Electronic copy available at: http://ssrn.com/abstract=2495278

Robustness of Renewable Energy Support Schemes Facing Uncertainty andRegulatory Ambiguity

Ingmar Ritzenhofena,∗, John R. Birgeb, Stefan Spinlera

aWHU – Otto Beisheim School of Management; Kuehne Foundation Endowed Chair in Logistics Management;Burgplatz 2, 56179 Vallendar, Germany

bUniversity of Chicago Booth School of Business; Jerry W. and Carol Lee Levin Professor of OperationsManagement; 5807 South Woodlawn Avenue, Chicago, IL 60637, United States

Abstract

Renewable portfolio standards, feed-in-tariffs, and market premia are widely used policy instru-

ments to promote investments in renewable energy sources. Regulators continuously evaluate these

instruments along the main electricity policy objectives of affordability, reliability, and sustainabil-

ity. We develop a quantitative approach to assess these policies and their robustness to exogenous

changes along these dimensions using a long-term dynamic capacity investment model. We compare

their robustness in the light of uncertain renewable feed-in and ambiguous future regulation. We

implement the robustness analysis employing different risk measures and find that renewable port-

folio standards deliver most robust results, while feed-in-tariffs achieve target renewable buildup

rates at least cost.

Keywords: Renewable portfolio standards, Feed-in-tariffs, Power generation, Scenario reduction

JEL: Q4, Q2, L9, L5

1. The Role of Renewable Energy Support Schemes

Installations of renewable energy sources (RES) for electricity generation continue their growth

path with investment of 214 Billion USD globally in 2013, equivalent to capacity additions of 80 GW

(REN21, 2014). This investment boom is fueled by different RES support schemes (RESSS) such

as renewable portfolio standards (RPS) and feed-in-tariffs (FIT) including market premia (MP)

(Couture et al., 2010), which have been adopted by 79 and 98 countries, states, and provinces,

respectively (REN21, 2014). In recent years, researchers have focused on assessing the effectiveness

of these policy tools in driving RES investment (e.g., Mormann (2012)). However, ongoing public

∗Corresponding author. Tel.: +49 261 6509 434Email addresses: ingmar.ritzenhofen@whu.edu (Ingmar Ritzenhofen), john.birge@chicagobooth.edu (John

R. Birge), stefan.spinler@whu.edu (Stefan Spinler)

Electronic copy available at: http://ssrn.com/abstract=2495278

debate shows that RESSS are now assessed more broadly for example in terms of market integration

of RES, total cost, and electricity price effects. At the same time, given the uncertain future of

electricity markets (Most and Keles, 2010), RESSS have to be designed in a way to deliver on these

policy objectives under uncertain and ambiguous conditions.

In our paper, we propose a quantitative approach to assess the structural impact and robustness

of the main RESSS RPS, FITs, and MPs on electricity markets via a long-term dynamic capacity

investment model. We compare these policies along the three dimensions of energy policy high-

lighted by the International Energy Agency (IEA) – affordability (or cost competitiveness), security

of supply, and sustainability (IEA, 2011), as well as their robustness to deliver the desired results

under uncertain and ambiguous conditions. This modeling approach applies in general settings for

dynamic decision models with uncertainty and ambiguity. With capacity factors as the uncertain

stochastic parameters in this model, decisions become path-dependent since high capacity factors

in earlier periods depress electricity prices that then affect future investment decisions. Since such

path dependencies can create large unwieldy models, we simulate a broad range of potential real-

ization paths and then aggregate them using scenario reduction techniques into a tractable number

of scenarios for computation. To obtain further robustness (see, e.g., Birge (2014)) we use multiple

sample batches to derive a corresponding set of performance criteria that we use for comparisons.

For the ambiguous parameters of regulatory policy, we assume that investors consider all possible

outcomes prior to their investment decision.

In addition to this modeling contribution, the combination of uncertain and ambiguous pa-

rameter decisions provides new insights into RES support policy analysis. As an extension to the

observations that appear in a deterministic model (Ritzenhofen et al., 2014) that RPS generally

provide the lowest volatility in electricity prices and that FIT achieve RES buildup targets at the

least cost, we observe several new characteristics with uncertain capacity factors and ambiguous

policy actions. First, the uncertain capacity factors lead to minor changes in total cost of less than

3%. The uncertainty of total cost is greatest for FIT followed by MP and RPS, since the effect

of convex prices (in the capacity factor) is fully mitigated under the market-based RPS, partly

dampened under the more rigid MP, and not cushioned at all under FIT. Second, we observe that

average electricity prices are most uncertain under FIT given the same rationale. Third, our results

indicate rising electricity price volatility in all schemes but greater relative increases for RPS and

MP than for FIT, which is consistent with the convex nature of electricity prices (in the capacity

factor) and already higher capacity factors under FIT in the deterministic case. Comparing results

2

under policy ambiguity with those under uncertainty, we observe, however, different effects. Main-

taining the support rates, we observe that total cost increases across RESSS when accounting for

the possibility of the abolition of the federal investment tax credits (ITCs). Before the abolition of

ITCs, the impact of regulatory ambiguity on investment decisions and thereby electricity markets

is small as it only affects expected prices. After an ITC policy change, two counteracting effects

emerge: incentives for RES investment are reduced by the lost ITC and potentially increased by

higher electricity prices in later periods. This is the case for MP, as increasing electricity prices

partly compensate the lost ITC support, thus inducing RES investment at a slower pace. For

FIT, RES investment decisions are isolated from market price signals, which reduces RES instal-

lations leading to a significant drop in electricity price volatility, higher CO2 emissions, and lower

RES generation. As opposed to these price-based schemes, the exogenously set RES quota under

the quantity-based RPS ensures the level of RES generation and support rates adjust in the REC

market to accommodate the higher net RES cost, which is partly compensated by technology shifts.

The remainder of this paper is organized as follows: in Section 2, we review related work. In

Sections 3 and 4, we describe the modeling methodology. We introduce a numerical study in Section

5 and discuss results and policy implications in Section 6. In Section 7, we conclude and outline

areas for further research. The paper builds on and extends work by Ritzenhofen et al. (2014) by

explicitly modeling alternative future capacity factor scenarios and ambiguous policy environments.

2. Theoretical Background and Contribution to the Literature

Global RES expansion is primarily supported through RPS and FIT schemes as well as MP

as a derivative of FIT (REN21, 2014). RPS exist in the US and the UK while FIT or MP are in

place in countries such as China, France, and Germany. These schemes are the most wide-spread

policy instruments to promote RES expansion and represent converse approaches to inducing RES

investment. Under RPS, the regulator defines target RES volumes and lets the market determine

required support levels. On the other hand, regulators fix support rates under FIT schemes and let

the market determine the new RES buildup. MP represent an intermediate approach combining

elements of the aforementioned schemes. Consequently, a comparative analysis of these schemes not

only allows us to shed light on the main RES support policies in place, but also provides insights

on structural differences between RESSS in general.

In recent years, numerous researchers investigated the impact and functionality of the different

RESSS for which REN21 (2014) provide an overview. Haas et al. (2004) survey different RESSS

3

employed in European countries showing that FITs are the preferable mechanism to promote RES

investments in terms of effectiveness without investigating their efficiency. Ragwitz et al. (2006)

empirically show that FIT have the greatest impact on RES investment while countries relying on

RPS schemes exhibit high RESSS costs and low buildup rates. In a more recent work, Mormann

(2012) discusses the qualitative benefits and disadvantages of RESSS, concluding that FIT have

“the greatest conceptual capacity to leverage investment in [...] renewable energy” (p.734). Couture

et al. (2010) summarize the research around FIT and their design options. Couture and Gagnon

(2010) conclude that FIT schemes are considered the most effective RESSS. They compare different

FIT design options and find that market-independent, fixed FIT increase investment security, lower

cost of capital and thereby attract a diverse set of investors. On the other hand, market-dependent,

premium price schemes such as the MP increase market integration of RES capacity. Research on

RPS is abundant, too, with a greater focus on the US. Chen et al. (2009) compare studies on

the impact of RPS schemes on electricity rates in US states and find that RPS introductions

often lead to moderate retail rate increases. In their empirical study, Yin and Powers (2010) find

a significantly positive effect of RPS policies on in-state RES investment. Overall, the existing

literature on RESSS suggests that FITs are best in increasing RES investment. However, there are

only few reports incorporating effects beyond the mere RES buildup and immediate price changes.

