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Valuation of Energy Storage: An Optimal Switching
Approach
Mike Ludkovski
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 mludkov@umich.edu,
http://www.umich.edu/∼mludkov
Ren´e Carmona
Department of Operations Research and Financial Engineering, also with Bendheim Center for Finance,
Princeton University, Princeton, NJ 08544 rcarmona@princeton.edu,
We consider the valuation of energy storage facilities within the framework of stochastic control. Our two
main examples are natural gas dome storage and hydroelectric pumped storage. Focusing on the timing
flexibility aspect of the problem we construct an optimal switching model with inventory. Thus, the man-
ager has a constrained compound American option on the inter-temporal spread of the commodity prices.
Extending the methodology from Carmona and Ludkovski (2005), we then construct a robust numerical
scheme based on Monte Carlo regressions. Our simulation method can handle a generic Markovian price
model and easily incorporates many operational features and constraints. The main challenge is dealing
with the path-dependent storage levels, for which two numerical approaches are proposed. The scheme is
compared to the traditional quasi-variational framework and illustrated with several concrete examples. We
also consider related problems of interest, such as supply guarantees and mines management.
Key words : gas storage; optimal switching; least squares Monte Carlo; hydro pumped storage; impulse
control, commodity derivatives
History : First Version: August 2005; This Version: May 25, 2007
Acknowledgments
We thank the participants of BIRS Workshop 07w-5502 “Mathematics and the Environment” and Zhenwei
J. Qin for many useful comments and feedback. The paper was previously circulated under the title “Gas
Storage Valuation: An Optimal Switching Approach”.
1. Introduction.
While classical financial contracts such as stocks and bonds are paper assets, ownership of com-
modities entails physical storage. As a result, the modern commodities industry incorporates an
extensive storage infrastructure, including natural gas salt domes, liquified natural gas (LNG) stor-
age tanks, precious metal repositories and hydroelectric reservoirs. In the last decade, with the
ongoing deregulation of these industries, storage facilities have also acquired an important role
1
Carmona and Ludkovski: Optimal Switching for Energy Storage
2
in the commodity financial markets. Storage allows for inter-temporal transfer of the commodity
and permits exploitation of the fluctuating market prices. The basic principle is to ‘buy low’ and
‘sell high’, such that the realized profit covers the intermediate storage and operating costs. Since
the profitable opportunities are driven by price volatility, the storage facility grants its owner a
calendar straddle option.
Traditionally, storage facilities have been owned by major players in the respective industries
who had the enormous capital typically needed to build and maintain them. However, with the
liberalized markets, all participants have nowadays the opportunity to rent a storage facility with
an eye towards speculation on prices and aggressive profit maximization. For example, with respect
to natural gas de Jong and Walet (2003) write that “natural gas storage is unbundled, ... [and]
offered as a distinct, separately charged service. ... Buyers and sellers of natural gas have the
possibility to use storage capacity to take advantages of the volatility in prices”.
The aforementioned price volatility can be either systemic or speculative. For instance, the
natural gas market exhibits strong seasonality since the main consumer group is households that
use gas for winter heating. Thus, natural gas demand (and prices) has a systematic spike in the
cold season, often on the order of 30%-50%. In contrast, in say silver, the volatility in prices is
almost entirely speculative, but nevertheless can sometimes lead to price swings of 100% within
one year, c.f. the Hunt brothers episode in late 1970s (Pirrong 1996).
The seasonality effects present in certain markets lead to intrinsic value of storage that can be
locked-in by a static purchase and sale of forward contracts. For instance, a typical July-January
forward spread in natural gas is on the order of $1/MMBtu and can be easily realized by a simple
one-time transaction. However, the presence of an increasingly liquid short-term market permits
further dynamic optimization. Intuitively, the manager holds timing options that allow her to
optimally exploit opportunities that appear as market prices evolve. Capturing this optionality is
crucial in the present competitive markets, especially with the entry of non-energy players who rent
facilities with the sole goal of maximizing profit (as opposed to old-fashioned participants who also
have strategic aims). Moreover, with the growing importance of energy commodities, sophisticated
valuation of energy storage becomes an integral aspect of functioning financial markets.1
Thus, it becomes necessary to compute the financial extrinsic value of such flexibility. Namely,
how much should one pay to gain control of a storage facility for a period of Tyears? The simple
question above hides the associated modeling difficulties. Indeed, the owner faces a multitude of
1For instance, the $5 billion loss reported by Amaranth LLC. in Fall 2006 was due to a poorly managed bet on the
March-April calendar spread in natural gas, which is in turn driven by actions of storage managers during the winter.
