2015-03-25 Exergy Discounting with Laplace Transform · 2015-03-27 · Exergy Discounting with the Laplace Transform openVPP Net Present Value with Complex Interest Rates Publications

Exergy Discountingwith the Laplace Transform

Two-dimensional Interest Rates Applied on a Physical Numéraire

Gunnar Kaestle

International Ruhr Energy Conference 2015

Essen, 2015-03-25

Gunnar Kaestle

INREC 2015, Essen 2Exergy Discounting with the Laplace Transform

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Agenda

� Laplace-Transformation as Net Present Value Calculation

� Energy Discounting using an Absolute Numéraire

� Exergy as Valuable Part of Energy

� System Dynamics of Energy Systems

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Laplace-Transform in System Analysis

� Integraltransform from time domain to frequency domain

- f(t): function in time domain

- s : complex number s=a+bj, j² = -1

- F(s): transformed f(t)

� Helpful tools for analysing stability of linear systems

- Bode plot: gain and phase shift

- Pole-zero plot: rational transfer function G(s)=P(s)/Q(s)

dtetfsFst

∫∞

−=

0

)()(

real part

imaginary part

x

x

x

x

double zero

conjured pole pairs

Stability zone for poles

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Calculus of Net Present Value in Economics

� Net Present Value = Integration of a weighted cash flow r(t)

� Weighting via discount rate i (real number)

with ρ=ln(1+i)

� Common is

discrete form

as a sum

Σ ((1+i)-t Rt)

ρ⋅−+⋅−+−===+

−titit

eeeit

)1ln()1ln()1(∫

∞

=

−⋅+=

0

)(r)1()(NPV

t

tdttii

Simple Example for Net Present Value Calculation

-1,5

-1

-0,5

0

0,5

1

1,5

0 2 4 6 8 10 12 14 16 18 20

Time in Years

Va

lue

in

Mo

ne

tary

Un

it /

W

eig

hti

ng

Fa

cto

r Weighting Factor with

Discount Rate i=0,08

Cash Flow r(t)

Cumulated Discounted

Cash Flow

Internal rate of return 19,4%

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Net Present Value with Complex Interest Rates

� Publications on NPV calculus with Laplace transform

e.g. by Robert Grubbström, 1967 and Steven Buser, 1986

� Discount rate ρ was used as a real number

�

� Mathematical rules make calculation of NPV easier

� How to interpret imaginary or complex numbers as interest rate ρ ?

� What does complex NPV as cash flow aggregated via ρ mean?

∫∞

=

−⋅=

0

)(r)(NPV

t

tdtte

ρρ

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� Kernel function e-ρt = e-(a+bj)t = e-at x e-bjt = A-t x e-jωt

- Real component determines exponential trend

- Imaginary component determines cycle

Further research:

� Transfer functions of value flow

- Goods and services compared with price vector

- Phase shift between price and quantity of a commodity

- Cf. pork cycle and cobweb theorem

� Modelling of combined production & trading networks

- Graph (V,E) with nodes for transformation (production)

and evaluation of usefulness (trade)

- Input/Output relations from a System Dynamics view

Cash-Flow-Analysis in the Complex Plane

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� „reactive and active cash flows“ = flow of goods x price signal

- reactive cash flow: initial investment and refinancing by depreciation

- active cash flow: value increase by entrepreneural activities

� Lagging behaviour due to realisation time of real options

Reactive and Active Cash Flows

Source: Hanau, 1928 Source: Wikimedia, 2011

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Agenda





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� Discounted integral of energy flow P(t) as physical yardstick

� Energy investment (cumulated exergy used for construction)

� Useful end energy pays back invested energy

� Energetic NPV with internal interest rate = „natural“ energetic yield

Energy Discounting

Source: Bruce Hannon, 1982

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Return on Energy Invested with Periodic Reinvestment

-1,5

-1

-0,5

0

0,5

1

1,5

0 5 10 15 20 25 30 35

Time in Years

Value Unit Weighting Factor

with i=0,194Energy Flow P(t)

Cumulated NPV

ω

Re

Im

rotating vector withobserved quantityin the Re-domain

A·e-jωt

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Net Present Value in the Complex Interest Plane

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� Decreasing energy return on energy invested of fossil fuels

� Available net energy declines („high hanging fruits last“)

The Net Energy Cliff

Source: Wikimedia

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Reinvestments in Energy Infrastructure I

Source: Charlie Hall

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Reinvestments in Energy Infrastructure II

Source: Charlie Hall

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Agenda





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� Energy = Exergy + Anergy

� Exergy: valuable part of energy, transformable to physical work

- Exergy content of electrical and chemical energy = 1

- of thermal energy = Carnot‘s efficiency

(ability to perform work W = ηc x Q, ηc =1- Ti/ Ts)

� From Energy to Exergy Savings!

