"Pum ping W ater to Com pete in Electricity Markets" C. - PowerPoint PPT Presentation

"Pum ping W ater to Com pete in Electricity Markets" C. Cram pes and M. Moreaux Energy Centre Workshop, February 2007

1) Introduction Energy Centre Workshop, February 2007 2

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2) Model setting * Steady state with two elementary periods t = 1, 2 * Two plants: > ≥ . ' '' thermal plant with cost ( ) c q , c 0, c 0 + ≤ + H H hydro plant constrained by pumped water q q f f 1 2 1 2 α pumping technology: f thermal kWh are necessary to pump the quantity of water α > ). f that will be transformed into f hydro kWh (where 1 + T H u q ( q ) * Final consumers: strictly concave t t t * We successively consider efficient mix first best monopoly open-loop Cournot game Energy Centre Workshop, February 2007 6

3. Efficient production schemes When the management of the two plants is integrated, the operator chooses a mix that solves + α + + α T T min c q ( f ) c q ( f ) 1 1 2 2 = H T ( q , q , f , t 1,2) t t t + ≤ + μ H H q q f f s.t. 1 2 1 2 + ≥ = γ H T q q q t 1,2 t t t t ν ≥ ≥ ≥ = H T q 0 , q 0 , f 0 t 1 , 2 t t t t Energy Centre Workshop, February 2007 7

* Lemma 1 q > Assume that 0 . Then cost minimization implies: t + = + = + T H H H q q q and q q f f ; t t t 1 2 1 2 × = H f q 0 . t t * Implication: electricity from the hydro-producer at any period t is actually coming from t − . water pumped during period 1 And since thermal marginal cost is increasing … Energy Centre Workshop, February 2007 8

* Lemma 2 ∀ t and ' = ≠ ≥ > implies that = . H q 0 t ( , ' t t 1,2 and t t ' ), q q 0 t t ' t ' * Implication: = = = = = H H q q q q 0 f f 0. if , then and consequently t t ' t t ' t t ' Actually, the hydro-system is a storage device or transfer device of the energy produced at one period to the next at a cost represented by the fraction of the energy lost during the transfer. Energy Centre Workshop, February 2007 9

Hence, it must be used if and only if, without transfer, the marginal cost differential q exclusively during period t is higher than resulting from the production of quantity t the marginal loss implied by the transfer. Lemma 3 ≥ > = ≠ ≤ α q q 0 , ( , ' t t 1,2 and t t ' c q '( ) c q '( ) Suppose ). Then is a necessary t t ' t t ' = = f f 0. and sufficient condition for t t ' Energy Centre Workshop, February 2007 10

( ) q q q 2 2 m 1 "hydro zone" period 2 is peak > q q 2 1 Let ( ) "no-hydro zone" q q be 2 m 1 the solution to = α c q '( ) c q '( ) 2 1 Example: if = + c q '( ) c cq 45 ° 0 q 1 c ( ) ( ) = α − + α 0 q q 1 q 2 m 1 1 c Energy Centre Workshop, February 2007 11

"no-hydro zone" q 2 "hydro zone" α = 1 c ( ) ( ) = α − + α 0 1 q q q 2 m 1 1 c q 1 q 2 "no-hydro zone" α = ∞ q 1 Energy Centre Workshop, February 2007 12

4. Integrated management * First best: ∑ + − + α T H T max u q ( q ) c q ( f ) t t t t t = H T = t 1,2 ( q , q , f , t 1,2) t t t + ≤ + ≥ ≥ ≥ = H H H T s.t. q q f f , q 0 , q 0 , f 0 t 1 , 2 t t t 1 2 1 2 = = = H As t 2 is the peak period, f q 0 in any case. 2 1 Proposition 1: First best dispatch is: = = = q = = ' u ' u H T u u q ( ) c q ( ) either all-thermal: f q 0 and q defined by , t 1 , 2 1 2 t ti t ti ti < α ' u ' u if u ( q ) c ( q ) . 2 2 i 1 i = + α ' T ' T H or a mixed solution: u ( q ) c ( q q ) and 1 1 1 2 α +α = + = T H ' T H ' T c ' ( q q ) u ( q q ) c ( q ) otherwise. 1 2 2 2 2 2 Energy Centre Workshop, February 2007 13

' α u €/kWh < α c ' ' u ' u u ( q ) c ( q ) 2 2 2 i 1 i NO HYDRO ' u 1 c ' quantities u q u q 2 i 1 i Energy Centre Workshop, February 2007 14

α c ' ' u > α €/kWh ' u ' u ( ) ( ) u q c q 2 2 2 i 1 i welfare loss THERMAL + HYDRO welfare gain c ' ' u 1 + T T H q q q quantities 2 2 2 + α T T u q q f q u q 1 1 1 2 i 1 i Energy Centre Workshop, February 2007 15