Carmona and Ludkovski: Optimal Switching for Energy Storage 3
optionalities and constraints that interact in a nonlinear fashion. First, the purchases and sales
can be done immediately, using spot prices, or forward-in-time using forward prices. Second, the
bought commodity must be put on the inventory. As a result, inventory capacity limits, as well as
storage costs, delivery charges and other operational and engineering constraints become crucial.
However, the latter are intrinsically path-dependent in terms of the storage strategy adopted by
the manager. Finally, the manager may be exposed to margin requirements (if the commodity is
bought with credit), mechanical break-downs and other external events.
To overcome these challenges the current literature on commodity storage has largely proceeded
in two different directions. The basic practitioner methods have been based on the traditional
option-pricing approach. Thus, one makes (often drastic) simplifications to shoehorn the problem
into the option pricing framework. For instance, gas storage can be reduced to a collection of
calendar Call options, paying out the spread between gas prices today and k∆t, k = 1,...,T/∆t
years from now (Eydeland and Wolyniec 2003). Once this is done, the extensive existing machinery
of derivative pricing can be imported. One gains intuition and computational speed but ignores
key operational constraints (such as dynamic capacity limits), as the calendar Calls are priced
independently of each other. Furthermore, the method is ad hoc, requiring heuristic adjustments to
correct for model assumptions. Alternatively, various stochastic programming algorithms (Nowak
and R¨omisch 2000, Fleten et al. 2002, Doege et al. 2006) have been considered, especially for
hydrothermal systems. These methods maintain the flexibility of incorporating realistic constraints,
but instead discretize the set of future scenarios. While powerful, stochastic programming suffers
from non-scalability with respect to number of scenarios and time-steps used.
To properly account for the interdependence between the timing optionality of the manager in
choosing the purchase and sale times and the inventory constraints, one must consider the full
stochastic control framework. This leads to a Bellman dynamic programming equation for the
value function. From here one may apply the Hamilton-Jacobi-Bellman theory, translating the
problem into a quasi-variational partial differential equation (pde) formulation. This has been
recently done in Ahn et al. (2002), Thompson et al. (2003) and solution is then obtained via
standard numerical solvers. However, the path-dependency due to presence of inventory implies
that the pde is degenerate (convection-dominated) and therefore extra care is necessary. Moreover,
the implementation is necessarily price-model dependent and consequently not robust.
In this paper, we also adopt the stochastic control formulation. However, in contrast to the pde
methods above, we proceed to a probabilistic solution based on optimal stopping problems. This
perspective allows us to obtain an efficient simulation-based numerical method for valuing energy
Carmona and Ludkovski: Optimal Switching for Energy Storage
4
storage on a finite horizon. The method is flexible and not tied to a particular class of asset prices;
in fact we abstract from asset dynamics and take as exogenous the (multi-dimensional) Markov
price process for the commodity. Thus, use of complex price dynamics, such as jump-diffusions,
several factors, etc. has only a marginal impact on the efficiency of the algorithm.
Compared to previous approaches, our method has several advantages. First, we maintain rigor-
ous modeling of the operational constraints, while considering the entire set of future scenarios of
commodity prices. Moreover, in contrast to pde solvers which suffer from the curse of dimensional-
ity, our scheme can easily handle multi-dimensional settings. In terms of performance, our scheme
is competitive with the pde solvers in one-dimension and is clearly superior in higher dimensions
(which are essential in a realistic model, see Section 7). Thanks to its scalability, the algorithm is
easily extendable and therefore suitable for realistic use. Thus, our main contribution is a robust
numerical method that remains on firm theoretical grounds of stochastic control while bridging
the gap to practitioner needs.
To be concrete, from now on we focus on the representative example of controlling a natural
gas salt dome facility; other applications are addressed in Sections 6 and 7. The rest of the paper
is structured as follows. Section 2 describes the stochastic control model we use and its relation
to existing literature. Section 3 summarizes the theoretical solution method which is then imple-
mented in Section 4. After outlining the numerical scheme, we proceed to illustrative examples in
Section 5. Sections 6 and 7 discuss hydroelectric pumped storage and several problems in natural
resource management and demonstrate that our methodology is applicable to a wide variety of
real options encountered among commodity derivatives. Finally, Section 8 concludes and outlines
future projects.
