2nd International Conference

On Next Generation Wind Energy, (2nd ICNGWE)

Lund, Sweden, August 24-26, 2016

Sensitivity analysis of atechno-economic optimal wind-energy

converter

Mario Holl*, Tim Janke, Peter F. PelzTechnische Universitat Darmstadt, Chair of Fluid Systems, Otto-Bernd-Straße 2, 64287, Darmstadt,Germany*Corresponding author: mario.holl@fst.tu-darmstadt.de

Max PlatzerAeroHydro Research & Technology Associates, Pebble Beach, CA, USA

ABSTRACTA conceptual wind energy converter is introduced, which makes vast ocean power accessible. In

a previous publication this concept has been energetically and economically investigated using themethod of multi-pole system analysis (MPSA). This method consists of the three steps of system(i) modeling, (ii) analysis and (iii) optimization. An assessment of the optimal system concerningrobustness is currently missing and presented in this paper. The screening method of elementaryeffects, variance-based sensitivity analysis and the recently published PAWN-method are presentedin theory and applied to the energy converter. As indicated by the results, we complete the MPSAmethod by incorporating sensitivity analysis as the (iv) step to ensure the overall quality of themodeling process.

1

NOMENCLATURE

AbbreviationsAPF annuity present-value factor T sensitivity index PAWN-methodCDF cumulative distribution function ST total effectCRF capital recovery factor T t-distributionDBSA density-based sensitivity analysis U ocean current speed in [m/s]EE elementary effects u dimensionless ocean currentKS Kolmogorov-Smirnov statistics V vessel speed in [m/s]LCOH levelized costs of hydrogen v dimensionless vessel speedMPSA multi-pole system analysis Var varianceO&M operation and maintenance w mass specific work in [m2/s2]OAT One-at-a-time wrel relative wind speed in [m/s]PDF probability density function X input random variables vectorRMSE root mean squared error x input variables vectorROI return on investment X random variable

x multi-pole model input vectorLatin symbols x realization of random variable

A component matrix y output variables vectora sample vector Y random output variableA area in [m2] y multi-pole model output vector

a dimensionless area Z monetary flow in [e/s]B sample matrix z rate of interestb sample vectorB operation time in [s] Greek symbolsC sample matrix Γ gamma function

C yearly costs in [e/s] Π system efficiency factorCTaylor Taylor coefficient α0 absolute wind directionc wind speed in [m/s] δ capacity factorcL sail lift coefficient ε mass conversion ratecD vessel drag coefficient ζ axial induction factorD sample matrix η efficiency factorD vessel displacement in [kg] λ O&M costsE expected value µ sensitivity index elementary effectsF distribution function σ standard deviationf specific costs ε scaling uncertainty

G yearly profit in [e/s] φ degrees of freedom t- distributiong decision variable vector % density in [kg/m3]H caloric value in [m2/s2]h decision variable vector SubscriptsI investment costs in [e] 0 initialL vessel length in [m2] avail availableM mass conversion rate Comp compressorM number of input factors Des desalinatorN number of model evaluations El electricN normal distribution Elec electrolyseurn operational lifetime in [s] g gaseousO order of magnitude Gen generatorP power in [kg m2/s] l liquidp pressure in [kg/(ms2)] S shaft

R yearly revenue in [e/s] T turbiner number of elementary effects calculations V vesselS main effect

2

1 INTRODUCTIONNearly seventy percent of the earths surface is covered with water. Based on this fact, the general

higher wind speeds and capacity factors in ocean areas, the substantial share of global wind powerresources can be found in ocean regions. Consequently, the conversion of this energy source meetsmore and more the interest of energy providing companies and governments.The current technological approach for the conversion of ocean wind-power focuses on offshore windturbines, which still have to face the problems of grid connection, durability due to high load fluc-tuations and high construction and installation costs. The construction of offshore wind turbines ismainly limited by factors like water depth and accessibility. In this paper we introduce a conceptualenergy converter, called energy ship, which partially resolves the above mentioned challenges. Theconcept involves a wind-powered vessel equipped with a hydrokinetic turbine. The kinetic wind-energyis used to propel the vessel. The relative speed of ocean current and vessel speed defines the availablepower of the hydrokinetic turbine, which is then partially converted into electricity. As a consequenceof the mobility of the energy converter, the electrical energy needs to be stored in the hull of the vessel.In this paper the energy is stored chemically by the electrolytic splitting of water into hydrogen andoxygen. Beside the obvious disadvantage of energy storage, the mobility of the concept offers theadvantages of

• higher capacity factor and wind speeds due to system mobility

• centralized maintenance in harbors

• compact system design caused by the substitution of wind by water for power generation

• mature and available system components

The concept has been first proposed by Salomon [1] in 1982, followed by Meller [2], Holder [3] andGizara [4]. More quantitative analysis were performed by the fourth author [5–9] and Kim [10–12].In 2015 and 2016 the first, third and fourth author presented a physically based upper limit for theconversion of wind power in mechanical energy [13–15]. This upper limit has been derived similar to thework of Betz [16] for wind turbines and the work of the third author [17] for tidal turbines. Followingthe initial energetic analysis, the first and third author introduced a general method for holistic systemanalysis using multi-pole formalism [18]. The method is called multi-pole system analysis (MPSA)and consists of the three steps (i) system modeling, (ii) system analysis and (iii) system optimization.Using this method the first and third author evaluated the energy ship concept under energetic andeconomic aspects using empiric scaling laws. Through the third step of MPSA the optimal systemdesign and operation of the energy ship has been derived by minimizing the levelized costs of hydrogen(LCOH). Nevertheless, currently one cannot estimate the uncertainty of the optimal LCOH if inputand/or model quantities change. Also, one cannot assume these quantities to be constant, especiallynot in economics, since prices and costs change. Consequently, the two questions to be answered are:

1. What is the distribution of the model output if we consider uncertain input quantities?

2. What are the parameters, which mainly cause the output uncertainty?

The first question is answered by performing an uncertainty analysis. The purpose of an uncer-tainty analysis is to capture all uncertainties of a model and, finally, combine them in the overalluncertainty. The second question is answered by performing a sensitivity analysis, which aims atrevealing interactions of model parameters and identifying parameters which significantly cause theoverall uncertainty. Hence, the uncertainty analysis is always part of the sensitivity analysis.

Consequently, the paper is organized as follows: Section 2 summarizes the content of the MPSAof the energy ship concept. In section 3 the methods of sensitivity analysis are introduced andcompared with respect to their properties and fitness to the energy converter model. Useful methodsof sensitivity analysis are chosen. Section 4 presents the application of the chosen methods on theenergy ship concept. The results are discussed in section 5. Section 6 closes the paper with a summaryof the content and the gained knowledge.

3

2 MULTI-POLE SYSTEMANALYSIS (MPSA) OF THE ENERGY CONVERTERMulti-pole system analysis is a method for holistic system analysis which has been recently pre-

sented by the first and third author. The method consists of the three steps (i) system modeling, (ii)system analysis and (iii) system optimization. In the following, the MPSA of the energy ship, limitedto energy-, mass- and monetary flows, is summarized. A more detailed version can be found in [18].

2.1 Model of the energy shipIn general, systems compose of components which interact through various fluxes. Multi-pole

formalism is a general approach for component modeling with multiple input and output quantities,e.g. four pole theory in electrical engineering or the mechanical equivalent by applying the electrome-chanical analogy. This approach is projected but fluxes like energy-, mass-, and monetary flows areused. According to table 1 all input quantities are listed in the column matrix x with the dimensionn × 1. The output quantities are listed in the output column matrix y with the same dimension. Acomponent is described through a square matrix A with the dimension n× n.

Table 1: Multi-pole notation on component and system level.