In this work, we aim at assessing RESSS along the dimensions of affordability, security of sup-

ply, and sustainability of electricity supply, as well as the robustness of their performance under

uncertainty and ambiguity. For this purpose, we develop a dynamic capacity investment model to

realistically reflect the behavior of electricity markets. As emphasized by Kettunen et al. (2011),

this requires us to account for investor behavior and their profitability considerations. Thus, we

cannot assume a centralized command-and-control cost-minimization perspective as used in well-

established models such as Takriti and Birge (2000). This is reconfirmed by Hobbs (1995) high-

lighting the importance of developing comprehensive models accounting for “rate feedback,” i.e.,

market price formation and reactions of supply and demand to it. Furthermore, we need to reflect

the long-term nature of investments in the electricity sector while maintaining the hourly granu-

larity of time steps to reflect short-term price developments in the presence of ramping constraints

of generators, intermittent RES feed-in, and strategic behavior occurring in moments of capacity

scarcity. To meet these prerequisites, we have to maintain chronological load and price information.

Given the resulting number of periods and the dimensionality when accounting for uncertainty or

ambiguity, we cannot resort to frequently used approaches such as linear programs as in Bloom

4

(1983) and dynamic programs as in Petersen (1973) focusing on cost-minimization. Instead, we

build on approaches applied in the context of RESSS such as Eager et al. (2012). They employ a

representative agent approach for a Monte Carlo simulation to assess conventional capacity invest-

ment for given levels of RES supply. In their system dynamics model, Fagiani et al. (2013) simulate

the development of a stylized electricity market with different RESSS to assess under which con-

ditions RPS are preferable to FIT. The modeling approach and objectives of these two works are

generally similar to our requirements, but exhibit several discrepancies: first, Eager et al. (2012) use

exogenously given RES buildup rates, while it is crucial to derive RES installations endogenously

when assessing RESSS. Second, they use load duration curves and thereby omit chronological load

information essential to estimate the impact of rising intermittent RES under ramping constraints

and strategic bidding. Third, Fagiani et al. (2013) critically assume that electricity markets develop

towards a least-cost optimum in the medium term – an assumption that is debatable. Addition-

ally, they only compare aggregate cost instead of developing a deep understanding of the trade-offs

between cost minimization, security of supply, sustainability, and robustness.

We assess the impact of uncertainty and ambiguity on the performance of RESSS by comparing

their robustness, i.e., the variability of their performance in light of changing conditions. Objective

robustness – i.e., the deviation from the target or optimal solution (Cornuejols and Tutuncu, 2006) –

can be assessed building on models as in Malcolm and Zenios (1994), who propose a stylized robust

optimization to plan the “capacity expansion of power systems under uncertain load forecasts”

(p.1040). They emphasize that non-robust optimal solutions are highly sensitive to uncertainties.

This is also highlighted by Mulvey et al. (1995), who stress applications of robust optimization in

the power sector. In our model, we use Value at Risk (VaR) and Conditional Value at Risk (CVaR)

as constraints for investment decisions and report these risk measures. This is in line with Gabrel

et al. (2014), who emphasize that VaR and CVaR are well-suited risk measures in energy systems

to protect “the decision-maker against [...] ambiguity and [...] uncertainty” (p.471).

To the best of our knowledge, we are the first to quantitatively assess the structural impact of

RESSS in light of uncertainty and regulatory ambiguity, while accounting for profitability-driven

investor behavior, a multi-decade time horizon at hourly granularity, ramping, and strategic bidding

to reflect actual market behavior. We contribute to the literature by (i) quantitatively comparing

the impact and robustness of different RESSS using this novel and more realistic modeling approach

and (ii) providing policy makers and investors with a tool to estimate the impact of these RESSS on

key policy dimensions and market conditions. We find that policy makers face trade-offs between

5

minimizing total cost of electricity generation, security of supply, sustainability, and variability of

results. While FIT can induce RES investment at lower cost than MP and RPS, they lead to higher

price volatility and generally more uncertain market outcomes. RPS, on the other hand, deliver a

lower variance of results across all policy dimensions but incur higher total cost. In the following,

we explain the model and results in more detail.

3. Deterministic Base Model

First, we describe the methodology of our deterministic model based on Ritzenhofen et al.

(2014) used to compare RESSS. Subsequently, we extend this approach to reflect uncertainty and

ambiguity and analyze the robustness of RESSS. We compare the impact of RPS, FIT, and MP

on electricity markets along the key policy dimensions of affordability, security of supply, and

sustainability. We estimate affordability using total cost of electricity supply and electricity prices,

evaluate security of supply based on electricity price volatility, and assess sustainability using CO2-

emissions. The model and its results are extensively verified and validated using a framework

suggested by Gass (1983) (see Appendix).

3.1. Structure of Model

For the dynamic capacity investment model running for 60 years at hourly granularity we choose

a representative agent approach. Thus, we can account for investor behavior for each generation

technology for representative plants reflecting likely reactions to the RESSS analyzed and changing

market developments. Additionally, the hourly granularity allows us to reflect daily and seasonal

fluctuations in demand and supply, which is particularly critical accounting for intermittent genera-

tion. In every year, investors decide per technology on plant retirements because of age, divestitures

given negative profitability, or new investments for positive project values and thereby update the

installed electricity generation capacity. These decisions are initially based on a simplified forecast

and then on continuously updated price and volume expectations reflecting all information available

at that time and take into account possible future market outcomes. After the updates, all plants

bid into the electricity market to get dispatched to meet hourly demand. Additionally, they either

bid in the renewable energy certificate (REC) market or receive MP or FIT payments depending

on the RESSS. We run this logic multiple times for the entire time horizon always updating in-

stalled capacities, resulting prices, and quantities. We continue running this until investor behavior

6

and prices converge towards a fixed point solution, i.e., a situation in which none of the investors

changes his behavior in anticipation of reactions of other market participants.

3.2. Estimation of Future Generation Portfolios for Initial Forecast

We derive a first forecast by initially assuming that the generation portfolio ensures the least-

cost supply of electricity to meet demand while complying with annual RES buildup targets and

abstracting from operational constraints and the currently installed generation portfolio. All pa-

rameters used feature subscripts in the following order: i for the technology, j for the plant number,

t for the year, and y for the hour within a year. Subscripts are dropped when no differences exist

along a specific dimension. First, we estimate future annual electricity load using the chronological

hourly load curve adjusted by expected changes in electricity demand. We then derive the residual

chronological hourly load curve by deducting generation of intermittent RES of a representative

year. We map this against the screening curve – a joint total cost function for all technologies

showing the least expensive generation technology per runtime Qt – to forecast the least-cost gen-

eration portfolio per year. Based on these capacities, we compute hourly electricity prices Pt,y and

annual REC prices PRECt by intersecting the respective demand and supply curves as explained in

Sections 3.4 and 3.8. We use these heuristically derived prices and capacities as a starting point.

3.3. Estimation of Future Generation Portfolios for Actual Updates

Similar to the forecast, we first update the generation portfolio and then compute electricity

and REC prices in the actual updates for every year. First, all existing plants reaching the end of

their useful life LFi are retired and the installed capacity Ki,t is updated accordingly. Second, we

compute the profitability of each plant Πi,j,t and investors retire all unprofitable plants. We derive

the marginal cost for all technologies n with ηi,t being the heat rate, P fuelt the fuel price, ιi the CO2

intensity, PCO2t the CO2 price, and MCotheri,t other marginal cost for operation and maintenance:

MCi,t = ηi,t ∗ P fuelt + ιi ∗ PCO2t +MCotheri,t . (1)

We estimate plant profitabilities using technology-specific average prices per year Pi,t from the

forecast, adjusted by currently observed prices in the actual iteration Pi,t−1 from the previous period.