� Transformation examples (X: Exergy unit, Ti=0°C)

- Boiler (Ts = 90°C) 100 X → 100 * 0,25 X = 25 X

- CCGT 100 X → 60 X

- Heat Pump (Ts = 50°C) 100 X → 400 * 0,15 X = 60 X

- Power-to-Gas 100 X → 60 X

Exergy – what‘s that again?

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� How to minimise exergetic losses?

� Where are the sinks for

exergy?

� Fuel into Waste Heat

- 3 EJ Electricity Sector

- ~2 EJ Transport (ICE)

� Fuel into LowEx Heat

- 2,1 EJ HH + Com

- 0,9 EJ Industry

(Data from AGEB 2012)

Energy Flow in GermanySource:

AGEB

Primary Energy

Transformation

losses

Industry Transport Households Commercial

End Energy

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Marginal productivities / factor elasticities, USA 1900-1998

Source: Ayres, 2005

UB: physical work derived from exergy input`technicalprogress’ (akaSolow residual) explained byimprovementsin exergy conversion to physical work

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Agenda





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� Norbert Wiener: Cybernetics

� Kybernètès (gr.) = helmsman

� Science of Control Loops & Controlled Systems

� Cybernetics or Control and Communication

in the Animal and the Machine, 1948

� Jay Forrester: System Dynamics

� Simulation & Analysis of Complex Systems

� Including socio-economic models

� World3 – Meadows et al: Limits of Growth, 1972

Two Control Theory‘s Fathers

Source: Wikimedia

Source: Wikimedia

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� Generating unit (controller) reacts on price signal

� Market (plant/system) reacts on generation

� Closed loop structure, but only useful for dispatchable units

Price Signals in Closed Control Loops

controller

(generating

unit)

System

(market)

set value

(marginal costs)

market price p(t)

droop

(yield

margin)

actuating

variable

(feed-in)

market participants

(generators, loads,

storages, disturbance)

-

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� Discrete switching costs KS for on/off cycles cause

- delayed start-up (higher market price)

- delayed shut-down (negative profit margin)

� Width of hysteresis KS/h is determined by the expected length h of

the following On or Off cycle

Market Distortion by Hysteresis

price signal p(t)

in €/MWhmarginal

costs kvar

Active Power P

KS/h

ON

OFF

0

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� Discrete switching costs KS for investment expenses

� Width of hysteresis KS/a is determined by depreciation strategy

and return on capital employed

� Wide range insensitive to market price changes

� Distinctive non-linear behaviour

Deficiency to Adjust in the Long Term

Yearly profit margin

in €/(kW a)

Fixed costs

Kfix, infl p.a.

Investment

KS/aDisinvestment

0

Capital Expenditure

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Hysteresis in Different Time Domains

� Discrete expenses caused

by changes of system state

(„switching costs“)

� Sunk costs are not relevant

for decisions, only cost

influencible in the future

� On/Off cycles short term,

Build/Decommission

decisions long term

Short term hysteresis between

Start-up: Marginal costs + switching costs are covered

Shut-down: Marginal costs – switching costs are not covered

Long term hysteresis between

Investment: full costs are covered by market price

Decomission: variable and fixed costs (influenceable, w/o

capital costs) are not covered by market price

Start-up

Shut-down

Plant on

standy

Plant

running

Green field:

investment

opportunity

completed

decomis-

sion

Investment

Demolition

Kfix,infl Kfix + Kvar

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� Problem: „Prices for PV system fall faster than the feed-in tariffs can

be reduced.“ (Matthias Kurth, former BNetzA president)

� Feedback loop with proportional controller

� Market success RONI determines the FIT reduction

� Autonomous reaction of disturbances (price drop in PV equipment, changing interest rates, minimization of overhead costs, etc.)

� Political specification of target value, but not FIT reduction path

Self-organizing Incentives in Energy Policy

controller(degression of new FIT)

System(market)

set value(2,5 GWp/a)

rate of new installations(RONI in GWp/a)

droop(deviation: set

target vs. actual value)

actuating

variable(tariff in ct/kWh)

disturbing effects(learning curve, interestrates, transaction costs)

-

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Energy Markets as Non-linear Control Loop

PV & Wind

PRL

Load Prognosis:

endogenous (time)

& exogenours

factors (weather)Merit Order

Tech-nology

pel

Fuel

pBSKfix, ηel etc

PPV+Wind

Pric

e o

f Ele

ctric

ityBulk-Storage

DispersedStorage

Con-sumer

Dispatch.