A1 𝑥 ⋮ ⋮

A𝑘 𝑦 ⋯ ⋮ ⋮

A1 ⋮ ⋮ 𝑥 𝑦

A1

A𝑘

⋮

⋮

⋮

⋮ ⋮

⋮

⋮

𝑥 𝑦

𝑥 = 𝐀1𝑦 𝑥 = 𝐀𝑖

𝑘

𝑖=1

𝑦 𝑥 = 𝐀𝑖

𝑘

𝑖=1

𝑦

COMPONENT LEVEL SERIES-CONNECTED SYSTEM PARALLEL-CONNECTED SYSTEM

By using multi-pole formalism one can easy identify interactions of all modeled fluxes by consider-ing the entries of the component matrix. An entry on a secondary diagonal implies an interaction oftwo fluxes. The contrary is given by an identity matrix, which represents an ideal component with nointeractions. Another advantage of using multi-pole formalism is that arbitrarily interconnected sys-tems can always be combined to an easy mathematical expression, e.g. k series connected componentscan always be combined by multiplying the component matrices

x =

k∏i=1

Ai y. (1)

In a similar way k parallel connected components can be taken together through matrix addition

x =k∑i=1

Ai y. (2)

The energy ship concept consists of the components: vessel, turbine, generator, desalinator, electroly-seur, compressor and hydrogen tank. The energy-, mass-, and monetary flow through the system aremodeled. The multi-pole model is shown in figure 1.

4

DESALINATOR GENERATOR 𝑃avail 𝑃Hydr 𝑃S 𝑃El 𝑃Elec

TANK 𝑃Comp

𝑚 H2O,s 𝑚 H2O,s 𝑚 H2O,f 𝑚 H2

𝐺

VESSEL TURBINE ELECTROLYSEUR COMPRESSOR

𝑅

𝑚 H2O,s

SOURCE

+

𝐼0 = 𝐼𝑖

7

𝑖=1

CRF + 𝜆 𝐶 = CRF + 𝜆 𝐼0

𝐶

𝐼1

𝐼2 𝐼3 𝐼4 𝐼5

𝐼7 𝐼6

SALE

Figure 1: Multi-pole model of the energy ship concept.

One can see in the model that energy- and mass flow interact within the components desalinator,electrolyseur and compressor since they need a certain amount of power for desalination, eletrolyticsplitting and compression. The initial investment costs I0 are multiplied with the capital recovery fac-tor CRF to consider capital interest and λ to consider maintenance costs. Thus, the initial investmentcosts turn into yearly costs. Another interaction of mass- and monetary flow occurs through the saleof the hydrogen. It can be seen in figure 1 that all components are connected in series and thus, thesystem can be combined by matrix multiplication.

2.2 Detailed system analysisIn the second step of the MPSA, the system is separately analyzed considering all modeled aspects.

Since the investigation is limited to energetic and economic aspects, the analysis can be separated intothese two steps.

2.2.1 Energetic analysisA physical model of the energy converter is presented in figure 2 (left). The sail area is denoted

as A, the wetted vessel area as AV, the turbine area as AT and the ocean current as U .

𝑉 + 𝑈

𝑊V

𝑤

𝑆

𝑇

𝐻

𝜚

𝑉 + 𝑈

VESSEL AREA 𝐴V

TURBINE DISK AREA 𝐴T

STREAM TUBE

SAIL AREA 𝐴

𝑉 + 𝑈 𝜁

𝑉

𝐴

SAIL AREA 𝐴

𝜚l

𝜚g

COUPLING

VESSEL PROPULSION

𝐿

𝑥 𝑦

𝑧

𝑥

𝑦

𝑧

𝑊T

𝛼

𝛽0

𝛼0 𝑐

𝑉

𝑤rel

𝑆

SAIL DIRECTION

VESSEL SPEED

𝛽

𝑇

𝐻

Figure 2: Physical model of the energy converter (left) and velocities and forces at a sail section(right).

The velocity triangle at a sail section is shown in figure 2 (right). The absolute wind speed isdenoted as c, the relative wind speed as wrel and the vessel speed as V . The sail area as well as thevessel displacement scale with the length of the vessel. In a previous work [18] we analyzed severalsailing boats and derived empirical scaling functions. They can be seen in figure 3.

5

101

102

101

102

103

104

𝐴 ∝ 𝐿2.04

VESSEL LENGTH 𝐿 in m

SAIL

AR

EA 𝐴

AN

D W

ETTE

D V

ESSE

L A

REA

𝐴V

in m

2

𝐴V ∝ 𝐿1.81

Figure 3: Geometric scaling of the sail area A and the wetted vessel area AV in logarithmic scale [18].

Similar to the work of Betz [16] for wind-turbines, we presented an upper limit for energy conversionof the energy ship in [13]. This limit is based purely on axiomatic conversion laws and thus representsa physical barrier of energy conversion. The so-called coefficient of performance CP is defined as theratio of the mechanical turbine power PS and the available power Pavail and determines the efficiencyof the fluid system, consisting of the vessel and the hydrokinetic turbine.

CP :=PS

Pavail=ηT(v + u)3aT(1− ζ2)(1 + ζ)

2(%+ u3aT), (3)

with the turbine efficiency factor ηT, the dimensionless vessel velocity v := V/c, the dimensionlessocean current u := U/c, the axial induction factor ζ, the dimensionless density ratio of air andwater % := %g/%l and the area ratio aT := AT/A. Dimensionless quantities are used here for modelparameter reduction. According to Eq. 3 one can expect that the coefficient of performance maximizesfor maximal dimensionless turbine areas. In table 2 the fluid system as well as all other componentsare presented in multi-pole notation. Energy-, mass- and monetary balances are inserted.

6

Table 2: Energy-, mass-, and monetary balances for all components of the energy system.

GENERATOR DESALINATOR ELECTROLYSEUR COMPRESSOR

ENERGY BALANCE

MASS BALANCE

𝜂Gen𝑃S = 𝑃El 𝑃El = 𝑃Elec +𝑤Des𝑚 H2O,f 𝑃Elec = 𝑃Comp +𝐻0,H2

𝜂Elec𝑚 H2

𝑃Comp = 𝑤Comp𝑚 H2

𝜀Des𝑚 H2O,s = 𝑚 H2O,f 𝜀Elec𝑚 H2O,f = 𝑚 H2 𝑚 H2

= 𝑚 H2 𝑚 H2O,s = 𝑚 H2O,s

COST BALANCE

𝑃S 𝑃El

𝑚 H2O,s 𝑚 H2O,s

𝐶 𝐶

𝑃El 𝑃Elec

𝑚 H2O,f 𝑚 H2O,s

𝐶 𝐶

𝐶 = 𝐶 𝐶 = 𝐶 𝐶 = 𝐶 𝐶 = 𝐶

ILLUSTRATION

𝑃Elec 𝑃Comp

𝑚 H2O,f 𝑚 H2

𝐶 𝐶

𝑃Comp

𝑚 H2

𝐶 𝐶

𝑚 H2

MULTI-POLE NOTATION

𝑃S𝑚 H2O,s

𝐶 =

1/𝜂Gen 0 00 1 00 0 1

𝑃El𝑚 H2O,s

𝐶

𝑃El𝑚 H2O,s

𝐶 =

1 𝑤Des 00 1/𝜀Des 00 0 1

𝑃Elec𝑚 H2O,f

𝐶

𝑃Elec𝑚 H2O,f

𝐶 =

1𝐻0,H2𝜂Elec

0

0 1/𝜀Elec 00 0 1

𝑃Comp

𝑚 H2𝐶

𝑃Comp

𝑚 H2𝐶

=0 𝑤Comp 0

0 1 00 0 1

0𝑚 H2𝐶

𝑃avail 𝑃hyd

𝑚 H2O,s 𝑚 H2O,s

𝐶 𝐶

𝑃S

𝑚 H2O,s

𝐶

FLUID SYSTEM

𝐶P𝑃avail = 𝑃S

𝑚 H2O,s = 𝑚 H2O,s

𝐶 = 𝐶

𝑃avail𝑚 H2O,s

𝐶 =

1/𝐶P 0 00 1 00 0 1

𝑃S𝑚 H2O,s

𝐶

The mass specific work of the component i is denoted as wi, the efficiency factor as ηi and the massconversion rate as εi. In the energy conversion chain, the salty water flow mH2O,s is converted intoa fresh water flow mH2O,f and finally split into oxygen and hydrogen mass flow mH2 . It can be seenthat no additional costs are incurred through the operation of the energy conversion system. This istypical for renewable energy systems. Hence, the monetary flow, consisting of the initial investmentand capital interest as well as operation and maintenance costs, is traced through the system. Thesecosts need to be balanced by the revenue for cost neutrality. The revenue is achieved through the saleof hydrogen.