2. Stochastic Model.
Natural gas storage is currently the most widespread class of commodity storage infrastructure in
the US2(FERC 2004). A variety of storage options, including depleted gas fields, aquifers, salt
domes and artificial caverns are available. In 2006 over 400 such facilities existed in the US and a
substantial portion are contracted out for periods of 6-60 months. In the near future, the industry
will expand even more with the rolling out of LNG technology and associated storage in North
America (see Geman (2005) for the general trends and organization of the gas universe.). In this
article we will specifically focus on the case of salt domes which permit the highest rates of injection
and withdrawal and therefore contain the most timing optionality (see Table 3).
2Throughout we focus on the North American markets and use imperial system units.
Carmona and Ludkovski: Optimal Switching for Energy Storage 5
A salt dome is an underground natural cave that can store several billion cubic feet of gas (Bcf ).
It is connected via pumps to the national pipeline system which allows to inject/withdraw gas at
a deliverability rate of 0.1−0.4Bcf per day. Taking the point of view of the renter, or manager
of such a cave, we now wish to maximize economic value by optimizing the dispatching policy, i.e.
dynamically deciding when gas is injected and withdrawn, as time and market conditions evolve.
We assume that the manager is rational and risk-neutral and aims to maximize total expected
revenue over the finite horizon of her rental. We moreover assume that the respective financial
markets are liquid and the manager is a price-taker (the situation of price impact is treated in
Section 7).
The ingredients of our model can be now listed as:
•Time horizon T, with a stipulation for the final state of the facility, see (7).
•Market gas prices given by a Markov continuous-time stochastic process (Gt), Gt∈Rd, quoted
in dollars per million of British thermal units (MMBtu), with 1 Bcf ≡106MMBtu.
•Level of inventory in storage denoted by Ct.
•Finite cave capacity represented by cmin ≤Ct≤cmax.
•Constant discount (interest) rate r.
•Three possible operating regimes of the storage facility: injection, storage and withdrawal.
•Denote by ain(Ct) the injection rate, quoted in Bcf per day. Injection of ain(Ct) Bcf of gas,
requires the purchase of bin(Ct)≥ain(Ct) Bcf on the open market.
•Similarly the withdrawal rate is labelled aout(Ct) and causes a market sale of bout(Ct)≤
aout(Ct) Bcf.
•Capacity charges Ki(t, Ct) in each regime that represent direct storage costs, delivery charges,
various O&M costs and seepage losses.
The case bi6=aiindicates gas loss during injection/withdrawal (typically on the scale of 0.25% −
1% for salt dome storage). The transmission rates ai, bithemselves are fixed by the physical char-
acteristics of the facility; they are a function of Ctand are based on gas pressure laws (Thompson
et al. 2003).
Remark 1. Typically, Gtwould represent the price at time tof the near-month forward con-
tract, which is by far the most liquid contract on the market3. However, given a variety of quoted
gas prices (spot, balance-of-the-month, futures, etc.), we remain agnostic about the precise inter-
pretation of the (Gt) process. The driving process (Gt) may also include longer maturity forwards.
3Recent daily volume on NYMEX has been over 90,000 contracts, with more than 50% of the trades in the near-
month.
Carmona and Ludkovski: Optimal Switching for Energy Storage
6
Unfortunately, forward selling is problematic, since the sale price is locked-in in advance, while
the inventory only changes at delivery time. We assume for simplicity that any purchase or sale is
immediately reflected in the current inventory.
Label the three regimes above as i∈ {−1,0,1}and denote by ψi(Gt, Ct) the payoff rate (in
$/year) from running the facility in regime i. Then ψi’s and the corresponding volumetric changes
in inventory are given by
Inject: ψ−1(t, Gt, Ct) = −Gt·bin −K−1(Ct), dCt=ain(Ct)dt,
Store: ψ0(t, Gt, Ct) = −K0(Ct), dCt=a0(Ct)dt,
Withdraw: ψ1(t, Gt, Ct) = +Gt·bout −K1(Ct), dCt=−aout(Ct)dt.
(1)
In principle, the facility can also be operated at a sub-maximal transfer rate, however when the
monetary reward is linear in the pumping rate as in (1), it is always optimal to inject/withdraw
at maximum speed. This is the so-called ‘bang-bang’ property of stochastic control problems
(Øksendal and Sulem 2005).
