Optimal subsidies for renewables and storage capacities

Transcription

1 Optimal subsidies for renewables and storage capacities Mathias Mier 1 Carsten Helm 2 1 ifo Institute 2 University of Oldenburg IEW Gothenburg, 19 June 2018

2 Electricity is dicult... but we solved the problem... by creating a new problem! Technological constraints solved by perfect old world Climate change is the biggest market failure the world has ever seen. Nicholas Stern Climate change (carbon externality) Policy aim: 100% renewables Not feasible (Heard et al. 2017) due to reliability of intermittent renewables like wind and solar Feasible (Brown et al. 2018) with storage

3 In a nutshell Real world observations Fossils cause pollution but carbon prices are (too) low Renewable energies are (highly) subsidized but storage not Carbon externality is (mainly) used to motivate this (other: induced technological change) Setup Carbon tax is political not feasible but subsidizing investments in renewables and storage capacity No dynamic investment issues (no induced technological change) to focus on the carbon externality No uncertainty, no periodic load, no ramping issues Intermittency, dynamic control storage problem At which point a regulator should stop promoting renewables and start subsidizing storage? 1 Renewables must be subsidized 2 Storage must be taxed if renewables capacity is small

4 Literature Pollution control (e.g. Baumol and Oates 1975, Polinsky 1979 Can, Requate and Unold 2003 EER, Bläsi and Requate 2010 PFM)... Without dynamic investment issues a (Pigouvian) carbon tax is ecient (1st best) Renewable support policies (e.g. Fabra and Reguant 2014 AER, Liski and Vehvilänen 2016 WP)... So far we know no literature about storage support policies (partly Ambec and Crampes 2017 WP) Electricity markets with renewables (e.g. Joskow 2011 AER PP, Borenstein 2012 JEPer, Ambec and Crampes 2012 REE, Helm and Mier 2018 WP) Storage (e.g. Gravelle 1976 EJ, Crampes and Moreaux 2010 EE, Steen and Weber 2013 EE)

5 Model Three technologies, j = r,f,s 1 Renewables r 2 Fossils f 3 Storage s Two policy instruments Carbon tax τ 0 (1st best) Capacity subsidy σ j (2nd best) Three stage game (solved by backward induction) 1 Subsidy choices of a regulator σ r,σ s 2 Capacity choices of perfectly competitive rms q j 3 Production y j and demand x choices of rms and consumers for each t [0,T ] Inverse demand is (weakly) decreasing, p x 0 Firms take total production, total capacity and prices as given

6 Stage 3 - Production choices 1. Renewables r y r (t) α (t)q r, where α (t) [0, 1] (intermittency) Marginal production costs of 0 T π ry := max p (t)y r (t)dt y r (t) 0 2. Fossils f y f (t) q f (fully dispatchable) Damage of δ T 0 y f (t)dt (carbon externality) Marginal production costs of b f T π fy := max (p (t) b f )y f (t)dt y f (t) 0

7 Stage 3 - Production and demand choices 3. Storage s with conversion losses η (t) (optimal control problem) η (t) = η s (0, 1] at times of storage (y s < 0) η (t) = η d 1 at times of destorage (y s > 0) S (0) = S (T ), initial and terminal condition 0 S q s, level constraint T π sy := max p (t)y s (t)dt such that y s (t) 0 ds = η (t)y s (t) dt Consumer surplus maximization w := max x(t) T 0 WTP max p(t) x ( p)d pdt

8 Stage 3 - Availability of renewables Representative cycle from 0 to T (solar night-day or wind seasonality) α (t) is (weakly) monotonically increasing up to α peak and then (weakly) monotonically decreasing

9 Stage 3 - Explaining storage level 1 Destorage period for low α until the storage is empty 2 Intermediate period (no storage) 3 Storage period for high α until the storage is relled

10 Stage 3 - Explaining prices 1 Destorage period with constant p d 2 Intermediate period (no storage) with p s p p d 3 Storage period with constant p s < p d

11 Stage 2 - Capacity choices Prot maximization with capacity costs and subsidies π r (σ) := maxπ ry (c r (Q r ) σ r )q r, q r π f (σ) := maxπ fy c f (Q f )q f, q f π s (σ) := maxπ sy (c s (Q s ) σ s )q s. q s Total capacity Q j = n j q j Capacity costs c j (Q j )q j with c j(q j)q j q j Subsidies σ j q j for j = r,s = c j (Q j ) and c f Q f 0