2.2.2 Economic analysisWithin the economic analysis the cost structure of the initial investment as well as the revenue

are investigated. This information is included in the economic model of net present value NPV todetermine the profitability of an investment and hence specifies the economic quality of the system.The method of net present value discounts all yearly monetary flows Zt to a specific point of time toconsider the time value of money. The rate of interest is denoted as z and the operation time as n. Ifthe initial investment I0 is excluded of the sum and the monetary flow is separated into yearly costsCt and revenues Rt one obtains

NPV =

n∑t=0

Zt(1 + z)t

= −I0 +

n∑t=1

Rt − Ct(1 + z)t

. (4)

With the assumption of periodic constant monetary flows, the discounted monetary flows can becalculated with the annuity present-value factor APF, which is the reciprocal of the capital recoveryfactor CRF. The yearly costs are calculated as a percentage of the initial investment Ct = λI0. Thus,the net present value becomes

NPV = −I0 + APF(R− λI0), (5)

with the annuity present-value factor

APF =1

CRF=

(1 + z)n − 1

z(1 + z)n. (6)

Using Eq. 5 one can derive the periodic profit function through multiplication with the capital recoveryfactor

7

CRF ·NPV︸ ︷︷ ︸G

= R− (CRF + λ)I0︸ ︷︷ ︸C

, (7)

with the periodic profit denoted as G, the periodic revenue denoted as R and the periodic costs denotedas C. The initial investment consists of the investment costs of all components. Based on literatureand market surveys, the vessel and turbine costs scale with the wetted vessel area and the turbine arearespectively. The desalinator, electrolyser and compressor scale with their consumed power. Figure 4shows the associated scaling functions.

100

101

102

103

104

105

AR

EA S

PEC

IFIC

PR

ICES

𝑓 𝑖≔

𝐼 𝑖

𝐴𝑖

in €/m

2

AREA 𝐴𝑖 in m2

𝑓T ∝ 𝐴T−0.357

𝑓V ∝ 𝐴V0.455

101

102

103

102

103

104

105

PO

WER

SP

ECIF

IC S

TOR

AG

E P

RIC

ES𝑓 𝑖≔

𝐼 𝑖

𝑃𝑖

in €/kW

POWER 𝑃𝑖 in kW

𝑓Comp ∝ 𝑃Comp−0.106

𝑓Elec ∝ 𝑃In−0.395

𝑓Des ∝ 𝑃Des−0.451

COMP

ELEC

DES

TURBINE

VESSEL

Figure 4: Economic empiric scaling functions for all components in logarithmic scale [18].

The investment costs of the tank scale with the stored mass and the hydrogen pressure,fTa = 1e/(kg bar). The revenue is obtained through the sale of the hydrogen and depends on thecapacity factor δ := B/T with the operation hours B and the yearly hours T , and the market price ofhydrogen fH2

R = δTmH2fH2 . (8)

Now that the system has been analyzed energetically and economically, one can present the system incombined multi-pole notation showing all linkages between energy-, mass-, and monetary flows

Pavail

mH2O,s

C

=

0

1

CPηGen

(Ho,H2

ηElec+wDes

εElec+ wComp

)0

01

εDesεElec0

0 δTfH2 −1

0

mH2

G

. (9)

One can see that all fluxes are coupled by the hydrogen mass flow. The ratio of the respective outputand input flux define the system efficiency Π, the mass conversion rate M and the return on investmentROI

Π :=mH2Ho,H2

Pavail=

CPηGenHo,H2

wComp +Ho,H2

ηElec+wDes

εElec

, (10)

M :=mH2

mH2O,s= εDesεElec, (11)

ROI :=G

C=

δTfH2

Ho,H2(CRF + λ)I0ΠPavail − 1. (12)

It can be seen that the return on investment is a function of the system efficiency. Hence, the economicprofitability cannot be investigated without considering the system efficiency. This is the reason, whywe call this kind of investigation techno-economic system analysis.

8

2.3 System optimizationIn the first step the energy ship concept was modeled using multi-pole formalism to show linkages

of all considered fluxes. In the second step the system is analyzed energetically and economically usingempirical scaling laws. Thus, a multitude of possible realizations of the energy systems has been de-scribed. Through the process of system optimization, the optimum of all possible systems is identified.The optimization is based on the broad system description and thus, different optimal systems can bederived. We present this here for the example of an energetic and a techno-economic optimal system.We also show that these two optimal systems differ in optimal system design and operation. In gen-eral, optimization problems consist of an objective function and associated constraints. The energeticoptimal system results from the maximization of the efficiency factor Π, whereas the techno-economicoptimal system results from the minimization of the levelized costs of hydrogen LCOH. The ROI isnot used as objective function, because it depends on the actual market price of hydrogen fH2 whichcannot be assumed constant. The LCOH results from the condition that NPV = 0 and representsthe critical price of hydrogen, so that no profit is obtained. By comparing the LCOH and the marketprice of hydrogen fH2 , one can easily assess the economic quality of the system by means of the NPV

NPV

< 0 , for LCOH > fH2

= 0 , for LCOH = fH2

> 0 , for LCOH < fH2 .

(13)

The structure of the two optimization problems is shown in Eq. 14. The detailed optimizationproblems are shown in [18]. The left optimization problem corresponds to the energetic optimalsystem, the right one to the techno-economic optimal system.

max Π min LCOH

s.t. Π =CPηGenHo,H2

wComp +Ho,H2

ηElec+wDes

εElec

s.t. LCOH =C

δTmH2

(14)

......

lbi ≤ gi ≤ ubi ∀ i = 1 . . . n lbj ≤ hj ≤ ubj ∀ j = 1 . . . k

with the decision variable vectors g and h and their respective upper (ub) and lower bound (lb).If several variables are considered constant the optimization results can be presented graphically.Figure 5 (left) presents the energetically optimal system design. It can be seen that the energeticallyoptimal energy system consists of a vessel with maximum length and maximum turbine area. Thisis reasonable due to the thermodynamic principle that the efficiency factor scales with machine size.That this machine design is economically unsustainable is shown by the techno-economic optimalsystem in figure 5 (right). One can see that for a specific vessel and turbine design the LCOH becomesa minimum. Beside this optimal design point the costs grow disproportionally in contrast to theenergetic efficiency and thus the LCOH increases.

9

VES

SEL

LEN

GTH

𝐿 in

m

DIMENSIONLESS TURBINE AREA 𝑎T ≔ 𝐴𝑇/𝐴 0 0.02 0.04 0.06 0.08 0.1

0

20

40

60

80

1000.45

0.4

0.35

0.3

Π

Πmax

0 0.02 0.04 0.06 0.08 0.10

20

40

60

80

100

DIMENSIONLESS TURBINE AREA 𝑎T ≔ 𝐴𝑇/𝐴

VES

SEL

LEN

GTH

𝐿 in

m

LCOH in €/kgH2

LCOHmin

Figure 5: Energetic and techno-economic optimal system design [18].