Thereby, we mirror behavior of investors, who typically use forecasts and recently available actual

price data as the basis for their decisions. Using exponential smoothing, we weigh these prices

with the factor (1 − α) and α, respectively. We update forecasted prices continuously with the

7

exponentially smoothed prices P smi,t . We use plant-specific electricity volume forecasts to estimate

the quantity Qi,j,t and obtain the annual profit Πi,j,t of each plant with fixed cost FCi,t:

Πi,j,t = −FCi,t + (α ∗ P smi,t + (1 − α) ∗ Pi,t−1 −MCi,t) ∗Qi,j,t. (2)

If Πi,j,t < θi the plant is divested and the installed capacity is updated. The threshold value θi

can be zero or a negative value depending on investor preferences and expectations of future returns.

In a third step, all market participants decide upon new investments. We assume a number invmax

of different investor groups. These investors inv with inv ∈ 1, ..., invmax have symmetrically and

discretely distributed discount rates dinv given their diverging costs of capital (Ehrenmann and

Smeers, 2013) and assess the profitability of new plants using net present value (NPV).

NPVinv,i,t = −ICi,t +LFi

∑tLF =1

Πi,t/(1 + dinv)tLF (3)

If NPVinv,i,t ≥ Θi, the threshold set by investors, investment takes place. Again, Θi can be

set at zero or a higher positive value applying real options theory and given uncertainty of future

returns. In light of capacity constraints of authorities and firms capable of building power plants

and manufacturing the required equipment, a maximum buildup of plants per year mnew,maxi exists.

A share of φinv investors applies dinv and only constructs the corresponding share of the maximum

investment per technology Knew,maxi when their NPV exceeds Θi. Based on these updates of the

generation capacity and the number of plants mi,t, we can then estimate electricity and REC prices.

3.4. Electricity Price Formation

Electricity prices are established as an equilibrium of supply and demand. We assume ex-

ogenously given linear hourly price-demand curves with a slope βt reflecting the price elasticity of

demand and its ordinate D0t,y as Pt,y(Lt,y) =D0

t,y−βt∗Lt,y. We model the supply curve based on the

hourly available capacity of each generation technology Kavi,t,y and its marginal cost MCi,t forming

the merit order of supply. We define Kavi,t,y =Ki,t ∗ϑi,t,y as a function of the installed capacity after

all retirements, divestitures and new investments Ki,t and its exogenously given availability ϑi,t,y.

The latter is the average annual capacity factor for all non-intermittent generation technologies

and therefore constant over y, but changes every hour y for the intermittent renewable technologies

wind and solar. The electricity market is cleared at the quantity Lt,y and price Pt,y. We run this

mechanism for every hour and year. If electricity demand cannot be met by existing capacity, load

8

reductions are enforced and costs amounting to the loss of load equivalent LOLE are incurred.

3.5. Ramping Constraints

We approximate ramping constraints by technology and adjust the electricity supply curve

accordingly. The maximum available capacity Kavi,t,y depends on the utilized capacity per technology

in the previous period Kuti,t,y−1 and the maximum ramp rate γupi :

Kavi,t,y(Kut

i,t,y−1) =Kuti,t,y−1 + γ

upi ∗Kplant

i ∗mi,t. (4)

Dispatchable plants can adjust their output downwards at any time but incur additional cost

for more abrupt reductions given the additional strain on equipment. With γdowni as the ramp down

rate, we assume that plants adjust their output from one hour to the next at no additional cost

as long as more than their “minimum capacity” is utilized with Kmini,t,y (Kut

i,t,y−1) =Kuti,t,y−1 − γdowni ∗

Kplanti ∗mi,t. Below Kmin

i,t,y , plant operators incur additional ramping cost RCi,t and reflect them in

their bidding behavior as follows: MCi,t,y(Kuti,t,y,K

uti,t,y−1) = ηi,t∗P

fuelt + ιi∗PCO2

t +MCotheri,t −RCi,t.

These ramping constraints can only be implemented using chronological electricity load data, which

is one reason why we extend the approaches of traditional capacity expansion models often purely

relying on non-chronological and more aggregate information such as Hobbs (1995).

3.6. Strategic Bidding

In electricity markets, strategic behavior and exercise of market power are common and typically

increase with augmenting supply shortage (Eager et al., 2012). This is why we model strategic

bidding as a function of capacity scarcity and define electricity prices with strategic bidding P εt,y

as the previously established prices plus a percentage markup ε depending on the capacity margin

Kmargt,y defined as the hourly excess capacity. Electricity prices P εt,y are then defined as

P εt,y(Kmargi,t,y , Lt,y) = Pt,y ∗ (1 + ε(Kmarg

t,y , Lt,y)) with∂ε

∂Kmargi,t,y

< 0 and∂ε

∂Lt,y> 0. (5)

3.7. Support Schemes

We incorporate three RESSS into the model. Under an RPS scheme, obliged entities have

to obtain a number of RECs corresponding to a quota λt of their total electricity production or

sales. RECs can be procured through own RES production or by buying RECs on the market. In

case of non-compliance with this quota, obliged entities pay a penalty κ set by the regulator. We

9

model the REC market with annual clearance corresponding to the compliance period. Demand is

exogenously given through regulation by the following piece-wise price-demand function:

PRECt =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

[−∞, κ] ∀8760

∑y=1

LRESt,y ≥ λt ∗8760

∑y=1

Lt,y

κ ∀ λt ∗8760

∑y=1

Lt,y ≥8760

∑y=1

LRESt,y .(6)

The supply curve for RECs is defined as a bid stack with r as the number of RES technologies.

All existing eligible RES plants bid their entire expected electricity production at

(MCi,t − P avrgi,t )+ ∀ i = 1, ..., r. (7)

RES electricity production beyond this point requires new investment. Investors bid ωinv,i,t as

the levelized NPV gap for new RES generation with

ωinv,i,t =(Θi −NPVinv,i,t)ϑi,t ∗ 8760 ∗Kplant

i

∗ dinv ∗ (1 + dinv)LFi

(1 + dinv)LFi − 1∀ i = 1, ..., r, ∀ inv = 1, ..., ninv. (8)

This formula illustrates three key properties of REC markets. First, negative prices do not

occur in REC markets as opposed to electricity markets. Second, REC prices decrease (increase)

with increasing (decreasing) electricity prices. Third, REC prices crucially depend on whether

new RES capacity is required. To determine annual REC prices, we intersect demand and supply

curves. The results from the REC market are fed back into the profitability calculations and into

the electricity markets. The bidding behavior of RES plants into the electricity market changes to

MCmarkett,y (Lt,y) =MCi,t,y(Lt,y) −PREC,expt ∀ i ≤ r with PREC,expt as the expected REC price. The

annual profit function of RES plants is updated accordingly with this new stream of revenue.

Under FIT schemes, eligible RES producers receive a fixed payment guaranteed for a pre-defined

period of time per unit of electricity. The tariff level is fixed by the regulator. Under most FIT

schemes, the RES electricity is not marketed by the RES producers themselves, but bought and then

marketed by the regulator or a related institution (Couture and Gagnon, 2010). The profit function

of RES generators is modified as follows: ΠFITi,j,t = −FCi,t + (FITi,t −MCi,t) ∗Qi,j,t. Additionally,

RES electricity is now fed into the market at a rate ξt,y with ξt,y ≤ min(MC1,t,y, ...,MCn,t,y).

Furthermore, investment is only induced by NPV considerations as shown in Section 3.3. Under

MP, RES generators participate in the electricity market as under RPS and receive a fixed premium

MPi,t per unit of electricity sold. The level of MPi,t is set by the regulator. The MP changes the

10

profit function of RES generators to ΠMPi,j,t = −FCi,t+(α∗P smi,t +(1−α)∗Pi,t−1+MPi,t−MCi,t)∗Qi,j,t.

Similar to FIT schemes, investment in RES is only induced through profitability considerations and

not through a quota. The bidding behavior of RES generators changes in a similar way as under

the RPS scheme as shown in Equation 7.