12 Stage 1 - Subsidy choices Welfare = consumer surplus + prots damage costs subsidy costs T maxw (σ) = w + π (σ) δ σ r,σ s 0 y f (t)dt σ j Q j j w, value function of consumer surplus maximization in stage 3 π (σ) := j n j π j (σ), value function of producer surplus maximization in stage 2 Applying envelope theorem yielding FOCs for j = r,s Q r Q T s σ r + σ s = δ σ j σ j 0 y f (t) dt σ j

13 Remember - Explaining prices 1 Destorage period with constant p d 2 Intermediate period (no storage) with p s p p d 3 Storage period with constant p s < p d Notation: d (destorage), s (storage), 1 (case 1), 2 (case 2) Obvious solution is the only solution if Q r σ r Q s σ s + Q r σ s Q s σ r 0

14 Really, subsidizing renewables but taxing storage? Very low renewables, i.e. Q r + Q f < x (b f ) Renewables and fossil fully used during destorage, storage and in case 1 0 = p(t) d,1,s Q δ r dt p(t) d,1,s Q dt c σ r Q r f σ j f Q f p(t) d Q + δ s dt + p(t) s Q s dt p(t) d,1,s Q dt c σ s Q s f σ j f Q f

15 Still, storage subsidies might be ecient! Low renewables, i.e. Q r < x (b f ) and Q r + Q f > x (b f ) Fossil not fully used in case 2 and during storage 0 = p(t) σ r δ d,1 Q dt r dt d,1 p(t) d,1 Q dt c + α (t)dt + dt (a hq r ) Q r f 2 s Q r σ j f Q f p(t) + σ s δ d Q dt s dt d,1 p(t) d,1 Q dt c + dt (a hq r ) Q s f s Q s σ j f Q f

16 Subsidizing storage if renewable capacity is high! Medium and high renewables, i.e. Q r > x (b f ) Renewables fully used but fossils not at all in case 3 and during storage 0 = σ r δ dt d,1 + σ s δ dt d,1 p(t) d,1 Q r dt p(t) Q dt c + α (t)dt Q r f 2 σ j f Q f p(t) d Q s dt Q s p(t) Q dt c f σ j f Q f d,1 d,1

17 Simplied double linear model Inverse demand is linear, i.e. p (t) = A x(t) γ Renewable availability follows a symmetric triangle from 0 to T = 2 with α (t) = t for all t 1 and α (t) = 2 t for all t > 1

18 Main results Starting intuition: 2nd best should be a subsidy for renewables and a subsidy for storage since storage makes renewables more competitive But we found that total renewables capacity matters if Q r + Q f < x (b f ) (very low), then σ r > 0 but σ s < 0 if Q r < x (b f ) and Q r + Q f > x (b f ) (low), then σ r > 0 but σ s 0 if Q r > x (b f ) (medium/high), then σ r,σ s > 0 (starting intuition) Simplied double linear model if Q r < x (b f ) (very low/low), then σ r = δ = δ ( ) T α (t)dt and σ 1 0 r = δ η s 1 η d if Q r > x (b f )(medium/high), then σ r = δt 2 2 = δ d,1,2 α (t)dt and σ s = δ 1 η d

19 Concluding remarks General model: Marginal subsidies must be equal to marginal damages Renewables must be subsidized Storage must be taxed for very low renewables, inconclusive results for low but subsidized for medium/high Simplied model: Technologies must be subsidized in high of the mitigated pollution damage Conversion losses of storage are dirty if (very) low renewables Storage is clean if only renewables are used during the storage period (medium/high renewables) (Other: No subsidy-tax-symmetry) Restrictions Monotonically increasing/decreasing α (t) to abstract from multiple destorage/storage cycles No technological change (but similar eects regarding the carbon externality: renewables subsidies should fall in Q r )

20 Outlook - German power sector in May 2018 Left axis: demand (black), solar production (yellow), wind production (green); right axis: day-ahead price (red) Maybe we should start subsidizing storage capacities soon... at least if the carbon externality is our motivation only!

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