* Private monopoly: Standard result as regards outputs: under-provision of energy Concerning the energy mix, all cases are possible "hydro zone" q 2 1 4 "no-hydro zone" 3 2 q 1 Energy Centre Workshop, February 2007 16

5. Cournot competition control sets: � T decides on T T q and q 1 2 � H decides on H H q and q 1 2 � Who decides on f and f ? 1 2 alternative institutional arrangements concerning f and f : 1 2 � T decides on its total output � H is an eligible consumer: it decides on inflows � f and f are strategic variables chosen by the social planner: TBC 1 2 Energy Centre Workshop, February 2007 17

5.1. The thermal firm controls its total output: def = + α T T Let y q f t t t The thermal firm solves ( ) ( ) ∑ + − α − H T T T max p q y f y c y t t t t t t ( ) = T T t 1,2 y , y 1 2 The hydro firm solves ( )( ) ∑ + − α − α H T H max p q y f q f t t t t t t = H = t 1,2 ( q , f t 1,2) t t + ≤ + μ H H s.t. q q f f 1 2 1 2 ≥ ≥ = ν H 0 , 0 1 , 2 q f t t t Energy Centre Workshop, February 2007 18

• hydro firm’s FOCs: + − α + ν − μ = = H H ' H q : p ( q f ) p 0 t 1 , 2 t t t t t t − α − α − α + μ + ν = = H ' f : ( ) 0 1 , 2 f p q f p t t t t t t t > and f > would require H Having both q 0 0 t t ( ) + − α = μ = α + − α H ' H ' ( ) ( ) p q f p p q f p t t t t t t t t μ = or α = that is 0 1 > and f > cannot be simultaneously true. H Therefore, in this institutional setting, q 0 0 t t Analysis of the Cournot equilibrium: TBC Energy Centre Workshop, February 2007 19

5.2. The hydro producer is an eligible consumer: The thermal firm solves ( )( ) ( ) ∑ + + α − + α H T T T max p q q q f c q f t t t t t t t ( ) = T T t 1,2 q , q 1 2 The hydro firm solves ( )( ) ∑ + − α H T H max p q q q f t t t t t = H = t 1,2 ( q , f t 1,2) t t + ≤ + μ H H s.t. q q f f 1 2 1 2 ≥ ≥ = ν H 0 , 0 1 , 2 q f t t t Energy Centre Workshop, February 2007 20

Two types of equilibrium when H is active most likely: = = > = H H q 0, f q 0, f 0 1 1 2 2 but we cannot exclude > > > = H H 0, 0, 0, 0 q f q f 1 1 2 2 Energy Centre Workshop, February 2007 21

• hydro firm’s FOCs: + − α + ν − μ = = H H ' H q : p ( q f ) p 0 t 1 , 2 t t t t t t − α + μ + ν = = f : 0 1 , 2 f p t t t t def • Note the difference with the case where T controls = + α T T y q f t t t − α − α − α + μ + ν = H ' f f : p ( q f ) p 0 t t t t t t Energy Centre Workshop, February 2007 22

> > > = H H 0, 0, 0, 0 q f q f 1 1 2 2 • Having both > and f > requires H q 0 0 t t + − α = μ = α H ' p ( q f ) p p t t t t t ( ) − α = α − < α H ' H or ( q f ) p 1 p , which means q f t t t t t t • Not possible at the peak period but it can be the solution at the off-peak period: ( ) > when f > requires + − α > μ = α H ' q 0 0 p 0 f p p 1 1 1 1 1 1 ( ) ( ) ( ) − α > α − H ' T T or q p q 1 p q 2 1 1 1 1 Energy Centre Workshop, February 2007 23

• actually > helps to maintain the cost of water α H q 0 p low; 1 1 • Most likely when H q is expected to be large 2 α is close to 1 ' p is large 1 Energy Centre Workshop, February 2007 24

= = > = H H q 0, f q 0, f 0 1 1 2 2 H illustration: choice of the thermal unit given q 2 peak period off peak period residual total demand demand c ' ( q ) 1 c ' ( q ) 2 q q 2 1 α H H q 2 q 2 Energy Centre Workshop, February 2007 25

Conclusions and extensions • key role of hydro production units thanks to flexibility • pump storage, a way to store electricity: costly but efficient under some circumstances • inefficient use of pump storage under competition • extensions o hydro capacity constraints o restrictions to thermal flexibility (ramping rates, warming-up) o more competition with several firms, each controlling both technologies o random natural inflows o … Energy Centre Workshop, February 2007 26