Many possibilities exist for the form of (Gt) and there is much recent debate (see e.g. Eydeland
and Wolyniec (2003)) about appropriate models for gas prices. A standard choice is an Itˆo diffusion
described by a stochastic differential equation (SDE)
dGt=µ(t, Gt)dt +σ(t, Gt)·dWt,(2)
where Wtis a d-dimensional Brownian motion and σ(t, g) is a non-degenerate volatility matrix.
A canonical example (see e.g. Jaillet et al. (2004)) is a one-dimensional exponential Ornstein-
Uhlenbeck process, namely
dGt=Gtκ(θ−log Gt)dt +σ dWt,(3)
or d(log Gt) = κθ−σ2
2κ−log Gtdt +σ dWt, G0=g.
This models the mean-reversion (to the average level eθ) in gas prices documented by Eydeland
and Wolyniec (2003), while keeping log Gtconditionally Gaussian. Upward jumps in (Gt) can also
be considered and may be used to take into account price spikes. The jury is still out whether such
jump-diffusion models are appropriate for natural gas. Other possibilities for (Gt) could include
regime-switching, stochastic mean reversion levels, latent factors, L´evy processes, etc. Our method
is independent of the assumed model for (Gt), and in general we only make the following technical
Assumption 1
(A) (Gt)is a d-dimensional, strong Markov, non-exploding process in Rd.
Carmona and Ludkovski: Optimal Switching for Energy Storage 7
(B) The information filtration F= (Ft)on the stochastic basis (Ω,F,P)is the natural filtration
of (Gt).
(C) The reward rate ψi: [0, T ]×Rd×[cmin, cmax ]→Ris a jointly Lipschitz-continuous function
of (t, g, c)and satisfies
E"sup
t∈[0,T ]|ψi(t, Gt, Ct)|2G0=g, C0=c#<∞,∀g , c.
For notational clarity we suppress from now on the dependency of ψiand the coefficients of (2) on
time t.
Remark 2. Above we have stated the model in continuous-time. This is to conform to clas-
sical financial stochastic control models; since the final implementation is computer-based and
consequently performed in discrete-time, one could also work in discrete-time from the beginning.
2.1. Control Problem.
The flexibility available to the manager is specified via the set Uof possible storage policies u.
For t∈[0, T ], ut∈ {−1,0,1}denotes the (dynamically chosen) operating regime of the facility.
It is convenient to write u= (ξ1, ξ2,...;τ1, τ2,...) where the variables ξk∈ {−1,0,1}denote the
sequence of operating regimes taken by u, while τk≤τk+1 ≤Tdenote the switching times. Thus,
ut=Pkξk[τk,τk+1)(t), where by convention τ0= 0, ξ0=i0is the initial facility state.
Given the initial inventory C0=cand the storage strategy u, the future inventory ¯
Ct(u) is
completely determined. Namely, ¯
Ct(u) satisfies the ordinary differential equation
d¯
Cs(u) = aus(¯
Cs(u)) ds, ¯
C0(u) = c. (4)
In the sequel we will also use the notation ¯
Ct(c, i),c+Rt
0ai(¯
Cs(c, i)) ds.
Each change of the facility’s regime incurs switching costs. In particular, moving the facility
from regime ito regime jcosts Ki,j =K(i, j;t, Gt, Ct). This represents both the effort —one must
dispatch workers, coordinate with the outgoing pipeline, stop/start the decompressors, etc.—and
the time needed to change the operating mode. We assume that the switching costs are discrete:
Ki,j > for all i6=jand some > 0, and Ki,i = 0. For actual salt dome facilities the switching
costs are economically negligible; however, in other applications, such as hydro pumped storage,
switching costs may be significant. Also, strictly positive switching costs are needed for technical
reasons in our continuous-time model in order to guarantee existence of optimal finite switching
strategies (i.e. to rule out chattering, where the owner would repeatedly change the regimes back-
and-forth over a very short amount of time). Since the ultimate computations are in discrete time,
switching costs can be set to zero on implementation-level.
Carmona and Ludkovski: Optimal Switching for Energy Storage
8
A necessary condition for uto belong to the set Uof admissible strategies is to be F-adapted,
right-continuous and of P-a.s. finite variation on [0, T ]. F-adaptiveness is a standard condition
implying that the agent only has access to the observed price process and cannot use any other
information. Finite variation means that the number of switching decisions must be finite almost
surely. Thus, P[τk< T ∀k>0] = 0. Other restrictions on uarise from engineering constraints;
for example the finite storage constraint requires that ¯
Ct(u)∈[cmin, cmax ] for all t≤T. Further
possibilities are explored in Section 7.2; in the meantime we assume that U(t, c, i), representing
the set of all admissible strategies on the time interval [t, T ] starting in regime iand with initial
inventory c, is a closed subset of U.
