Stochastic dynamic programming approach to managing power system uncertainty with distributed storage

Abstract

Wind integration in power grids is challenging because of the uncertain nature of wind speed. Forecasting errors may have costly consequences. Indeed, power might be purchased at highest prices to meet the load, and in case of surplus, power may be wasted. Energy storage may provide some recourse against the uncertainty of wind generation. Because of their sequential nature, in theory, power scheduling problems may be solved via stochastic dynamic programming. However, this scheme is limited to small networks by the so-called curse of dimensionality. This paper analyzes the management of a network composed of conventional power units and wind turbines through approximate dynamic programming, more precisely stochastic dual dynamic programming. A general power network model with ramping constraints on the conventional generators is considered. The approximate method is tested on several networks of different sizes. The numerical experiments also include comparisons with classical dynamic programming on a small network. The results show that the combination of approximation techniques enables to solve the problem in reasonable time.

Appendix

1.1 Nomenclature

1.1.1 Sets

N :

Buses

L :

Transmission lines

M :

Wind farms

G :

Conventional generators

$G_n$ :

Conventional generators at bus n

S :

Storage devices

$O_n$ :

Transmission lines that leave node n

$I_n$ :

Transmission lines that enter node n

$\mathbf{R}$ :

Real numbers

$\mathbf{R}^+$ :

Non negative numbers

1.1.2 Parameters

T :

Length of the planning horizon

t :

Time period

${\hat{D}}_{nt}$ :

Forecast of the load, in period t at bus n

${\underline{p}}_g$ :

(resp. ${\bar{p}}_g$) Lower (resp. upper) bound on generator g output

${\underline{\lambda }}_g$ :

(resp. ${\bar{\lambda }}_g$) Ramp down (resp. ramp up) limit on generator g

${\tilde{W}}_t$ :

Random vector of outputs from the wind turbines in period t

$w_{mt}$ :

A particular realization of the stochastic process $\{{\tilde{W}}_{mt}\}$

${\bar{e}}_l$ :

Bound on power trough line l

$B_l$ :

Susceptance of line l

${\underline{s}}_n$ :

(resp. ${\bar{s}}_n$) Lower (resp. upper) bound on the level of the storage at bus n

$c_n$ :

(resp. $d_n$) Efficiency coefficient of charging (resp. discharging) of the storage device at bus n

$\delta _{n}=$ :

$\left\{ \begin{array}{ll} 1 &{} \hbox {if there is a storage facility at node n},\\ 0 &{} \hbox {otherwise}. \end{array} \right.$

1.1.3 Decision variables

$p_{gt}$ :

Power generation from generator g in period t

$e_{lt}$ :

Power flowing through transmission line l in period t

$\theta _{nt}$ :

Phase angle of bus n in period t

$s_{nt}$ :

Level of stored energy at bus n in the beginning of period t

$\Delta ^+_{nt}$ :

(resp. $\Delta ^-_{nt}$) Positive (resp. negative) variation in the level of charge from the beginning through the end of period t

$\kappa ^+_{nt}$ :

(resp. $\kappa ^-_{nt}$) Power excess absorbed (resp. delivered) by the storage unit located at bus n

$\Delta _{nt}$ :

Power absorbed or delivered by the storage unit at bus n

1.1.4 Functions

$CP_{gt} (p_{gt})$ :

Cost function of generator g in period t

$CS_{nt}(\Delta _{nt})$ :

Cost associated with varying the stored energy at bus n

1.1.5 Operators

${\mathbb {E}}$ :

Mathematical expectation

|X|:

Cardinality of the set X