3.8. Iteration and Convergence

The calculations described so far represent one iteration for the entire time horizon. These

iterations do not yet reflect reactions of market participants to updated prices and installed gener-

ation capacities. We account for market reactions by repeating all calculations several times until

results converge. We use average electricity prices as well as average total capacity per iteration as

convergence criteria. While a single aggregated parameter could indicate convergence despite very

different underlying model results, this is unlikely for two different criteria. We compute relative

changes δ between iterations as illustrated in the following example for the average electricity price

and require that these changes fall below a threshold ζ.

δP,it =P avrgit − P avrgit−1

P avrgit−1

< ζ with P avrgit =

toutput

∑t=1

8760

∑y=1

Pt,y ∗Lt,ytoutput

∑t=1

8760

∑y=1

Lt,y

(9)

4. Robustness Analysis

The base model is extended to reflect uncertain and ambiguous inputs. We analyze and com-

pare the robustness of different RESSS, i.e., how performance changes vis-a-vis uncertainty and

ambiguity along the main policy dimensions. Electricity markets are influenced by multiple uncer-

tain parameters affecting both the demand and supply side such as uncertain load, learning rates,

fuel cost, and capacity factors in particular for RES generation. Ambiguous parameters comprise

imminent regulatory changes – e.g., the abolition of existing regulation such as production tax cred-

its (PTCs), technological breakthroughs reducing cost, or geopolitical changes resulting in import

interruptions of fuels. While some factors such as changes in the cost structure can be assessed

reasonably well using sensitivity analysis (see Ritzenhofen et al. (2014)), other uncertainty and

ambiguity factors require a more comprehensive assessment as decision rules change and various

elements of the market are affected directly. This is why we select the variability in average annual

capacity factors as the source of uncertainty and the potential abolition of ITCs as the source of

ambiguity. We explain these choices in more detail in the following.

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4.1. Description of Capacity Factor Uncertainty

Existing research emphasizes that intermittent power generation from solar and wind is both

uncertain at an (sub-) hourly (Papavasiliou and Oren, 2013) and annual level (NREL, 2014; Zhou

and Smith, 2013). While the (sub-) hourly fluctuations are highly relevant for unit commitment or

dispatch models, investment decisions are rather affected by changes in electricity generation over

the entire life-cycle of a plant (Dean, 2010). This is particularly relevant as wind speeds and solar

radiation and thus capacity factors for the respective generation technologies change between years

(Dean, 2010; Castro et al., 2011). This is why we extend our model, which already reflects hourly

intermittency using a sample year, to account for the annual variability in electricity generation

from wind and solar plants. We capture this in the capacity factor ϑi,t and focus on its variation

by introducing an annual capacity factor multiplier per technology χi,t. We use tildes to illustrate

stochastic variables. While the distribution for example of wind speeds at a specific site can

be characterized well by the frequently used Weibull distribution (Lun and Lam, 2000; Zhou and

Smith, 2013), this approach cannot be easily used for large areas with multiple sites with potentially

correlated capacity factors and over long time horizons (Castro et al., 2011; Kim and Powell, 2011;

Papavasiliou and Oren, 2013). We assume ϑi,t and therefore χi,t to be normally distributed with

χi,t ∼ N(υi,t, σ2i,t) and uncorrelated both for ease of exposition and based on first-hand data from

the market used in Section 5 (EIA, 2014b; CEC, 2013). However, it is easily possible to adjust the

model to account for different distributions and correlations by reformulating the decision rules in

Section 4.2 and adjusting the scenario specifications in Section 4.3.

4.2. Impact on Decision Rules

We adjust decision rules of investors and utilities in the electricity market model to account for

the solar and wind power capacity factor uncertainty.

4.2.1. VaR for Investment Decisions

In the deterministic model, we use a “static” NPV criterion for investment decisions. To reflect

the uncertainty for the intermittent technologies wind and solar, we complement this by a VaR

criterion as suggested in Gabrel et al. (2014). Using VaR allows us to quantify the loss aversion

exhibited by many investors and utilities. We choose VaR instead of the frequently preferred risk

measure CVaR (Artzner et al., 1999) for computational ease in our specific case. Additionally, the

potential drawbacks such as lack of coherence of VaR do not apply here, as normally distributed

capacity factors lead to normally distributed project values, thus rendering VaR a coherent risk

12

measure (Rockafellar and Uryasev, 2002). Since we abstract from potential financing or liquidity

constraints during the lifetime of such projects, we do not have to explicitly account for the multi-

period coherence of the risk measure used (Eichhorn et al., 2005; Artzner et al., 2007). We assume

that investors intend to limit losses from new projects to a threshold ΘV aRi with a confidence level

of ϕ. Therefore, they only invest in new projects if

V aRinv,i,t(ϕ) ≥ ΘV aRi and NPVinv,i,t(dinv) ≥ Θi. (10)

Furthermore, we assume that χi,t is identically and independently distributed for all t ≤ T and i ≤

rint as the number of intermittent RES technologies resulting in χi,t ≡ χi ∼ N(υi, σ2i ). Adjusting the

NPV formula accordingly and denoting the average variable profit per unit of electricity generated

as V Pi,t, NPVs are also normally distributed with

NPV inv,i,t = −ICi,t +LFi

∑tLF =1

−FCi,t + V Pi,t ∗ χi ∗Qi,j,t(1 + dinv)tLF

with V Pi,t = P avrgi,t −MCi,t. (11)

NPV inv,i,t ∼ N⎛⎝⎛⎝υi

LFi

∑tLF =1

V Pi,t ∗Qi,j,t(1 + dinv)tLF

⎞⎠

−ICi,t +LFi

∑tLF =1

−FCi,t(1 + dinv)tLF

, σ2i

LFi

∑tLF =1

(V Pi,t ∗Qi,j,t(1 + dinv)tLF)2⎞⎠

(12)

This allows us to establish a simple VaR decision rule with V aRinv,i,t(ϕ) = NPV−1inv,i,t(ϕ).

Following this logic, bids in the REC market are adjusted as ωinv,i,t is now defined as the maximum

of the NPV and the VaR gap so that both investment criteria are met. The VaR gap is computed

in analogy to Equation 8 by replacing NPVinv,i,t with V aRinv,i,t and Θi with ΘV aRi .

4.2.2. Changes in REC Demand

For the MP and FIT cases, no further changes in decision rules are required. However, in the

RPS case, given the uncertainty in electricity generation from intermittent RES, REC demand is

no longer equal to the quota set by the regulator. In the deterministic model we assume that RES

production exactly meets the RES quota with

λt ∗8760

∑y=1

Lt,y =r

∑i=1

8760

∑y=1

Li,t,y =r

∑i=1

8760

∑y=1

Ki,t ∗ ϑi,t,y. (13)

13

However, given uncertain electricity generation from wind and solar plants, annual RES gen-

eration becomes uncertain, too. Utilities have to account for this uncertainty when developing a

strategy to minimize their expected cost for RPS compliance. This is primarily a cost-minimization

task and often independent of the afore-described investment decisions. Therefore, a newsvendor

approach with a critical fractile rule can be applied to determine the amount of RECs to be pur-

chased. For entities under RPS regulation, any REC shortfall is fined with an amount κ, thereby

representing underage cost. Excess RECs – assuming that RECs cannot be banked – are forfeited

and represent overage cost of PREC,expt . Based on overage and underage cost we can establish a

newsvendor rule to determine the optimal REC order quantity. Given normally distributed capacity

factors, annual RES generation LRESt = ∑ri=1∑8760y=1 Li,t,y is also normally distributed with

LRESt ∼ N⎛⎜⎝

rint

∑i=1

⎛⎝υi

8760

∑y=1

Ki,t ∗ ϑi,t,y⎞⎠+

r

∑i=rint+1

8760

∑y=1

Li,t,y,rint

∑i=1

⎛⎜⎝σ2i

⎛⎝8760

∑y=1

Ki,t ∗ ϑi,t,y⎞⎠

2⎞⎟⎠

⎞⎟⎠. (14)

We can now determine the critical fractile C.F.t needed to compute the optimal order quantity

for RECs Ξt, i.e. in our case the demand for RECs. With Ξt REC demand increases for most

critical fractile calibrations given high underage cost κ and comparatively low overage costs PRECt .