Nevertheless, the optimal system design shown in figure 5 (right) is obtained by assuming nominalinput arguments. In the next step the uncertainty of the LCOH will be investigated if input argumentsare considered uncertain by performing a sensitivity analysis. Using methods of sensitivity analysis,the uncertain input factors, which cause the output uncertainty, will be determined.

3 METHODS OF SENSITIVITY ANALYSISSo far the system has been considered as a deterministic model, consisting of specific input and out-

put quantities. This truly does not represent reality sufficiently, so that we now loosen the assumptionof deterministic input quantities and end up with a model under uncertainty.

Assume a model y = f(x) with M input quantities and the single output y. The function f(x)projects the multidimensional parameter space onto a one dimensional one, f : RM → R. According totable 3, this represents a deterministic model. The model turns into a model under uncertainty, if theexact input variables x turn into non-specific variables and thus become the random variable vectorX. Consequently, the mathematical function f(X) forms the random output variable Y . In proba-bilistic sensitivity analysis one assumes that we have information about the model input probabilitydistribution. An analytical calculation of the output uncertainty is not always possible and thus, inuncertainty analysis, Monte-Carlo-Simulations are used to determine the output random variable Y .In general, the distribution of all random variables X are discretized N times, so that a matrix withN ×M realizations of the input variables is generated. The model is then evaluated N times, so thatone obtains N realizations of the output y, which can be used for an empirical output distributionestimation. Table 3 shows the basic principle of Monte-Carlo-Simulation. In this paper the methodof Latin Hypercube Sampling is used for random number generation. An overview of this and similarmethods can be found by Han [19] and Kurowicka [20] but will not be further addressed here.

Table 3: Mathematical representation of a deterministic model, a model under uncertainty and theprinciple of Monte-Carlo-Simulation.

𝑥1

𝑥𝑀

⋮

𝑦 𝑦 = 𝑓(𝑥1, … , 𝑥𝑀) 𝑥𝑖 ⋮

𝑋1

𝑋𝑀

⋮

𝑌 𝑌 = 𝑓(𝑋1, … , 𝑋𝑀) 𝑋𝑖 ⋮

𝐱 ∈ ℝ𝑀 𝑦 ∈ ℝ 𝑓:ℝ𝑀 → ℝ 𝐗 ∈ ℝ𝑀 𝑌 ∈ ℝ 𝑓:ℝ𝑀 → ℝ

A DETERMINISTIC MODEL A MODEL UNDER UNCERTAINTY PRINCIPLE OF MONTE-CARLO-SIMULATION

𝑦1 𝑦2

𝑦𝑁 𝑥11

𝑥𝑀1

⋮

𝑥𝑖1 ⋮

𝑥12

𝑥1𝑁

𝑦𝑗 = 𝑓(𝑥1𝑗, … , 𝑥M

𝑗)

∀𝑖=1…𝑀

⋮

⋱

The above mentioned method of Monte-Carlo-Simulation is used to determine the output uncer-

10

tainty and thus belongs to the uncertainty analysis. The methods of sensitivity analysis focus onassigning the output uncertainty to the different sources of input uncertainties and thus provide theadvantages of

• validating model quality

• determining technical model errors

• revealing priorities in research

• reducing design complexity

• identifying dominant input factors

• explaining the cause and effect relationship

According to Saltelli [21] methods of sensitivity analysis should be quantitative with low computationalcosts, global, and model-independent. Borgonovo [22] additionally demands methods of sensitivityanalysis to be moment-independent, referring to the statistical moment of the random variables dis-tribution. Borgonovo and Plischke give a detailed and current review of recent advances in sensitivityanalysis in [23]. In table 4 several often used methods of sensitivity analysis are listed and evaluatedaccording to the above named criteria. Only a screening method, the method of elementary effects,variance-based methods and density based methods will be explained in more detail, since they arethe methods used here for the sensitivity analysis of the energy converter.

Table 4: Methods and requirements of sensitivity analysis.

COMPUTATIONAL COSTS

GLOBAL MODEL-INDEPENDENT

MOMENT-INDEPENDENT

ONE AT A TIME (OAT)

SCREENING

REGRESSION

VARIANCE-BASED

DENSITY-BASED

REQUIREMENTS OF SENSITIVITY ANALYSIS METHODS OF SENSITIVITY ANALYSIS

𝐱 LOW

HIGH

HIGH

MODERATE

MODERATE 𝐱

𝐱

𝐱

𝐱

𝐱

𝐱 𝐱

3.1 Screening: Elementary EffectsThe screening method of elementary effects EE is based on the work of Morris [24] and similar

to one-at-a-time methods. The screening method has been further developed by Campolongo et al.[25, 26]. Assume aj = (aj1, . . . , a

jM ) and bj = (bj1, . . . , b

jM ) as two vectors with realizations of the

random input variable vector X. The EEji for the input variable i and the vector sample j is definedas

EEji :=y(aj1, . . . , a

ji−1, b

ji , a

ji+1, . . . , a

jM )− y(aj1, . . . , a

jM )

bji − aji

, (15)

and, thus, addresses the change in output if input variable i is changed. This is done for every inputfactor and repeated at r points in the M -dimensional input space. This method is called radialsampling. Finally the sensitivity degree µ∗i of the input variable i is defined as the average of allelementary effects

µ∗i :=

∑rj=1 |EEji |r

. (16)

11

This quantity indicates the sensitivity of the output variable if the input variable i changes. Thestandard deviation of the sensitivity degree is given by

σ∗i =

√∑rj=1(EEji − µ∗i )2

r − 1, (17)

and indicates the degree of interaction of the input variable i. The method of elementary effects isused in this paper for factor fixing purposes, so that irrelevant variables can be determined and fixed.Only the dominant variables will be further investigated by the computationally expensive methodsof sensitivity analysis. The computational costs of the elementary effects screening method can beestimated to OEE (r(M + 1)).

3.2 Variance-based sensitivity analysis (VBSA)Variance-based sensitivity analysis VBSA has been used in the context of scientific modeling by

Cukier et al. [27] for the first time in 1973. Further contributions to the method are attributedto Sobol [28], Iman and Hora [29] and Wagner [30]. VBSA defines uncertainty as the alteration ofvariance and thus refers to a statistical moment of input and output distributions. According totable 4, the method is not moment-independent. The VBSA studies how the output variance can beassigned to the uncertain input factors. One has to distinguish between the theory of VBSA and theactual calculation, because in application variances can usually only be estimated. Thus, we will firstconsider the theory of VBSA in more detail, as they are presented by Saltelli [21, 31].

3.2.1 Theory of VBSALet us go back to the above assumed model. Using Monte-Carlo-Simulations the output distribu-

tion of the random variable Y is calculated. Thus, the variance of the output distribution, denotedas Var(Y ), can be estimated. Let us now further assume that we repeat the same calculation butconsider the random variable Xi as a fixed realization x∗i , Xi = x∗i . Thus, we are able to estimatethe resulting variance, taken over X∼i (all input factors but Xi). This variance is called conditionalvariance Var(Y |Xi = x∗i ) but yet depends on the realization x∗i . If one takes the mean for all realiza-tions the dependence on x∗i disappears and one obtains an estimation for the expected value of theconditional variance, denoted as E

(Var(Y |Xi)

). The central core of variance-based sensitivity analysis

is the variance decomposition. According to Eq. 18 the total variance can be decomposed into thevariance of the conditional expected value Var

(E(Y |Xi)

)and the expected value of the conditional

variance E(Var(Y |Xi)

)Var(Y ) = Var

(E(Y |Xi)

)+ E

(Var(Y |Xi)

). (18)

If this equation is divided by the total output variance, one obtains the definition of the sensitivityindex Si as the ratio of the conditional and unconditional variance

1 =Var

(E(Y |Xi)

)Var(Y )︸ ︷︷ ︸:= Si

+E(Var(Y |Xi)

)Var(Y )

. (19)

This sensitivity index is called main effect or first-order effect, because it indicates how much one couldreduce the output variance if the input factor Xi could be fixed. This procedure can be continuedto determine the second-order effect etc. The total variance can thus be composed of the differentconditional variances. This equation is the so-called ANOVA-HDMR decomposition

Var(Y ) =∑i

Var(E(Y |Xi)

)+∑i

∑j>1

Var(E(Y |Xi, Xj)

)+ · · ·+ Var

(E(Y |X1, X2, . . . , XM )

). (20)

By dividing Eq. 20 by the total variance one finds, that all order effects sum up to one.