Units

MarketPLast

Taxes, Dues(national)

Grid Fee (regional)

consum

er

pric

ep

el

pel (EPEX)

GridGridGridGrid

Disturbances (Prognosis errorGen.+Cons., Outages)

Physical measurable values(f,U,P,Q,etc.), ∑P=0, fsoll=50Hz

MVMVMVMVLVLVLVLVLVLVLVLV

Balancing Power with PID-behaviour& different time constants

LocalBalancingOptions

Physical Domain

Monetary DomainRedis-patchSchedules PKnoten(t)

within acountinggrids („copper disk“)

Time Domain for theEnergy Market:

- Future (Years)- Day Ahead- Intra-Day

Time Domain for Capacity Call: initial response 0-30s (Differential), primary control 30s-5min (Proportional),

secondary control 5min-¼h (Integr.), tertiary control ¼h-1h

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Any questions on System Dynamics of Energy Systems?

Either now or later:

[email protected]

Tel. +49 5323 997725

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Backup

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� Cap & trade system of the EU Emission Trading Scheme

� Prices for CO2 certificates have fallen due to oversupply

� Lost functionality as a signal for CO2 mitigation measures

� ETS market should react on changing framework

� Time lag for realising investments in the energy sector is large

� Hybrid instrument „breathing cap“: inner loop quota based, outer price based

Self regulation in EUA Trading System

controller

for cap

reduction

System(EU ETS)

set value

(carbon price

path in €/t)

EUA price p(t)

droop(deviation)

actuating

variable(no. of

certificates)

disturbances(economic recession, RES deployment, import of certificate)

-

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� How does the market react on different controller outputs?

� Method known from system analysis to extract from input/output

data basic model parameters

� Needed to model the whole system of market & controller

� Capital Asset Pricing Model useful? Incentive to invest in

comparision to other options.

Estimation of System Behaviour

risk

rate of

return efficiency border

x x

x x

x

xx xx

xx

xxx

x

investment options

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� Long investment cycles and large economical inertia

� Hysteresis caused by substancial share of capital (sunk) costs

� Disturbing effects including market failure are natural

- Capacity mechanisms?

- Long term allocation?

� Compensation by supervisory control structures

- Monetary incentives (RES, CHP)

- Reduction of barriers

- Targeting the optimum from a macro economics‘ perspective

� Money is a renewable ressource (cf. Steve Keen‘s analysis of

commercial banks), but exergy is not (2nd law of thermodynamics)

� Energy as an absolute Numéraire?

System Analysis in Energy Economics

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Price Duration Curve, EPEX Spot DE 2014

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Yearly Income Spot Market, DE 2014

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Yearly Profit Margin, DE 2014

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� Sigmoid-type, bell-type and exponential curves with similar beginning

Hubbert Cycles from Fossil Fuels vs. Renewable Energy Input

Source: Patzek, 2008

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Robert Grubbström: On the Application of the Laplace Transfrom to CertainEconomic Problems; Management Science; Vol. 13, No. 7; 1967; pp. 558-567.

Steven Buser: LaPlace Transforms as Present Value Rules: A Note; The Journal of Finance; Vol. 41, No. 1; March, 1986; pp. 243-247.

Arthur Hanau: Die Prognose der Schweinepreise; Vierteljahreshefte zur Konjunkturforschung, Sonderheft 7; Verlag Reimar Hobbing; Berlin; 1928.

Tad Patzek: Exponential growth, energetic Hubbert cycles, and the advancement of technology; Archives of Mining Sciences; Vol. 53, No. 2; 2008; pp. 131-159.

Bruce Hannon: Energy discounting; Technological Forecasting and Social Change; Vol. 21, No. 4; August 1982; pp. 281–300.

Olaf Schilgen: Energy as the Numéraire of any Given Economy, Conference on thepolitical economy of economic metrics; World Economics Association (WEA); 2013.

Charles Hall, Kent Klitgaard: Energy and the Wealth of Nations – Understanding theBiophysical Economy; Springer; New York et al.; 2012.

Reiner Kümmel: The Second Law of Economics – Energy, Entropy, and the Originsof Wealth; Springer; New York et al.; 2011.

Robert Ayres, Benjamin Warr: Accounting for growth: the role of physical work; Structural Change and Economic Dynamics; Vol. 16, No. 2; June 2005; pp. 181-209.

Miguel Mendonca; Feed-in tariffs: accelerating the deployment of renewable energy; Earthscan; London; 2007.

Recommended Reading

Documents

2015-03-25 Exergy Discounting with Laplace Transform · 2015-03-27 · Exergy Discounting with the Laplace Transform openVPP Net Present Value with Complex Interest Rates Publications