C.F.t =κ

κ + PREC,expt

and Ξt = (LRESt )−1(C.F.t). (15)

4.3. Scenario Tree Structure to Model Uncertainty

After adjusting the decision structures in the model, we implement the uncertainty of intermit-

tent RES generation. The state space is defined by the realizations of the annual capacity factor

multipliers χi ∀ i = 1, ...rint, t = 1, ..., T following multivariate i.i.d. variable realizations with known

probabilities. However, we cannot leverage these properties to reduce the otherwise very large state

space as each state is additionally defined by an a priori unknown capacity vector Kzt comprising

all capacities Kzi,t for each scenario z. Kz

t1is path dependent and can only be computed running

the entire model for a given set of realizations of χzi,tt1t=1. Similarly to Takriti and Birge (2000),

we address these challenges by discretizing the stochastic variables into a large set of scenarios and

by applying scenario reduction techniques to ensure tractability and computational efficiency.

We build on work by Dupacova et al. (2000), who survey approaches for generating scenarios ap-

proximating the underlying stochastic process. We generate discrete scenarios using Monte Carlo

simulation and ensure that the model and all decisions herein remain non-anticipatory (Heitsch

14

and Romisch, 2009) in order to preserve the stochastic nature of the model. First, bidding in the

electricity market is based merely on actual installed capacity and previously utilized capacity.

Second, REC market bidding as well as NPV and profit decision rules are purely driven by actuals

and expected values while the VaR decision rule is based on the entire distribution of the uncer-

tain variables. Additionally, for each iteration, we randomly draw from the entire set of possible

scenarios. Third, demand in the REC market based on the critical fractile rule takes into account

the entire distribution of RES generation and is thereby also non-anticipatory.

We reduce the number of scenarios following an approach by Heitsch and Romisch (2009) and

Growe-Kuska et al. (2003). Based on the probability distributions described in Section 4.1 we

generate a large number of scenarios with equal probabilities using standard software for Monte

Carlo simulation such as @risk. Given the comprehensiveness of our model it is computationally

highly demanding to compute Kzt for all realization paths z of χi. Instead, we aggregate similar

scenarios in terms of realizations of χzi,t to a target number S of representative scenarios covering

the entire range of potential outcomes. Further details are provided in Section Appendix A in

the Appendix. For each iteration of the model, we randomly draw from these preserved scenarios

according to their probability distribution after combination of scenarios.

4.4. Regulatory Ambiguity

In recent years, electricity markets have been transformed driven in particular by new regulation.

One focus of regulatory change has been on RES as illustrated by the introduction, abolition, and

modification of RESSS such as the PTC in the US (DoE, 2014b). While it is known that such

changes can occur and sets of potential outcomes can be defined accordingly, it is often impossible

to attach reliable probability distributions to these scenarios. This distinguishes ambiguity from

conventional uncertainty (Ellsberg, 1961), e.g., of wind and solar capacity factors. We choose to

implement regulatory ambiguity surrounding ITCs, which exist on a federal level in the US and are

currently scheduled to expire or change at the end of 2016 (DoE, 2014c). Under ITCs, a share of

the investment dITC in eligible RES installations can serve as a tax credit and effectively reduces

the net investment. We assume that the ITC is currently in place and is likely to be abolished at

some a priori unknown future time tITC with tITC ∈ 1, ..., T or a subset of this. Thus, the net

investment depends on the ITC state reflected through the binary variable ∆(t, tITC) with

∆(t, tITC) ∈ 0,1 and ∆ = 1 ∀ t < tITC , ∆ = 0 ∀ t ≥ tITC . (16)

15

Therefore, the effective investment becomes ICITCi,t = ICi,t ∗ (1 −∆(t, tITC) ∗ dITC). We imple-

ment this ambiguity by randomly drawing from the set of ITC abolition times for each iteration.

Since eligibility for the ITC is known upon investment and thus directly influences investment

decisions when abstracting from substantial lead times, the ambiguity itself does not have a first

order impact on investment nor divestiture decisions. However, there is a second order impact on

the investment decisions as – contingent on the state of the ITC – future capacity installations and

therefore electricity prices and volumes might differ. We account for second order effects through

exponential smoothing (see Section 3.8), as changes in expected prices contingent on potential fu-

ture generation portfolio compositions are reflected here. Note that for other kinds of regulatory

ambiguity, which does not resolve when decisions such on investments and divestitures are made,

first order effects can exist too, requiring modifications of decision rules such as the implementation

of max-min/min-max rules in analogy to the analysis of the uncertainty factors. Based on these

considerations, we do not adjust decision rules here. Nonetheless, one could consider refining these

rules to reduce the set of potential scenarios contingent on prior developments and thereby induce

learning. However, this is beyond the scope of this paper and represents an area of future research.

5. Calibration to Californian Electricity Market

We apply the model to California (CA) as the biggest US state with a population of 38.3 million

people in 2013 and a GDP of 1.96 trillion USD in 2011 (EIA, 2014c). CA is the second biggest RES

electricity market with 28 TWh of non-hydro RES generation in 2011 (EIA, 2012) and an RPS in

place since 2002 (CEC, 2013). We account for key regulation and CA’s technology profile.

5.1. Regulation

Resource adequacy requirements (RAR) were established after the CA electricity market crisis of

2001. RAR oblige load-serving entities to contract a minimum capacity RARt of 115% of forecasted

hourly peak demand (CPUC, 2013) with ψ as the capacity margin: RARt =max(Lt,1, Lt,2, ..., Lt,8760)∗

ψ. We denominate the capacity eligible for compliance with RARt asKRARt = ∑ni=1Ki,t∗%i,t−Koutage

i,t

with %i,t being the technology factors and Koutagei,t the expected outages. %i,t is based on availabil-

ity factors for dispatchable generation and on three-year averages for non-dispatchable generation.

Under RAR it is required that KRARt ≥ RARt. Non-compliance is fined. In the model, we sus-

pend profitability retirements, if this causes KRARt to fall below RARt and force investment in the

16

dispatchable technology with the smallest NPV shortfall until KRARt ≥ RARt, if KRAR

t < RARtbeforehand. Investors are compensated for this profitability gap and these costs are accounted for

in the total costs of electricity supply. The CA CO2 cap-and-trade is included as a cost element in

the marginal cost function (see Equation 1). We use the price floor of these allowances of 10 USD

with an annual progression of 5% to estimate the cost (Association of Corporate Counsel, 2012).

PTC and ITC are incorporated as key federal regulatory elements. Under PTCs, RES producers

receive a tax credit per unit of RES electricity generated for a pre-defined time horizon. We im-

plement PTCs as a reduction in (net) marginal cost for RES producers. We use PTC amounts of

2.3 cUSD/kWh for wind and geothermal as well as 1.1 cUSD/kWh for biomass (DoE, 2014b). In

analogy, we include the afore-described ITC in the model amounting to 30% for solar and 10% for

geothermal (DoE, 2014b). Despite their uncertain future, we assume that both PTC and ITC will

be prolonged beyond their current expiration date.

5.2. Technologies, Imports, and Exports

We account for all technologies producing more than 1% of CA’s annual electricity generation.

We assume that no capacity additions in hydro or nuclear will occur given limited site availability

and regulation. While hydro potential for plants > 1 MW amounts to 4.3 GW in CA, most of these

sites are subject to environmental concerns (DoE, 2014a). Furthermore, no new nuclear power

plants have been built in recent years and heavy restrictions for new nuclear plants have been

introduced (NCSL, 2010). In recent years electricity imports have accounted for approximately

30% of CA’s electricity consumption (CEC, 2013). We cannot include a detailed reflection of

imports, since their generation technologies are not reported. Instead, we treat imports in analogy

to solar and wind generation using the same sample year with hourly data obtained from CAISO

(2014).