1 =∑i

Si +∑i

∑j>1

Sij + · · ·+ Sij...M . (21)

12

Nevertheless, the computational effort for order effect calculation grows exponentially with the numberof input arguments. This is the reason why in VBSA one calculates another sensitivity index, besidethe main effect Si. This sensitivity index is called the total effect STi and can be derived similar tothe main effect. According to Eq. 18 one can decompose the variance as

Var(Y ) = Var(E(Y |X∼i)

)+ E

(Var(Y |X∼i)

). (22)

Again, by dividing through the total variance, one obtains the definition of the total effect STi as theratio of the expected conditional variance and the unconditional variance. The other ratio indicatesthe change of variance if all input factors are fixed but Xi. Hence, the total effect STi has to describe,according to the variance decomposition, all influences and interaction effects of the input factor Xi

1 =Var

(E(Y |X∼i)

)Var(Y )

+E(Var(Y |X∼i)

)Var(Y )︸ ︷︷ ︸:= STi

. (23)

This calculation yields much lower computational costs and thus VBSA calculates main and totaleffects of each input factor. The main effect indicates the direct and linear impact of the consideredinput factor, whereas the total effect considers direct and indirect impacts. Because the total effect doesalways contain the main effect, the following applies STi ≥ Si. The sensitivity difference STi − Si is ameasure for the non-linear interaction of the input factor i with other input factors. For clarity, assumethree input quantities. The total effect for input factor i = 1 is than given by ST1 = S1+S12+S13+S123.

3.2.2 Calculation of VBSASimilar to the estimation of the output distribution in the uncertainty analysis, Monte-Carlo-

Simulations are used for the VBSA. Since analytic specifications of the variances are not possible,they are estimated. First, two matrices B and C are defined, both with the dimension N ×M . Athird matrix Di is formed and contains all columns of C except of the ith-column, which is taken frommatrix B. Using these matrices the output vectors yB, yC and yDi can be calculated. The estimatedmain effect Si and the estimated total effect STi can than be calculated as

Si =yB · yDi − q20yB · yB − q20

=1N

∑Nj=1 y

jBy

jDi− q20

1N

∑Nj=1(y

jB)2 − q20

, (24)

STi = 1− yC · yDi − q20yB · yB − q20

= 1−1N

∑Nj=1 y

jCy

jDi− q20

1N

∑Nj=1(y

jB)2 − q20

, (25)

with the squared average q20 = (1/N∑N

j=1 yjB)2. The computational costs of the VBSA can be specified

to OVBSA (N(M + 2)).

3.3 Density-based sensitivity analysis (DBSA)In contrast to VBSA, methods of density-based sensitivity analysis DBSA considers the entire

distribution and do not refer to a statistical moment. Thus, DBSA is moment-independent. WhileVBSA defines uncertainty as the alteration of variance, density-based methods use a geometric quan-tity, e.g. the distance or the enclosed area, of unconditional and conditional cumulative distributionfunctions CDF. There are several methods of DBSA, depending on the chosen geometric quantity.In this paper we use the DBSA method called PAWN (named by the authors [32, 33]). Again, wedistinguish between the theory and the calculation of the method, which was initially developed byPianosi, Sarrazin and Wagener [32, 33].

3.3.1 Theory of PAWNIn the context of our model, we denote FY (y) as the unconditional CDF of the output factor. The

conditional CDF is denoted as FY |Xi=x∗i (y) and does depend on the realization x∗i . In the next stepthe so-called Kolmogorov-Smirnov statistic KS, which is attributed to Kolmogorov [34], is calculatedas the maximal distance between the unconditional and conditional CDF

13

KS(x∗i ) := maxy|FY (y)− FY |Xi=x∗i (y)|. (26)

Thus, the KS statistic is a geometrically interpretable quantity. The actual sensitivity index Ti forthe input factor i is calculated as the maximum value or the median of KS statistics for all x∗i . Weuse the median in this paper since the impact of extreme but improbable realizations is otherwiseoverestimated.

Ti := medianx∗i

(KS(x∗i )

)(27)

Since the median is used here, the dependence on x∗i is lost. The sensitivity index Ti varies betweenzero and one. A low value implies no influence of the random variable Xi, whereas a value close toone does. Since the unconditional CDFs are affected by interaction effects, the sensitivity index canbe considered comparable to the total effect of the VBSA.

3.3.2 Calculation of PAWNThe CDFs are estimated using Monte-Carlo-Simulations, since they cannot be calculated analyt-

ically. Thus, they are also referred to as empirical CDFs. The estimated unconditional CDF FY (y)can be calculated using the results of the Monte-Carlo-Simulations of the uncertainty analysis. Theestimated conditional CDF FY |Xi=x∗i,j (y) is calculated by creating a ND × 1 vector, containing the j

conditioning samples of x∗i . In the next step ND matrices Bj of the dimension Nc ×M are created,consisting of the Nc samples of the input factors X∼i. In every matrix Bj the column Nc×i is replacedby the specific entry x∗i,j in order to keep this input factor constant. In this way the calculation of the

jth matrix leads to the conditional CDF FY |Xi=x∗i,j (y). The estimated KS statistic is calculated as

KS(x∗i,j) = maxy|FY (y)− FY |Xi=x∗i,j (y)|, (28)

and the estimated sensitivity index as

Ti = medianx∗i=x

∗i,1,...,x

∗i,ND

(KS(x∗i,j)

). (29)

The computational costs of the PAWN method can be specified to OPAWN (N +ND ·Nc ·M). Itshall be mentioned here, that the PAWN method is able to perform sensitivity analysis in a definedarea of the output distribution. Hence, it is possible to determine input factors, which mainly causeuncertainty in a specific range of the output distribution, e.g. the estimated KS statistic investigatesuncertainty subjected to the condition y > y0

KS(x∗i,j) = maxy>y0|FY (y)− FY |Xi=x∗i,j (y)|. (30)

To assess the quality and the precision of the respective estimator, the method of Bootstrapresampling is used. Based on the original Monte-Carlo sample of size N , a large number of newsamples of size N is drawn with replacement from the original sample. For each of these resamplesthe respective sensitivity indices are calculated. Using this method, one can calculate an empiricaldistribution and confidence intervals, which represent the estimation error.

4 SENSITIVITY ANALYSIS OF THE ENERGY CONVERTERIn terms of a probabilistic sensitivity analysis, the uncertainty of the input factors needs to be

known. The uncertainties of the input factors are stochastically modeled using three probabilitydensity functions.

i) Uniform distributionA random variable Xi ∼ U(a, b) is uniformly distributed in the interval [a, b] with a < b anda, b ∈ R with the probability density function PDF

fU (x) =

{1b−a , a ≤ x ≤ b0 , else

. (31)

14

A uniform distribution is used, if the input quantity can be assumed to appear in a specific intervalwith the same probability.

ii) Normal distributionA random variable Xi ∼ N (µ, σ) is normally distributed with the expected value µ and thestandard deviation σ and µ, σ ∈ R with the symmetric PDF

fN (x) =1

σ√

2π· exp

(−(x− µ)2

2σ

). (32)

A normal distribution is used to describe random processes, which scatter around the expectedvalue.

iii) t-distributionA random variable Xi ∼ T (φ) is t- distributed with the degree of freedom φ. The PDF is givenby

fT (x) =Γ(φ+12

)Γ√φπ(φ2

) (1 +x2

φ

)−φ+12

, (33)

with the gamma function Γ. The t-distribution plays an important role for the construction ofconfidence and prediction intervals and converges to the normal distribution for φ→∞. Thus, ifthe sample size is small and the standard deviation is unknown, a greater uncertainty is assumedin comparison to an assumed normal distribution.