Subject to those costs and the operational constraints, the facility manager then maximizes
the net expected profit. Given initial conditions at time t:Gt=g, Ct=cand initial operating
regime i, suppose the manager chooses a particular dispatching policy u∈ U(t, c, i). If we denote
by V(t, g, c, i;u) the corresponding expected profit until final date T, then
V(t, g, c, i;u) = EhZT
t
e−r(s−t)ψus(Gs,¯
Cs(u)) ds −X
τk<T
e−rτkKuτk−,uτkGt=g, Ct=ci.(5)
The first term above counts the total revenues and costs from managing the facility up to the
horizon Tand the second term counts the incurred switching costs. Our formal control problem is
thus computing
V(t, g, c, i)M
= sup
u∈U(t,c,i)
V(t, g, c, i;u),(6)
with V(t, g, c, i;u) as defined in (5). Besides the value function Vwe are also interested in explicitly
characterizing an optimal policy u∗(if one exists) that achieves the supremum in (6). It remains to
specify the terminal condition at T. Typical contracts specify that the facility should be returned
with the same inventory C0as initially held, and in a certain state, e.g. store. To enforce this
stipulation, various buy-back provisions are employed. A common condition is
V(T , g, c, i;C0) = −¯
K1·(C0−c)+−¯
K2·(C0−c)−−¯
K3i6=0,(7)
making the penalty proportional to the difference with stipulated inventory C0, with multipliers
¯
K1and ¯
K2used for under-delivery and over-delivery respectively, and adding a second penalty of
¯
K3if the final regime is not store. Another common choice is V(T , g, c, i;C0) = −¯
K1·g·(C0−c)+,
which penalizes for having less gas than originally and makes the penalty proportional to current
price of gas.
Carmona and Ludkovski: Optimal Switching for Energy Storage 9
Before proceeding, let us emphasize the path-dependent nature of (6). Observe that an optimal
policy u∗
t0at intermediate time t<t0< T depends on the current inventory Ct0. However, Ct0=
¯
Ct0(u∗) is itself a function of past strategy (u∗
s:t≤s≤t0). Conversely, current Ct0affects the
feasibility of future strategies {us, s ≥t0}through the corresponding constraints on U(t0, c, i). The
standard method of solving control problems is by dynamic programming and would proceed
backwards in time, from s=Ttowards s=t0. However, in our case to find the optimal action at time
t0we need to know optimal actions before t0, hence the path-dependency and the resulting challenge.
Of course, this was abstractly resolved by making Cta state variable in (6). Unfortunately, from
a numerical analysis point of view this is only a superficial fix as Ctis now neither exogenously
stochastic, nor directly controlled, creating a degenerate and numerically unstable variable.
3. Iterative Optimal Stopping.
Without inventory, (6) belongs to the class of Optimal Switching problems. These have been
recently extensively studied, both analytically (Zervos 2003, Pham and Ly Vath 2005, Dayanik
and Egami 2005), and numerically (Barrera-Esteve et al. 2006, Porchet et al. 2006). In particular,
one can exploit the idea of the authors’ earlier paper (Carmona and Ludkovski 2005) to represent
(6) as a sequence of optimal stopping (American option) problems. These sub-problems precisely
capture the timing flexibility of the manager.
Let Stdenote the set of all F-stopping times between tand T. Recursively construct the functions
Vk(t, g, c, i) with k= 0,1,..., 0 ≤t≤T,g∈Rd,c∈[cmin, cmax] and i∈ {−1,0,1}via
V0(t, g, c, i)M
=EhZT
t
e−r(s−t)ψi(s, Gs,¯
Cs(c, i)) dsGt=gi,
Vk(t, g, c, i)M
= sup
τ∈St
E"Zτ
t
e−r(s−t)ψi(s, Gs,¯
Cs(c, i)) ds
+ max
j6=ie−r(τ−t)n−Ki,j +Vk−1(τ, Gτ,¯
Cτ(c, i), j)oGt=g#.(8)
The results in Carmona and Ludkovski (2005) show that
Proposition 1. Let Uk(t, c, i)M
={u∈U(t, c, i): u= (ξ1,...,ξk;τ1,...,τk)}be the subset of admis-
sible strategies with at most kswitches. Then
1. Vkis equal to the value function for the storage problem with at most kswitches allowed:
Vk(t, g, c, i) = supu∈U k(t,c,i)V(t, g, c, i;u).