5.3. Data Used

Key data sources include EIA (2013a) for technology parameters, CAISO (2014) for price,

demand, and hourly generation data, as well as CEC (2013) for the existing generation park and

its age. We use multiple sources such as CAISO, CA Public Utilities Commission, CA Energy

Commission, and IEA to supplement and validate the data. Plant sizes reported are average values.

Future values of ICi,t are endogenously derived using exogenously given learning factors µi based

on NREL (2010). We use data from CAISO (2014) for wind and solar generation patterns. Based

on our sample year, we use average capacity factors ϑi,t for wind of 23%, 26% for solar, 57% for

17

Tech. Size Life Invest Fixed cost Marg. cost Heat rate Learning Init. capa.

i Kplanti LFi ICi,1 FCi,1 MCi,1 ηi µi Ki,1

MW years Mio. USD Th. USD USD/MWh GJ/MWh % GW

Wind 100 25 2.2 39.6 −2.3 − 0.6 6.5Solar 150 25 2.7 24.7 −0.0 − 2.1 2.3Geoth. 50 25 5.6 132.0 −1.6 − 0.7 2.6Biomass 20 25 4.1 105.6 56.9 14.2 0.8 1.6Hydro 500 100 2.9 14.1 0.0 − 0.0 13.5Nuclear 2,234 40 5.5 93.3 2.1 − 0.9 2.2CCGT 620 35 0.9 13.2 42.9 7.4 1.0 8.1OCGT 210 35 0.7 7.0 53.7 10.3 0.8 38.4LOLE inf 1 0.0 0.0 2,000.0 − 0.0 0.0

Table 1: Technology parameters

geothermal, and 81% for biomass. RES, excluding geothermal, hydro, and nuclear, feature a CO2

intensity ιi of zero. For geothermal, ιi amounts to 0.054 t/MWh and to 0.374 t/MWh for OCGT

and CCGT (EIA, 2013b; Air Resources Board, 2013). The initially installed generation capacity

and its age structure are based on CEC (2013). We derive parameters for the demand function from

CAISO (2014) and a short-term price-sensitivity of demand βt of 0.1 USD/(MWh)2 in line with a

survey provided by Lijesen (2007). In line with Kettunen et al. (2011), we assume heterogeneous

investors with symmetrically distributed discrete discount rates d with a share φ1 = 10% of investors

applying d1 = 6%, φ2 = 20% use d2 = 8%, φ3 = 40% require d3 = 10%, φ4 = 20% take d4 = 12%, and

φ5 = 10% have d5 = 14%. All investors use a risk-free rate drf of 3% for revenues from MP and FIT

given their isolation from market risk. Together, all investors can build up to 2 GW of capacity of

each technology per year except nuclear and hydro. This is consistent with empirical evidence, as

maximum annual capacity additions amounted to 3.4 GW for OCGT and CCGT combined and 1

GW for each RES technology in the period 2002 to 2012 (CEC, 2013). Only a share φinv of this is

invested, if only some investors obtain NPVs and VaRs beyond Θi and ΘV aRi .

For the uncertain capacity factor multiplier. χi ∼ N(υi, σ2i ), we assume υi = 1 ∀ i = 1, ..., rint, σi

for wind of 14.8%, and 10.6% for solar consistent with the literature (NREL, 2014; Dean, 2010).

We base these estimates on data from CEC (2013) for 2001 to 2012 and fit the capacity factor

distributions. Q-Q plots for wind and solar indicate a reasonable fit to normal distributions.

Based on the Akaike information criterion (AIC) and Chi-square statistics the normal distribution

outperforms alternative distributions for solar. For wind, the normal distribution ranks sixth (out

of 18 distributions tested) according to the AIC with only a 10% lower AIC value than the best-

18

fit distribution, and ranks second according to the Chi-square statistics. We initially generate

S = 100 scenarios to reflect the uncertain capacity factor realizations and use a target cardinality

of S = 10. RES support rates for MP and FIT are calibrated so that they induce similar levels of

RES electricity production and CO2 emissions for comparability with the RPS.

For regulatory ambiguity,. we assume that two states of the ITC exist – either it is in place or

abolished. Potential abolition times are ambiguous and investors can only define the set of possible

abolition times but not their probability distribution. E.g., it can be argued that the ITC will

expire within the next 20 years. Thus, the number of scenarios amounts to 20 and is sufficiently

small in light of 10-20 iterations per run and 100 runs per RESSS, so that no scenario generation

and reduction techniques are needed. RES support rates from the uncertainty case are maintained

for comparability.

6. RPS as most Robust Support Scheme

As described in Ritzenhofen et al. (2014) in more detail, all three RESSS increase RES instal-

lations and RES electricity generation in a deterministic setting. FIT and MP can deliver these

results at lower cost given the lower risk exposure of investors benefiting from the certainty of FIT

and MP payments, while RPS provide the lowest electricity price volatility and deliver consistent

results for varying market conditions, which lead to substantial under- or over-investment under

FIT and MP schemes. Based on these deterministic results, we expect RPS schemes to exhibit a

greater robustness vis-a-vis uncertain and ambiguous input parameters compared to MP and FIT

schemes, while the latter are likely to deliver target RES buildups at lower cost.

6.1. Robustness against Uncertainty

For all three RESSS, we report average results and their distribution based on 100 runs per

scheme under RES capacity factor uncertainty. In Table 2 we show key results across policy di-

mensions for the deterministic case, the uncertain RES capacity factor case, and the regulatory

ambiguity case. For all RESSS, total cost only changes by less than 3% between the deterministic

case and the uncertain case. Equation 15 increases RES capacity under RPS, which we match

through calibration of MP and FIT rates. This and the adjustment of decision rules in Equation

10 increase cost. On the other hand, additional RES capacity reduces the need for costly and po-

19

Dimension Scheme Unit Deterministic Uncertain RES Policycase capacity factors ambiguity

Total cost RPS Billion USD 371/− 359/1% 373/3%Total cost MP Billion USD 346/− 341/1% 352/2%Total cost FIT Billion USD 307/− 316/1% 330/3%Electricity prices RPS USD/MWh 61/− 60/1% 60/1%Electricity prices MP USD/MWh 60/− 60/2% 61/2%Electricity prices FIT USD/MWh 57/− 56/7% 59/7%Electricity price volatility RPS % 11/− 17/12% 14/17%Electricity price volatility MP % 12/− 15/29% 11/44%Electricity price volatility FIT % 34/− 36/52% 25/80%CO2 emissions RPS Million t 576/− 573/1% 570/1%CO2 emissions MP Million t 565/− 581/3% 604/4%CO2 emissions FIT Million t 580/− 578/5% 608/8%RES generation RPS TWh 88/− 90/8% 90/7%RES generation MP TWh 90/− 94/10% 80/16%RES generation FIT TWh 89/− 87/14% 68/26%

Table 2: Key results across policy dimensions (averages / relative standard deviation)

tentially idle conventional capacity for meeting RAR requirements and decreases electricity market

prices. Therefore, the overall effect is mixed and depends on the precise calibration of the model.

We now compare RESSS performance under uncertainty as illustrated in Figure 1 in the Box-

and-Whisker-Plot. The boxes illustrate the second and third quartile of results and the whiskers

show maximum and minimum values. Three results become immediately apparent: first, total cost

under the RPS scheme are higher than under MP and FIT based on median values, 5% VaR, and

5% CVaR. We report both risk metrics for consistency with measures reported earlier and since

only CVaR ensures coherence for these market outcomes, which are non-normally distributed as

opposed to the normally distributed NPV results. Second, RPS is the most robust scheme across all

policy dimensions with the smallest ranges for quartiles and whiskers. Third, variability of results

is partly skewed in particular for electricity prices and their volatility under FIT.