The model of the energy converter consists of M = 31 input factors. The estimation of the intervalsand the respective PDFs are based on data of empirical market surveys, expert opinions or literaturereviews. According to Sachs [35], the uncertainty of all used scaling functions are determined bycalculating the prediction interval of the respective regression function y

y0 ± εi = y0 ± t1−α/2,φ−2 · RMSE ·√

1 +1

φ+

(x0 − x)2∑φi=1(xi − x)2

, (34)

with the root mean squared error RMSE =√∑φ

i=1(yi−yi)2φ−2 , the average of the input factor x and

the t statistics, which is a function of the number of samples φ. Consequently, the prediction intervalis small, if the RMSE is small, the input factor x0 is close to the average x and the number ofsamples is high. Figure 6 shows the scaling functions of sail area and vessel costs with their respective95%-prediction intervals. For the optimal vessel length, the uncertainties can be seen.

100

101

102

100

102

104

106

108

VESSEL LENGTH 𝐿 in m

VES

SEL

INV

ESTM

ENT

𝐼 V in

€

𝐼 V ∝ 𝐿2.63

𝐼 V,lb

𝐼 V,ub

𝒯𝐼V14

𝐿opt 100

101

102

100

102

104

SAIL

AR

EA 𝐴

in m

2

VESSEL LENGTH 𝐿 in m

𝐴 ∝ 𝐿2.04

𝐴 lb

𝐴 ub 𝒯𝐴 48

𝐿opt

Figure 6: Estimated scaling functions for vessel costs (left, φ = 14) and sail area (right, φ = 48) withprediction intervals in logarithmic scale.

15

It can be seen on the example of the vessel investment scaling function that the upper and lowerbound of the prediction intervals grow for greater vessel lengths, because the prediction is extrapolatedand not based on empirical data. One can see that the prediction bounds are smaller in the area ofthe empirical data, because the prediction is based on real data. Table 5 lists the input factors, theirmeaning and the assumed PDF. The input factors are based on the physical model of the energyconverter presented in [18].

Table 5: Listing of all input factors with respective PDF. The † indicates variables, which are laterfixed as a result of the screening method.

INPUT FACTOR SYMBOL PROBABILITY DENSITY FUNCTION

ABSOLUTE WIND DIRECTION†

ABSOLUTE WIND SPEED DIMENSIONLESS OCEAN CURRENT

SAIL LIFT COEFFICIENT VESSEL DRAG COEFFICIENT

EFFICIENCY FACTOR TURBINE EFFICIENCY FACTOR GENERATOR

EFFICIENCY FACTOR ELECTROLYSEUR EFFICIENCY FACTOR COMPRESSOR†

INLET PRESSURE COMPRESSOR OUTLET PRESSURE COMPRESSOR VOLUME SPECIFIC WORK DESALINATOR†

LIFESPAN

RATE OF INTEREST

CAPACITY FACTOR

O&M VESSEL O&M TURBINE† O&M ELECTROLYSEUR†

O&M DESALINATOR†

O&M COMPRESSOR† O&M TANK†

SPECIFIC COSTS HYDROGEN TANK†

ASPECT RATIO† COEFFICIENT TAYLOR APPROXIMATION† SCALING FUNCTION SAIL AREA

SCALING FUNCTION VESSEL DISPLACEMENT

SCALING FUNCTION VESSEL COSTS SCALING FUNCTION TURBINE COSTS

SCALING FUNCTION ELECTROYSEUR COSTS† SCALING FUNCTION DESALINATOR COSTS†

SCALING FUNCTION COMPRESSOR COSTS†

ENER

GET

IC IN

PU

T

FAC

TOR

S EC

ON

OM

IC IN

PU

T

FAC

TOR

S SC

ALI

NG

FU

NC

TIO

NS

𝛼0

𝑐 𝑢

𝑐L 𝑐D

𝜂T 𝜂Gen

𝜂Elec 𝜂Comp

𝑝1 𝑝2 𝑤Des

𝑛

𝑧

𝛿

𝜆V 𝜆T 𝜆Elec

𝜆Des

𝜆Comp 𝜆Ta

𝑓Ta

𝐿WL/𝐿 𝐶Taylor 𝜖A

𝜖D

𝜖IV 𝜖IT

𝜖IElec 𝜖IDes

𝜖IComp

𝛼0 ~ 𝒰 100, 120

𝑐 ~ 𝒩 8, 1/ 20 𝑢 ~ 𝒰 0, 0.2

𝑐L ~ 𝒰 1, 1.8 𝑐D ~ 𝒰 3.74 ⋅ 10−3, 7.7 ⋅ 10−3

𝜂T ~ 𝒰 0.8, 0.95 𝜂Gen ~ 𝒰 0.8, 0.95

𝜂Elec ~ 𝒰 0.54, 0.74 𝜂Comp ~ 𝒰 0.5, 0.8 𝑝1 ~ 𝒰 1, 80 𝑝2 ~ 𝒰 50, 400 𝑤Des ~ 𝒰 3.75, 5.95

𝑛 ~ 𝒰 15, 25

𝑧 ~ 𝒰 0.03, 0.09

𝛿 ~ 𝒰 0.6, 0.9

𝜆V ~ 𝒰 0.04, 0.1 𝜆T ~ 𝒰 0.043, 0.128 𝜆Elec ~ 𝒰 0.01, 0.051

𝜆Des ~ 𝒰 0.1, 0.2

𝜆Comp ~ 𝒰 0.04, 0.1 𝜆Ta ~ 𝒰 0.01, 0.5

𝑓Ta ~ 𝒰 1, 4

𝐿WL 𝐿 ~ 𝒰 0.7, 0.95 𝐶Taylor ~ 𝒰 2.5, 3 𝜖A~ 𝒯 48

𝜖D~ 𝒯 38

𝜖IV~ 𝒯 14 𝜖IT~ 𝒯 25

𝜖IElec~ 𝒯 10

𝜖IDes~ 𝒯 20 𝜖IComp

~ 𝒯 10

Now, that the methods of sensitivity analysis have been presented and the uncertainty of theinput factors have been determined, we pursue the following strategy for the sensitivity analysis ofthe energy converter.

1. Screening: The most influential input factors will be determined using the screening methodof elementary effects as presented in section 3.1. The input factors with negligible influence arefixed in order to reduce computational costs. This procedure is called Factor Fixing.

2. Uncertainty analysis: Using the model with reduced parameters, the probability distributionof the model output is calculated with Monte-Carlo-Simulations.

3. VBSA and PAWN: To assign the model output uncertainty to the different sources of inputuncertainty, a VBSA and the PAWN method is performed. The ranking of all input factors interms of their influence on the output uncertainty is called Factor Priorization. As mentionedin section 3.3.2, the PAWN method is able to determine the most influencing input factors in agiven range of the output distribution. This is called Factor Mapping.

16

5 RESULTS AND DISCUSSIONThe results will be presented and discussed according to the above stated strategy. All calculations

presented here and all the results were obtained using the Matlab Toolbox SAFE [36].

