO
Aggregator 2

Aggregator 3

Spot 1 (AM)

Totalenergy

Aggregator 1

priceSpot 2 (PM)ISO can find each clearing price and balancing amounts
as crossing points of (total) bidding curves

But… Are such crossing points really solutions ofsocial cost minimization:??
5/28

Brief Summary: Bidding System Design
Socially optimal market clearing problem:
Find

primal:dual:

solving

Q1: What is a reasonable cost function
Prosumption??

should be a mixture of generators, batteries, renewables etc

Q2: Is it possible to construct bidding curves fromSocial cost should be minimized withandEnergy Market (ISO)??

found as crossing-points
total bidding curves

bidding curves

Agg 1

Agg 2

Agg 3
n spots

Bidding system design = Distributed algorithm design under pre-specified ISO operation

Contents
Part I: What is bidding system design?
relation between market clearing problem and convex optimization
relation between bidding curves and cost functionsdifficulty in bidding system design for multiperiod markets

Part II: How to design a bidding system for multiperiod markets?
basis transformation compatible with energy shift
sequential clearing scheme for energy shift markets
numerical examples

7/28

Prosumption Cost function
Constraints:
Agg

(e.g.

Internal decision variablesandGiven

Generation cost
Battery usage cost

generated power
battery charge/discharge

Theorem If)

are both convex, thenis convex with respect towhereExample 2 time spots (AM/PM) Constants:
s.t.
: supply-demand balance inside aggregator

not unique!

Optimal strategy for energy resources ensures convexity of.

8/28

Prosumption Cost function
Constraints:
Agg

Internal decision variables
generated power
battery charge/discharge

Theorem Ifand(e.g.

)

Given

Generation cost
Battery usage costare both convex, thenis convex with respect towhereUncertain renewables can be handled as robust optimization like:

whereis a scenario set of renewable generation

on-going work

(More interesting to see how magnitude of uncertainty affects economics!)
9/28

Derivation of Bid Functions
Example 2 time spots
Generators & loads:
Spec of generators:

[yen]
(A)
(A+B)

[kWh]

(A) 0~50 [kWh] 5 [yen/kWh](B) 0~50 [kWh] 10 [yen/kWh]Generation cost:Feasible generator outputs:

additively
decomposabledisjoint
decomposable!

Bid functions
10/28

Mathematics behind Bid Functions
convex[yen]

Bid functions (period-specific)[kWh]

Bidding curve[kWh]

Maximum profit

monotoneincreasing

convex[yen][yen/kWh][yen/kWh]

Legendre transformof cost function
Legendre transformation (convex conjugation): convex
11/28

Multiperiod Bid Function
Generation cost:Feasible generator outputs:
Ramp rate limit (Added):Multiperiod bid function

temporally correlated!
indecomposable!

: monotone increasingbidding curves
bidding hyperplanes
Multiperiod bid function is not compatible with current bidding system

Separability of Multiperiod Bid Function[yen]

Example Battery aggregator

5 [yen/kWh]

SOC constraints:[kWh]

not disjoint!
10 [yen/kWh]Cost function based on utility of final SOC:
not additively
decomposable!

Lemma The multiperiod bid function is separate iff
the cost function is additively decomposable and its domain is disjoint
i.e.

Negative fact!! Traditional bidding curves available just in very special cases

Brief Summary: Bidding System Design
Socially optimal market clearing :Bidding system design = Distributed algorithm design under pre-specified ISO operation

Theorem

is convex

whereMultiperiod bid function:
bidding hyperplanesmonotone increasing

additively decomposableLemmaseparate

disjoint

Bidding system design for multiperiod markets is not so simple!! 14/28

Contents
Part I: What is bidding system design?
relation between market clearing problem and convex optimization
relation between bidding curves and cost functionsdifficulty in bidding system design for multiperiod markets

Part II: How to design a bidding system for multiperiod markets?
basis transformation compatible with energy shift
sequential clearing scheme for energy shift markets
numerical examples

15/28

Ideas in Proposed ApproachA) Basis transformation towards better approximation
B) Approximation of bidding hyperplanes to bidding curves
highApproximate bidding curve

lowlarge variancetemporal correlation
reduced!

smallervarianceC) Sequential market clearing scheme
From optimization view: (A) preconditioning (B)-(C) updates of primal/dual variables

16/28

Energy Shift: A Key Property of Batteries[kWh]

Agg 1 (generators):
Spec of gens:(B) 10 [yen/kWh](A) 0~50 [kWh] 5 [yen/kWh](B) 0~50 [kWh] 10 [yen/kWh]

Agg 2 (loads):(A) 5 [yen/kWh]

Optimal price:[kWh]

Agg 3 (batteries):
PM: discharge (sell)
AM: charge (buy)New optimal price:

shiftsocial cost minimum(B) 10 [yen/kWh](A) 5 [yen/kWh]

Battery leads to price levelling-off!

Energy market with explicit consideration of energy shift??