Robustness of RPS compared to MP and FIT can be explained by the functionality of the REC

and electricity market. The REC market ensures that a pre-determined amount of RES electricity

is produced for all capacity factor realizations. For example, increases in RES electricity production

from existing plants given high capacity factors lower the rate of new RES investment and thereby

reduce REC prices (see Equation 7). At the same time, electricity prices drop given their convexity

in capacity factors, thereby decreasing price expectations for new investment and thus increasing

REC bids for new generation (see Equation 8). On the other hand, new investment under MP

20

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

362/362 349/349 322/324

Figure 1: Aggregate results for RPS, FIT, and MP under capacity factor uncertainty

and FIT is triggered by the threshold rules in Equation 10 and therefore changes easily. As shown

in Section 3.7, revenues of RES producers are isolated from the electricity market under a FIT

regime, as all RES electricity generated is dispatched into the market. This exacerbates the effect

of convex prices and is reflected in the asymmetric distributions of average electricity prices and

their volatility. While the range is narrow for RPS, electricity prices are skewed towards the low

end and their volatility towards the high end under MP and in particular under FIT, reflecting

how increasing intermittent RES supply depresses prices and increases volatility. In Figure 2, we

illustrate the trade-offs faced by regulators when designing RES policies. For example, if regulators

aim to achieve a CO2 target at least equal to the results under the RPS scheme, they could increase

FIT rates to boost RES generation and thereby ensure that even under non-favorable capacity factor

realizations the target is met. While total cost do not increase substantially, electricity prices drop

and their volatility rises sharply compared to the initial FIT calibration.

6.2. Robustness against Ambiguity

We account for regulatory ambiguity by investigating the possible abolition of ITCs within the

next 20 years. For such settings, probability distributions are usually unknown and only ranges

of results are reported. For comparability, we combine the results for the ambiguous case into

one average value and a standard deviation assuming equal weighting of all scenarios. Without

21

Total cost of electricity*Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**%

CO2 emissions*Million t

Dimension RPS FIT FIT+2% FIT+4% FIT+6%

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD and Million t of CO2, respectively

FIT+8%

300350400450

3040506070

0%100%200%

400500600700

2060100140

362/362 322/324 322/323 322/324 323/326320/321

583/584 622/632 597/608 580/591 576/584604/615

Figure 2: Effect of moderately increasing FIT support rate

additional information on ITC abolition times, this uniform probability distribution provides for an

unbiased initial guess and therefore a valid starting point for a this analysis. We observe that total

cost increases across schemes compared to the uncertain case driven by the potential abolition of

the ITC otherwise reducing net investments (see Table 2). This cost increase is most pronounced in

the VaR/CVaR, which rises from 362/362 to 387/387 Billion USD for RPS, from 349/349 to 360/361

for MP, and from 322/324 to 347/351 for FIT. These mean values and risk criteria highlight that

before the abolition of ITCs, the impact of regulatory ambiguity on investment decisions and thereby

electricity markets is small and only affects expected electricity prices. The impact amplifies with

earlier ITC policy changes and two counteracting effects emerge: the lost ITC reduces incentives for

RES investment, while these are increased by higher electricity prices in later periods. Compared

to the uncertainty case, RES investment drops under MP and FIT as profitability thresholds are

no longer met given the higher net upfront investment, which is partly compensated by higher

electricity prices under MP. However, for the RPS scheme with its exogenously set quota total RES

generation remains constant and part of the increase in cost from the ITC abolition is compensated

by a shift to wind power, which is not affected by ITC regulation. The RPS scheme continues to be

most robust followed by MP and FIT as similar dynamics apply as for the uncertainty case. This

illustrates that our methodology can be applied equally to uncertain and ambiguous settings and

that robustness performance of RESSS is similar under both settings.

22

6.3. Policy Implications

These results suggest that there is not a single “best” RESSS. For policy makers this implies

that the choice of a RESSS has to be matched carefully to specific policy objectives. On the one

hand, MP and FIT schemes have the potential to induce RES investment at lower cost. On the

other hand, RPS lead to lower electricity market price volatility and are more robust in light of

uncertain or ambiguous parameters such as RES capacity factors or regulatory changes. Further-

more, uncertainty not only increases performance variability along all policy dimensions but market

outcomes are also skewed as shown for electricity prices and their volatility. Consequently, pol-

icy makers should consciously make the trade-offs between these respective benefits and pitfalls.

Furthermore, our results indicate that policy makers should not only consider a single policy in

isolation but account for interactions between multiple policies – e.g. between the ITC and the

RPS – and potential changes to these inducing regulatory ambiguity.

7. Conclusions and Directions for Future Research

In this work, we develop a quantitative approach to investigate the structural impact of RESSS

on electricity markets under uncertainty and ambiguity. We measure performance and robustness

of RESSS along key policy dimensions. Moreover, we introduce appropriate risk measures and

use scenario reduction techniques to facilitate the implementation of the stochastic variables. This

approach can be extended to different electricity markets, to assess alternative electricity market

policies, or to account for alternative sources of uncertainty or ambiguity. In a numerical case

study, we show that for the CA electricity market RPS deliver more robust results along all policy

dimensions compared to a hypothetical MP or FIT and explain that this might come at higher cost.

While this novel approach provides for broad insights into the dynamics of electricity markets and

related policies such as RESSS, further research is needed to investigate the impact of more flexible

MP and FIT designs and to apply this framework to further markets, which is beyond the scope

of this paper. Additionally, further research is needed to investigate the impact of alternative risk

measures, other potential sources of uncertainty and ambiguity as well as to investigate different

ways investors learn how to deal with the uncertainty and ambiguity factors.

23

Acknowledgements

The authors would like to thank the University of Chicago Booth School of Business and the

Energy Policy Institute at Chicago (EPIC) for facilitating this collaboration. Moreover, the au-

thors gratefully acknowledge the valuable input from multiple members of the two above-mentioned

institutions.

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Appendix A. Scenario Reduction

Following an approach suggested by Heitsch and Romisch (2009) and Growe-Kuska et al. (2003),we reduce the number of scenarios as follows. Let χi ∀ i = 1, ..., rint be the stochastic variables withtheir realizations over time χui,tTt=1 ∀ i = 1, ..., rint jointly forming the scenarios χz ∀ z = 1, ..., S

with S being the number of scenarios. The probability of each scenario z is $z with ∑Sz=1$z = 1and Ωz being the probability distributions of the stochastic variables χi. We use the Kantorovichdistance DK (Growe-Kuska et al., 2003) to successively combine the most similar scenarios. Wereduce the number of scenarios S by the cardinality #Z of the index set of deleted scenarios Z tothe new number of target scenarios S. For a pre-specified Z ⊂ 1, ..., S, the Kantorovich distanceis defined as

DK(Ω,Ω′) = ∑z1∈Z

$z minz2∉Z

ct(χz1 , χ′z2) with ct(χz1 , χ′z2) ∶=T

∑t=1

∑rinti=1 ∣χz1i,t − χ

′z2i,t ∣

(1 + d)t (A.1)

with χz1 with χz1z1∉Z having minimal distances to χz2i . We discount with the risk-free rate drf

to attribute greater importance to early periods and reflect NPV-driven investor behavior. Wecompute the probability $z2

update of preserved scenarios χz2 with z2 ∉ Z as

$z2update =$

z2 + ∑z1∈J

$z1 . (A.2)

Therefore, we can formulate the “optimal reduction problem” (Growe-Kuska et al., 2003, p.2)as

min

⎧⎪⎪⎨⎪⎪⎩∑z1∈Z

$z minz2∉Z

ct(χz1 , χz2) ∶ Z ⊂ 1, ..., S,#Z = S − S⎫⎪⎪⎬⎪⎪⎭. (A.3)

Appendix B. Model Verification

In these parts of the Appendix, we show how we verify and validate the model at hand. Forthis purpose, we build on well-established approaches of Gass (1983) and Sargent (2012).

We continuously verify the model and attempt to ensure that the model “runs as intended”(Gass, 1983, p.609). This is done as follows. First, we develop a modular model structure asillustrated by Figure B.3 with each box and the computations for each decision rule being a separatemodule. Second, we test these modules separately, i.e. use simplified and stylized input datato verify whether each the relationships between inputs and outputs for each module behave asexpected.