5.1 ScreeningThe screening method of elementary effects is chosen to determine influencing and non-influencing

variables. The calculation was performed with r = 200 for all M = 31 input factors, which isequivalent to 6400 model evaluations. As a result the sensitivity index µ∗i is calculated, implying thelinear influence of the input factor i. The standard deviation σ∗i of the sensitivity index µ∗i indicates thedegree of interaction of the input factor i with other input factors. Figure 7 illustrates the sensitivityindex and the standard deviation for all input factors with the corresponding 95%-confidence intervals.

SENSITIVITY INDEX 𝜇𝑖∗ STANDARD DEVIATION 𝜎𝑖

∗

051015 0 5 10 15

FIX

𝜇∗𝑖,crit

𝜎∗𝑖,crit

𝑐L 𝜖A

𝑐D 𝜖IV

𝛿 𝑛 𝜂Elec 𝑐 𝜖𝐷 𝜆V

𝛼0 𝜖IComp

𝜆Elec 𝜆T 𝜆Ta

𝜖IT 𝑝2 𝑧 𝑢 𝜂Gen 𝜂T

𝐿WL/𝐿 𝜖IElec 𝑓Ta

𝑝1

𝜂Comp 𝜆Comp

𝜆Des 𝑤Des

𝜖IDes

𝐶Taylor

0 5 10 150

2

4

6

8

10

12

SENSITIVITY INDEX 𝜇𝑖∗

STA

ND

AR

D D

EVIA

TIO

N 𝜎𝑖∗

𝑐L

𝜖A

𝑐D

𝜖IV

𝛿

Figure 7: Factor fixing as a result of the elementary effect screening method with the boundariesµ∗i,crit = 2.21 and σ∗i,crit = 1.37. The fixed input factors are presented in the darker bars.

A factor is treated as non-influential if the condition µ∗i < µ∗i,crit ∧ σ∗i < σ∗i,crit is fulfilled. Usingthis condition, one guarantees that a factor is still considered influential, even if his linear impact(considered by µ∗i ) is small but its non-linear impact (considered by σ∗i ) is high. There is no rule forthe calculation of µ∗i,crit and σ∗i,crit. In this paper we calculate them as one fifth of the mean of thefirst three maximum values. The fixed factors are presented in the tornado chart shown in figure 7(right) by the darker bars and are also marked with a † in table 5. It can be seen that the uncertaintyof the investment as well as the operation and maintenance costs (O&M) of the storage componentsdo not have a great influence on the uncertainty of the LCOH. In contrast, the investment and O&Mcosts of the vessel are influential input factors. This is reasonable, since the vessel investment andmaintenance costs are one of the cost drivers of the energy system. Further, the efficiency factor of thecompressor is not influential, since the compressor consumes only a small portion of the total power.On the other hand, the efficiency factors of turbine, generator and electrolyser have a non-negligibleimpact, since the total mechanical power is generated by the turbine, converted into electric powerby the generator and the majority of the power is used to produce hydrogen in the electrolyser. Themost influential factors are the uncertainty of the lift coefficient cL, the drag coefficient cD and thescaling uncertainty of the vessel sail area εA, even though the total uncertainty is low due to a lotof data. According to the above stated condition and the aim of factor fixing, 14 input factors arefixed because of negligible impact. Thus, in the further analysis the amount of inputs is reduced toM = 17 to reduce computational costs. Nevertheless, one should keep in mind that the method ofelementary effects is local and only gives a qualitative estimation of the input factors influence.

17

5.2 Uncertainty analysisSubject of the uncertainty analysis is the estimation of the total output uncertainty under consid-

eration of the input uncertainties using Monte-Carlo-Simulations. The output factor of our proposedsystem is the LCOH. The simulations are performed with the reduced number of uncertain inputfactors according to the result of the previous screening. The PDF as well as the CDF of the LCOHare shown in figure 8.

0 10 20 30 40 500

0.02

0.04

0.06

LCOH in €\kg

PR

OB

AB

ILIT

Y

𝑓(LCOH)

LCOH in €\kg C

UM

ULA

TIV

E P

RO

BA

BIL

ITY

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

𝐹 LCOH = 10

1 − 𝐹 LCOH = 10

𝐹(LCOH = 12.5) = 0.5

𝐹(LCOH = 17.1) = 0.75

𝐹(LCOH = 9.1) = 0.25

𝐹(LCOH = 10) = 0.33

𝐹(LCOH)

Figure 8: Probability density function PDF (left) and cumulative distribution function CDF (right)of the LCOH.

As one can see in figure 8 (left), the output PDF, denoted as f(LCOH), is not symmetric but right-skewed. One can already suspect here that the VBSA should be scrutinized, because the uncertainty ofthe model output cannot be adequately described through the expected value and the output variance.The value with the highest probability is LCOH = 10 e/kg. The output CDF, denoted as F (LCOH),is gained by integrating the PDF. Thus, the probability of values smaller and greater as 10 e/kg canbe calculated as

F (LCOH = 10) =

10∫−∞

f(x) dx ≈ 33 %, (35)

1− F (LCOH = 10) =

∞∫10

f(x) dx = 1−10∫−∞

f(x) dx ≈ 67 %. (36)

The cumulative distribution function can be seen in figure 8 (right) with the 25%-quartile, 50%-quartile (median) and 75%-quartile respectively. Because of the extreme values in figure 8 (right), theaverage of the LCOH = 13.84 e/kg is greater than the median.

5.3 VBSA and PAWNFor the VBSA the model was evaluated 245 000 times. In contrast, the model was evaluated

410 000 times for the PAWN method. Nevertheless, one has to keep in mind that in each simulationof the PAWN method only the effect of one input factor is estimated. Thus, the number of modelevaluations per input factor is 24 000 for the PAWN method and 245 000 for the VBSA. The confidenceintervals of the sensitivity indices have been calculated using the Bootstrap resampling method. Forthe VBSA, the population has been resampled 5000 times. For the PAWN method, the populationhas been resampled 1000 times. Figure 9 compares the convergence of both methods. One can seethat the estimation of the PAWN sensitivity index Ti converges much faster in comparison to the totalsensitivity index STi of the VBSA.

18

0 100 000 200 000

0

0.2

0.4

0.6

0.8

1

NUMBER OF MODEL EVALUATIONS

ESTI

MA

TED

SEN

SITI

VIT

Y IN

DEX

𝑆 T

𝑖

NUMBER OF MODEL EVALUATIONS

ESTI

MA

TED

SEN

SITI

VIT

Y IN

DEX

𝑇 𝑖

0 10 000 20 000

0

0.2

0.4

0.6

0.8

1

Figure 9: Convergence comparison of the VBSA sensitivity index STi (left) and the PAWN sensitivityindex Ti (right).

According to the aim of factor prioritization, the estimated sensitivity measures STi and Ti ofthe input factors are sorted in descending order and shown in figure 10 with their respective 95%-confidence interval. The order of the sensitivity indices of the input factors agree within the VBSAand PAWN method, representing that the variance influencing input factors are of the same order asthe entire distribution influencing input factors. Nevertheless, it can be seen that the estimated totalsensitivity effects STi have huge confidence bounds, meaning that the estimation of the sensitivityeffect is highly inaccurate. Since the confidence intervals are that huge and partially underneath zero,no reliable statement about the real total sensitivity effect can be made. In contrast, the sensitivityindices Ti have small convergence intervals.

-0.2

0

0.2

0.4

0.6

ESTI

MAT

ED S

ENSI

TIV

ITY

IND

EX

𝑆 T𝑖

𝑇 𝑖

𝑐L 𝑐D 𝜖A 𝛿 𝜖IV 𝜂Elec 𝑧 𝜆V 𝑐 𝜖D 𝑢 𝑛 𝜂Gen 𝜂T 𝜖IT 𝑝1 𝑝2

Figure 10: Factor prioritization of the VBSA (left markers) and PAWN method (right markers) withrespective 95%-convergence intervals.