17/28

Fourier-Like Basis Transformation
Example 2 time spots
[kWh]

[kWh]

[kWh]

AMAMPMaverage (total) energy

PMshift energy (PM to AM): average (levelling-off) price: energy shift price

Explicit consideration of levelling-off price & energy shift price
Energy shift market:

primal:
dual:

18/28

Sequential Market Clearing
Step 1) Market clearing of average energy amounts
Multiperiod bid function:
Approximate bid function:
assumption (premise) ofprice levelling-off

ISO determines&by approximate bidding curves

Step 2) Market clearing of shift energy amounts
Approximate bid function:
ISO determines&cleared amountby approximate bidding curves

19/28

Example: Sequential Market Clearing(A) 0~50 [kWh] 5 [yen/kWh](B) 0~50 [kWh] 10 [yen/kWh]

Agg 1 (generators):

Utility [yen]Agg 2 (loads):

3 [yen/kWh]Final SOC
7 [yen/kWh] [kWh]Agg 3 (batteries):
Socially optimal market results (only god knows!!)
[kWh]

[kWh]

Optimal clearing price:(B) 10 [yen/kWh](A) 5 [yen/kWh]Agg 1

optimal price levels off

Agg 3(i.e.

)
20/28

Example: Sequential Market Clearing(A) 0~50 [kWh] 5 [yen/kWh](B) 0~50 [kWh] 10 [yen/kWh]

Agg 1 (generators):

Utility [yen]Agg 2 (loads):

3 [yen/kWh]Final SOC
7 [yen/kWh] [kWh]Agg 3 (batteries):
1) Market clearing of average energy amounts
Approximate bid function:[kWh]discharge (sell)[yen/kWh]charge (buy)21/28

Example: Sequential Market Clearing(A) 0~50 [kWh] 5 [yen/kWh](B) 0~50 [kWh] 10 [yen/kWh]

Agg 1 (generators):

Utility [yen]Agg 2 (loads):

3 [yen/kWh]Final SOC
7 [yen/kWh] [kWh]Agg 3 (batteries):
1) Market clearing of average energy amountsBalancing amounts:

Average price:2) Market clearing of shift energy amounts
Approximate bid function:[kWh][yen/kWh]
22/28

Example: Sequential Market Clearing
Agg 1 (generators):(A) 0~50 [kWh] 5 [yen/kWh](B) 0~50 [kWh] 10 [yen/kWh]

Agg 2 (loads):

Utility [yen]3 [yen/kWh]

Agg 3 (batteries):

Final SOC
7 [yen/kWh] [kWh]1) Market clearing of average energy amounts
Balancing amounts:

Average price:2) Market clearing of shift energy amounts
Balancing amounts:

Shift energy price:Theorem Socially optimal market clearing iff optimal price levels offi.e.
23/28

Example: Sequential Market Clearing(A) 0~50 [kWh] 5 [yen/kWh](B) 0~50 [kWh] 10 [yen/kWh]

Agg 1 (generators):
Agg 2 (loads):

Optimal clearing price:(without battery aggregator)
Agg 3 (batteries):

1) Market clearing of average energy amounts
Balancing amounts:(w/o)2) Market clearing of shift energy amounts[kWh]

Average price:

Clearing price:[yen/kWh]At least approximate clearing even if optimal price does not level off

24/28

Numerical Example
Charge/discharge efficiencies:95% (Agg 1)
94% (Agg 2)

SOC/inverter constraints: parameters

Agg 3 (9 types of generators)
Generation costs: 3, 6,…, 27 [yen/kWh]Load [kWh]Agg 1, Agg 2 (loads & batteries)

Time [h]Social cost [yen]Resultant social costs when varying battery penetration levels (16 time spots): time domain (sequential clearing): Fourier-like domain (sequential clearing)comparison (next slides)

Battery penetration level: socially optimal (only god knows)
Almost optimal market clearing
when high battery penetration!
price levelling-off
25/28

Sequential Clearing in Time Domain: socially optimal: approx clearing

large gap from optimal!

Generators (9 types)
Generation [kWh]Price [yen/kWh]Clearing priceCharge/discharge of batteries

Time [h]Time [h]Stored energy [kWh]Charge/discharge [kWh]Time [h]SOC of batteries

Time [h]26/28Sequential Clearing in Fourier-Like Domain: socially optimal: approx clearing

smaller gap from optimal

Generators (9 types)
Generation [kWh]Price [yen/kWh]Clearing priceCharge/discharge of batteries

Time [h]Time [h]Stored energy [kWh]Charge/discharge [kWh]Time [h]SOC of batteries

Time [h]27/28Concluding Remarks
Bidding system design for multiperiod electricity markets
Distributed algorithm design for convex optimization
Each aggregator submits bidding curves to ISO
ISO finds clearing price and balancing amounts by bidding curves

Proposed approach to bidding system design
Basis transformation compatible with energy shift markets
Sequential clearing scheme based on approximate bidding curves
A Distributed Scheme for Power Profile Market Clearing under High Battery
Penetration, IFAC WC 2017
Bidding System Design for Multiperiod Electricity Markets: Pricing of Stored Energy
Shiftability, CDC 2017 (to appear)

Thank you for your attention!