In addition, we conduct further tests of the correctness of the model mechanization as part ofthe sensitivity analyses described in Section Appendix C.

Appendix C. Model Validation

Following Gass (1983), we then test the technical, operational, and dynamic validity of themodel in order to “establish how closely the model mirrors the perceived reality” of the electricitymarkets analyzed.

For the validation we build on the different philosophy of science methods – rationalism, em-piricism, and positive economics – as suggested in (Sargent, 2012, p.17). We validate the setup,

28

Repeat Ttimes for ititerations

Cost-optimal conventional generation portfolio1

Electricity prices

Updated generation portfolio

Electricitymarket

Stoppingafter con-vergence

Initial forecast Actual iterations

REC pricesREC market

Investor decisionson capacity

Investors decisions include Plant retirements at end of useful life Divestitures given lacking profitability

based on initial forecast/prior iteration New investments given positive expected

NPV based on initial forecast/prior iteration

100 repetitions with randomly drawn capacity factor scenarios

Figure B.3: Structural setup of dynamic capacity investment model

structure, and logic of the model to address rationalism. Empiricism requires us validate our as-sumptions and all model results against real data (where available). By validating outcomes we alsoaddress positive economics, which are only concerned with correct model results. We use variousvalidation techniques suggested by Gass (1983) and Sargent (2012), which we summarize in TablesC.3 and C.4. Details on these validation items are provided in the following Sections.

29

Type of technical validation Aspects Description

– Model validation Objectives of model – Modeling objectives explicitly outlined in Sections 1 and 2– Assessment of affordability, security of supply, sustainabilityof electricity supply, and robustness under RESSS

Appropriateness of structure – Choice of model motivated by literature review (see Section 2) and modelingobjectives– Documentation of structure illustrated in Figure B.3

Structural limitations – Limitations and assumptions discussed in Sections 3, 4, and 7– Further limitations (only implicitly mentioned via assumptions): Abstractionfrom transmission constraints given spatial aggregation of our analysis; no modelingof reserve or balancing power markets given their limited size; no incorporationof other exogenous shocks like economic crises beyond those explicitly modeled

Documentation of model – Extensive documentation in Sections 3 and 4 and in Ritzenhofen et al. (2014)– Logical/mathem. validation Model logic – Clear model logic and structure (see Figure B.3)

– Clear reflection electricity market (see Figure C.4)Critical points – Critical assumptions highlighted: rational and profit-oriented behavior;

heterogeneous investors attributing no risk to MP and FIT payments; constructionof representative plants per technology; constant climatic conditions, i.e. stationaryuncertainty factor distributions

Replication of results – Ability to replicate results in multiple model runs (deviationof average total cost of < 0.5% between sample runs)– Model available upon request

– Data validation Input data documentation – Extensive input data documentation in this work and in Ritzenhofen et al. (2014)Empirical data comparison – Use of data from industry reports and actual market outcomes

Table C.3: Technical model validity

30

Type of operational validation Aspects Description

– Results validation Face validity – Discussion of results with researchers, policy makers,and practitioners from utilities, RES project developers,and equipment manufacturers

Hypothesis tests – Formulation and comparison to hypotheses (e.g., see Section 6)Operational graphics – Monitoring of results through detailed non-aggregated output

graphs (see Section Appendix E)– Experimental validation Sensitivity analysis – Sensitivity analyses in Ritzenhofen et al. (2014)

– Additional sensitivity analyses on keyparameters (see Figure 2 and Section Appendix D)

– Comparative validation Comparison to other studies – RPS case with similar results as in otherstudies such as CPUC (2013); first, mainly addition of wind power,then increasingly solar power; reduction in geothermal and biomassover time– No direct comparison available for FIT and MP as alternativestudies for CA mostly model the existing RPS scheme– Electricity price progression in line with EIA (2014a)

– Policy impact validation Comparison against public debate – Similar discussions in journals and magazineslike The Economist (2013, 2014) elaborating onthe various challenges of electricity markets– Key aspects of this work such as market integration explicitlyhighlighted in recent electricity market reforms forexample in Germany (Agora Energiewende, 2014)

Table C.4: Operational model validity

31

1 Renewable energy certificates; 2 Utilities, transmission and distribution operators or others depending on local regulation; 3 Households, industry, and others; sometimes these are also the obliged entities depending on regulation

Regulator

Renewable generators

Electricity market REC1 market

Conventional generators

E $

E

$

REC $

REC$E $

RECData

RECQuota

E $ $

Retailers / obliged entities2

Electricity consumers3

Market Endogenous player Exogenous player

Regulator (ex ante announcement of

all annual quotas)

Renewable generators

(annual invest & hourly dispatch per plant &

technology)

Electricity market (hourly dispatch)

REC1 market (annual clearance)

Conventional generators

(annual invest & hourly dispatch per plant &

technology)

Retailers / obliged entities2

(hourly aggregate demand curve)

Electricity consumers3

(not simulated explicitly)

Figure C.4: Reflection of electricity market in model

Appendix D. Sensitivity analyses

32

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

365/365 349/349 326/328

Figure D.5: Results with standard deviation for capacity factors increased by 5 percentage points

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

363/364 349/350 323/325

Figure D.6: Results with 50 instead of 100 runs – set of runs 1

33

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

362/362 346/346 322/324

Figure D.7: Results with 50 instead of 100 runs – set of runs 2

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

363/363 348/349 320/322

Figure D.8: Results with 75 instead of 100 runs – set of runs 1

34

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

362/362 347/348 321/323

Figure D.9: Results with 75 instead of 100 runs – set of runs 2

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

362/362 348/349 322/324

Figure D.10: Results with 100 runs for result replication – set of runs 1

35

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

363/363 348/349 320/322

Figure D.11: Results with 100 runs for result replication – set of runs 2

Total cost of electricity*

Billion USD

Electricity prices**USD/MWh

RES gene-ration year 20TWh

Electricity price volatility**

%

CO2 emissions*Million t

Dimension RPS MP FIT

* Sum over 20 years; ** Average over 20 years VaR/CVaR in Billion USD

300350400

305070

0%100%200%

400500600700

2060100140

362/363 342/344 323/325

Figure D.12: Results with 20 remaining scenarios instead of 10

36

0

50

100

150

200

0 5 10 15 20U

SD/M

Wh

Iteration

Average electricity price

RPS MP FIT

020406080

100

0 5 10 15 20

GW

Iteration

Average installed total capacity

RPS MP FIT

Figure E.13: Electricity price behavior for sample run under RES capacity factor uncertainty

Hydro Nuclear CCGT OCGTBiomass Geothermal Wind onshore Solar PV

0

20

40

60

80

100

0 2 4 6 8 10 12 14 16 18 20

GW

RPS

0

20

40

60

80

100

0 2 4 6 8 10 12 14 16 18 20

GW

MP

0

20

40

60

80

100

0 2 4 6 8 10 12 14 16 18 20G

W

FIT

Figure E.14: Installed capacity for sample run under RES capacity factor uncertainty

Appendix E. Operational Graphics

37

Hydro Nuclear CCGT OCGTBiomass Geothermal Wind onshore Solar PV

0

50

100

150

200

250

1 3 5 7 9 11 13 15 17 19

TW

h

RPS

0

50

100

150

200

250

1 3 5 7 9 11 13 15 17 19

TW

h

MP

0

50

100

150

200

250

1 3 5 7 9 11 13 15 17 19

TW

h

FIT

Figure E.15: Electricity generation for sample run under RES capacity factor uncertainty

98%

99%

100%

101%

102%

0 2 4 6 8 10 12 14 16 18 20

Fullfillment of RAR

RPS MP FIT

Figure E.16: RAR fulfillment for sample run under RES capacity factor uncertainty

0

50

100

150

200

0 5 10 15 20

USD

/MW

h

Iteration

Average electricity price

RPS MP FIT

020406080

100

0 5 10 15 20

GW

Iteration

Average installed total capacity

RPS MP FIT

Figure E.17: Convergence of results for sample run under RES capacity factor uncertainty

38