The huge confidence intervals of the total effects result from the varying values for the estimatedtotal effect. If the calculation of single total effects is recalled and one demands it to be negative, oneobtains

19

STi = 1−1N

∑Nj=1 y

jCy

jDi− q20

1N

∑Nj=1(y

jB)2 − q20

< 0. (37)

Since the estimated Var(Y ) > 0, this leads to

1

N

N∑j=1

(yjB)2 − q20︸ ︷︷ ︸est. Var(Y )

<1

N

N∑j=1

yjCyjDi− q20.︸ ︷︷ ︸

est. Var(E(Y |X∼i)

)(38)

One can see that negative values are obtained if the conditional variance is not reduced but exceedsthe unconditional variance. If this is the case, variance becomes a misleading measure for uncertaintysince the knowledge of a true value of an input factor should always reduce the reported uncertainty.Borgonovo et al. describe an example, that leads to the same conclusion in [37]. We assume thatthe non-symmetric, right-skewed distribution is the reason for this, since the distribution cannot beadequately characterized by its expected value and variance. The PAWN method does consider theentire output distribution and does not refer to a statistical moment, the confidence bounds are smallercompared to the ones of the VBSA. The results of the PAWN method allow a reliable statement interms of factor prioritization and, thus, will be the only results further considered. As shown in figure10 the dominating input factors are the lift coefficient cL, the drag coefficient of the vessel cD, thescaling uncertainty of the sailing area εA and the capacity factor δ. It is worth noting that the scalinguncertainty of the vessel investment εIV is only the fifth important input factor, even if the uncertaintyis great due to the small number of accessible data. The uncertainty of the sail area scaling law ismuch smaller but still causes greater uncertainty in the model output. It is recalled here that thePAWN method defines uncertainty as the median of all maximal distances between unconditional andconditional CDFs. Figure 11 presents, as an illustrative example, the variation of the CDFs for threeinput factors with varying influence.

0 10 20 30 40 500

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CU

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BIL

ITY

LCOH in €/kg LCOH in €/kg LCOH in €/kg

𝐹LCOH

𝐹LCOH | 𝑐L=𝑐L𝑖 𝐹LCOH | 𝜖IV

=𝜖IV,𝑖

𝐹LCOH 𝐹LCOH

𝐹LCOH |𝜖IT=𝜖IT,𝑖

Figure 11: Conditional and unconditional cumulative distribution functions of the lift coefficient cL(left), the scaling uncertainty of the vessel investment εIV (middle) and the scaling uncertainty of theturbine investment εIT (right).

One can see that the conditional cumulative CDFs of the lift coefficient FLCOH|cL=cLi(figure 11

left) indeed significantly vary among the unconditional CDF FLCOH. The scaling uncertainty of thevessel investment εIV as well as the scaling uncertainty of the turbine investment εIT have a muchsmaller influence. This can be graphically seen in figure 11 (middle and right), since the variation ofthe conditional CDFs is much smaller compared to the ones of the lift coefficient.

As can be seen in figure 8, large values for the LCOH appear with a certain probability. One canbe interested in analyzing which input factors mainly cause these large values of the LCOH. This

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kind of analysis is called factor mapping and is presented here for LCOH > 20 e/kg. Therefore,the Kolmogorov-Smirnov statistic, according to Eq. 30, is modified to

KS(x∗i,j) = maxLCOH>20

|FLCOH(LCOH)− FLCOH|Xi=x∗i,j (LCOH)|. (39)

Figure 12 shows the probability density function of the LCOH with the marked range of the part ofthe probability distribution considered here. Also, the results of the factor mapping are presented.

0

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Y

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LCOH in €/kg

1 − 𝐹 LCOH = 20 = 𝑓 𝑥

∞

20

dx

𝑓 LCOH

𝑐L 𝑐D 𝜖A 𝛿 𝜖IV 𝜂Elec 𝑧 𝜆V 𝑐 𝜖D 𝑢 𝑛 𝜂Gen 𝜂T 𝜖IT 𝑝1 𝑝2

Figure 12: Results of the factor mapping for the probability density distribution rangeLCOH > 20 e/kg.

It can be seen in the figure that the order of the input factors is mainly the same. Only some inputfactors, whose sensitivity indices are close together, switch their position, e.g. the rate of interest zand the O&M costs of the vessel λV. This indicates that the uncertainty of these input factors doesaffect the uncertainty in the here shown range of the probability density function more than in thetotal value range of the LCOH.

6 SUMMARY AND CONCLUSIONIn this paper we introduced an innovative energy converter, called the energy ship concept. The

method of MPSA has been used in the past to evaluate the energetic and economic quality of theenergy converter. The method consists for now of the steps (i) system modeling, (ii) detailed systemanalysis and (iii) system optimization. As shown in section 2 one can derive the optimal system designand operation using this method. Nevertheless, the robustness of the optimal solution found has notyet been assessed. In general, we formulated two questions that needed to be answered:

1. What is the distribution of the model output if we consider uncertain input quantities?

2. What are the parameters, which mainly cause the output uncertainty?

To answer these two questions, methods of sensitivity analysis were applied to the proposed sys-tem. Following this train of thought, in section 3 we described the theoretical basis of the sensitivityanalysis methods used in the present paper. The input factor uncertainty of the energy converter wasinvestigated in section 4. Also, the strategy of sensitivity analysis was stated. The results are pre-sented and discussed in section 5. Using the screening method of elementary effects, we were able to fixinput factors with negligible influence on the output uncertainty. This procedure is generally knownas factor fixing and allows one to reduce the number of input factors and thus, the computationalcosts for further sensitivity analysis. In the second step, in the uncertainty analysis, the total output

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uncertainty was calculated using Monte-Carlo-Simulations. Thus, the first of the above two questionscould be answered. In the last step, a VBSA and the PAWN-method, belonging to the density-basedmethods, was performed to assign the total output uncertainty to the different sources of input un-certainty. In summary, it can be stated, that the VBSA led to very little knowledge gain about theproposed model, because the estimated total effects STi are subjected to huge 95%-confidence inter-vals. Thus, a reliable statement of input factor sensitivity is not possible. We suppose that thesehuge confidence intervals are caused by the non-symmetric, right-skewed output probability densityfunction (see figure 8), which cannot be adequately characterized by its expected value and variance.Thus, defining uncertainty as the alteration of variance can cause misleading results. This is consis-tent with current research observations [23, 37]. In contrast, the PAWN method leads to statisticallysignificant and reliable results, since the method considers the entire distribution. Thus, the inputfactors were ordered according to their respective influence on the output distribution. This procedureis called factor prioritization and provides the information, which input factor uncertainty mainlycauses the model output uncertainty. The most influential input factors of the proposed model are thelift coefficient of the sail, the drag coefficient of the vessel, the scaling uncertainty of the sail area andthe capacity factor. Thus, the second question could be answered. The PAWN method does also allowto perform a factor prioritization in a specific range of the output probability density function. Thisis called factor mapping and is used if one wants to become aware of the dominant input factors ina specific range of the total uncertainty. In this paper, factor mapping was used to determine whichuncertain input factors cause large LCOH.

In general, the sensitivity analysis helps to understand and explain the cause and effect relationshipof a given model. Also, sensitivity analysis reveals priorities in research, e.g. one will pursue the designof the here presented energy converter with maximal lift coefficient and minimal drag as a result of thesensitivity analysis. Through the identification of non-influential input factors, one does not need toimprove the accuracy of these input factors, since the impact is negligible. Thus, sensitivity analysisevaluates and measures model quality. Because of these advantages and the important informationto ensure overall quality of the modeling process, we incorporate sensitivity analysis here as the (iv)step of the method MPSA.

ACKNOWLEDGMENTThe authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the financial

support in the framework of the Excellence Initiative, Darmstadt Graduate School of ExcellenceEnergy Science and Engineering (GSC 1070).

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