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Assessment of risk-averse policies for the long-term hydrothermal scheduling problem

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Abstract

The goal of the long-term hydrothermal scheduling (LTHS) problem is to determine an optimal policy that minimises the expected operational cost over a multi-annual planning horizon. In this problem, inflows are naturally random; thus, the development of policies requires the use of specialised techniques such as stochastic optimization algorithms. Although the optimal policy defined by means of the expected cost is economically efficient, the associated decisions can incur a high risk of energy deficit. To overcome such disadvantages, one alternative is to include a risk-measure approach in the LTHS modeling. In this study, we compared two different risk-measure strategies applied to the LTHS problem: (i) a convex combination between the conditional value at risk (CVaR) and the expected operational cost, (ii) a reservoir risk-curve. To accomplish the comparison, we consider the Brazilian power system with a 5-year planning horizon discretized in monthly stages. The policies supplied by the risk-averse strategies are simulated in a set of synthetic inflow scenarios. Thus, the results are obtained in terms of the expected long-term hydrothermal decisions, such as the total operational cost and the reservoir storage. In summary, the CVaR-based policies provided better operational results than the risk-curve one.

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Abbreviations

DP:

Dynamic programming

LP:

Linear programming

SP:

Stochastic programming

IID:

Independent and identically distributed

ISO:

Independent electric power system operator

EER:

Energy equivalent reservoirs

VaR:

Value at Risk

CAR:

Risk-averse curve (Brazilian case)

CVaR:

Conditional value at risk

LTHS:

Long-term hydrothermal scheduling

SDDP:

Stochastic dual dynamic programming

K :

Counter k = 1,...,nk

T :

Index of monthly stages over the planning horizon, \(t=\) 1,...,nt

\(R_{t}\) :

Set of realizations for the stochastic variable representation in stage t, \(R_{t}\) = {r:\(r\in \) Z \(_{+}\),1\(\le r\le \) nr \(_{t}\)}

\(C_{t}\) :

Set of approximations, at stage t,  for the recursive function of stage t+1, \(C_{t}\) = {c:\(c\in \) Z \(_{+}\),1\(\le c\le \) nc \(_{t}\)}

\(\alpha _{t}\) :

Percentile under risk of Conditional Value at Risk (CVaR) at stage t, \(\alpha _{t} \in \)[0,...,1]

\(\lambda _{t}\) :

Risk-aversion level in stage t, \(\lambda _{t} \in \)[0,...,1[

cf:

Incremental operational cost of thermal fuel plant (R$\(^{1}\)/MWh)

cv:

Cost of violation of minimum target reservoir storage level (R$/ hm\(^{3})\)

pb\(_{t,r}\) :

Probability associated with realization r in stage t

pl\(_{t}\) :

Electricity power demand at stage t (MW)

pr:

Productivity coefficient of hydro plant (MW/m\(^{3}\)/s)

vc\(_{t}\) :

Volume of risk-curve defined for the reservoir of hydro in stage t (hm\(^{3})\)

vz:

Coefficient to convert constant water flow (m\(^{3}\)/s) to monthly volume (hm\(^{3})\)

ZT \(_{t,r}\) :

Monthly risk-neutral cost function (R$)

ZV \(_{t,r}\) :

Monthly cost function with reservoir violation cost (R$)

ZR \(_{t,r}\) :

Monthly cost function with coherent risk-measure based on CVaR (R$)

ZF \(_{t}\) :

Recursive cost-to-go function in t (R$)

ur \(_{t}\) :

Value at Risk (VaR) of ZT \(_{t,r}\) : \(r\in R_{t}\)

vh \(_{t-1}\) :

Hydro plant initial volume in stage t (hm\(^{3})\)

yh \(_{r,t}\) :

Monthly reservoir inflow associated with the realization r (m\(^{3}\)/s)

av \(_{t }\) :

Auxiliary variable

pd \(_{t}\) :

Electric system power shortage (MW)

pt \(_{t}\) :

Thermal plant power generation (MW)

qh \(_{t}\) :

Hydro plant turbined outflow (m\(^{3}\)/s)

sh \(_{t}\) :

Hydro plant spillage (m\(^{3}\)/s)

vr \(_{t}\) :

Risk-curve violation in reservoir (hm\(^{3})\)

ur \(_{t+1}\) :

Value at Risk of ZT \(_{t+1,r}\) : \(r \in R_{t+1}\)

vh \(_{t}\) :

Hydro plant final volume at stage t (hm\(^{3})\)

zf \(^{LB}_{t }\) :

Lower bound approximation of ZF \(_{t}\)

pt:

Thermal plant power generation upper bound (MW)

qh:

Hydro plant turbined outflow upper bound (m\(^{3}\)/s)

vh:

Hydro plant reservoir storage upper bound (hm\(^{3})\)

zt \(^{UB}_{r,t}\) :

Monthly operational cost associated with realization r (R$)

zv \(^{UB}_{r,t}\) :

Monthly operational cost plus reservoir violation associated with r (R$)

zr \(^{UB}_{r,t}\) :

Monthly coherent risk-measure based on operational cost associated with r (R$)

\(\psi \)ur\(_{r,t}\) :

Lagrange Multiplier of (21) obtained from LP in stage t associated with realization r

\(\psi \)vh\(_{r,t }\) :

Lagrange Multiplier of (7) obtained from LP in stage t associated with realization r

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Authors and Affiliations

  1. Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil

    Paulo Vitor Larroyd & Erlon Cristian Finardi

  2. Plan4, Florianópolis, Santa Catarina, Brazil

    Vitor Luiz de Matos

Authors
  1. Paulo Vitor Larroyd
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  2. Vitor Luiz de Matos
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  3. Erlon Cristian Finardi
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Correspondence to Paulo Vitor Larroyd.

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Larroyd, P.V., de Matos, V.L. & Finardi, E.C. Assessment of risk-averse policies for the long-term hydrothermal scheduling problem. Energy Syst 8, 103–125 (2017). https://doi.org/10.1007/s12667-016-0191-y

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