The aim of hydroelectric plants is to convert the potential energy of stored water into elec-tricity. They can be reservoir, run-of-the-river or pumped-storage hydroelectric plants. A
pumping plant usually has two separate reservoirs located at two different levels, an up-per and a lower one and its main characteristic is that its machinery can be used either for pumping water from the lower to the higher reservoir consuming electricity, or for releas-ing water from the upper reservoir through the turbine generatreleas-ing electricity. Reversible turbine units allow the pumping and the generation using the same turbine and the same pipe joining the higher and the lower reservoir. Generation companies use this type of electric energy storage for load balancing. During periods of low demand (and low market price), for example at night, they pump the water to the upper reservoir by using electricity (at a low cost). During periods of high electricity demand (and high market price), power is generated by releasing the stored water through turbines in the same way as conventional hydropower station.
In Bloom and Gallant [11] is described and demonstrated the procedure for modeling pumped-storage systems and conventional units in a single medium-term period of gen-eration planning with pumped storage and conventional units while matching the LDC through the Bloom and Gallant formulation.
This procedure was extended in Marí and Nabona [51] to a multiperiod stochastic problem with N-DRs, which involves matching three different LDCs in each period (see §5.3), and it will be formulated here for problems (with no SPV generation) where a single LDC is to be matched in each period. It requires the increase of the LDC load of each period by the capacity cP of the pumped-storage unit, the use of two extra units of capacity cP, zero failure probability and zero cost: a pseudo-unit, not receiving revenue from the market, for compensating the extra load, with expected generation xνCmp and the other with expected generation xνPhydfor hydro generation from pumped water, and the consideration of a vari-able vνP with the pumped energy stored in the upper reservoir with 0 6 vνP 6 vP∀ν with vP being the energy capacity of the pumping-station upper reservoir. The energy spent in pumping is transformed into stored energy through an efficiency coefficient effP < 1 (effP= 0.75 in our data).
The total extra energy cPTi(ν) in the LDC of node ν will be split in three parts, each of which produced or consumed in non-overlapping hours within the period length Ti(ν): the energy of the compensating unit xνCmp, the hydro energy generated in the pumping station xPhyd, and the pumping by the other units. This pumped energy is
cPTi(ν)− xνCmp− xνPhyd> 0 ∀ ν ∈ N . (35)
In Figure 13, the left picture represents the LDC for a month, on the right picture it can be seen the same LDC but with the capacity cP increased. The green slice on the left picture represents the total extra energy added to the LDC and it will be split as it is shown in the right picture. The order in which these generations are situated within the extra load is relevant. GenCos will pump water to the upper reservoir in the base-load power hours because they have a low market price, while in the peak load or peak market price hours,
they will release hydro from pumped water. The compensating unit will be in the mid-dle of these two other generations in order to satisfy the extra load in hours with neither pumping nor generation from pumped hydro.
p
Figure 13: Representation of the LDC without pumping storage for one period (left), the LDC extended with the extra pumping load (right) and the linear market-price (dotted line) The stored energy in the upper reservoir could only be a fraction of the pumped energy, effP(cPTi(ν)− xνCmp− xνPhyd). Thus, assuming that the initial, and final, energy stored in the upper reservoir is v0P, the energy stored in the pumping-station upper reservoir must satisfy respectively for the root node, numbered one, for nodes ν not being the root nor the leaf nodes, and for the leaf nodes λ ∈L:
v1P=effP cPT1− x1Cmp− x1Phyd − x1Phyd+ v0P,
vνP=effP cPTi(ν)− xνCmp− xνPhyd − xνPhyd+ vpred(ν)P ∀ν > 1 and ν 6∈L ,
effP cPTi(λ)− xλCmp− xλPhyd − xλPhyd+ vpred(λ)P = v0P ∀λ ∈L ,
(36)
where pred(ν) is the predecessor node of node ν in the scenario tree, which is in the for-mer period of that of ν.
Both (35) and (36) are included in the model as non-LMCs.
3.4.1 The pumping load fee
Even though the pumped-storage scheme means a net energy consumption overall, the pumped-storage operation should produce a profit increase provided that both the
satis-conventional units and the hydro generation that is produced from it are remunerated at market price. It is assumed that the extra load (35) is also paid at market price, thus requir-ing that this fee be deducted from the profit function to maximize.
The pumping extra load fee, to be deducted from the profit objective function of each node νin problems considering medium-term pumping, can be calculated as:
cP ZTi(ν)
xνCmp+xνPhyd cP
bν+ li(ν)tdt = bν cPTi(ν)− xνCmp− xνPhyd +l2ci(ν)
P
cPTi(ν)2
− xνCmp+ xνPhyd2 ,
(37)
where bν is as in (40). The integration limits in (37) have been taken so that the pumping load-duration segment corresponds to the lowest market prices.
The model description above is for a single pumping-scheme in the system. It is easy to extend the model to several different pumping-schemes in the system, each having a different set of generation units devoted to pumping for each different scheme.