2. An optimal strategy u∗=u∗,k for Vk(0, g, c, i)exists, is Markovian and is explicitly defined by
τ∗
0= 0, ξ∗
0=i, and for `= 1,...,k by
(τ∗
`
M
= infns≥τ∗
`−1:V`(s, Gs, Cs(u∗), i) = maxj6=i−Ki,j +V`−1(s, Gs, Cs(u∗), j)o∧T ,
ξ∗
`
M
= arg maxj6=i−Ki,j +V`−1(τ∗
`−, Gτ∗
`−, Cτ∗
`−(u∗), i).(9)
Carmona and Ludkovski: Optimal Switching for Energy Storage
10
3. limk→∞ Vk(t, g, c, i) = V(t, g , c, i)pointwise, uniformly on compacts.
4. The limit V(t, g, c, i)is continuous and is the minimal solution of the Bellman equation
V(t, g, c, i) = sup
τ∈St
EhZτ
t
e−r(s−t)ψi(Gs,¯
Cs(c, i)) ds
+ e−r(τ−t)·max
j6=i−Ki,j +V(τ, Gτ,¯
Cτ−t(c, i), j)Gt=gi.(10)
Item (i) says that Vk(t, g, c, i) is the maximum expected profit to be had on the time period [t, T ]
conditional on the initial state (g, c, i) and at most kswitches remaining. This is useful because
according to item (iii), for any > 0, there is a Klarge enough such that an optimal control of VKas
defined in (9), generates an -optimal strategy for V. The key insight behind the proposition is the
Bellman optimality principle which implies that solving the problem with at most k+ 1 switching
decisions allowed is equivalent to finding the first optimal decision time τwhich maximizes the
initial payoff until τplus the value function at τcorresponding to optimal switching with kswitches.
Remark 3. We do not have full results showing the uniqueness or existence of optimal control
for the original value function V, which is a delicate impulse control problem. On a practical level
this makes no difference since an -optimal control is always available. Theoretically, it would be
interesting to find a good set of working assumptions to ensure optimality existence/uniqueness.
3.1. Quasi-variational formulation.
The presented storage model is a special case of stochastic impulse control problems. Hence one
can apply the generic quasi-variational method developed by Bensoussan and Lions (1984). The
verification theorem presented below states that a suitable smooth candidate function ϕ, which
dominates the switching barrier and solves the Kolmogorov pde in the continuation region is indeed
the value function of (6). The proof follows from standard techniques, see e.g. Øksendal and Sulem
(2005).
Proposition 2. Let LGdenote the infinitesimal generator of the Markov process (Gt). Suppose
there exists ϕ(t, g, c, i)such that for
DM
=[
i(t, g, c) : ϕ(t, g , c, i) = max
j6=i{−Ki,j +ϕ(t, g, c, j )},
ϕbelongs to C1,2,2([0, T ]×Rd×[cmin, cmax]) \D∩ C1,1,1(D)and satisfies the following quasi-
variational inequality (QVI) for each i∈ {−1,0,1}:
minϕ(t, g, c, i)−max
j6=i−Ki,j +ϕ(t, g, c, j ),
−∂tϕ(t, g, c, i)−LGϕ(t, g, c, i) + ai(c)·∂cϕ(t, g, c, i)−ψi(g , c) + rϕ(t, g, c, i)= 0,
ϕ(T , g, c, i) = −¯
K1·(C0−c)+−¯
K2·(C0−c)−−¯
K3i6=0.
Then ϕ=Vis the optimal value function for the storage problem (6).
Carmona and Ludkovski: Optimal Switching for Energy Storage 11
If the process (Gt) is an Itˆo diffusion as in (2), then LG=µ(g)∂
∂g +1
2σ2(g)∂2
∂g2is a second-order
differential operator. The derived parabolic pde system with a free boundary can then be solved
using standard tools, see for example (Wilmott et al. 1995, Chapter 7). In the context of gas storage
this approach has been explored by Ahn et al. (2002). As a simplest choice, consider the basic
finite differencing (FD) algorithm. We set up a uniform space-time grid with steps ∆t, ∆gand ∆c
in the respective variables, and on this grid solve
ϕt(t, g, c, i) + µ(g)ϕg(t, g , c, i) + σ(g)2
2ϕgg (t, g, c, i)−ai(c)·ϕc(t, g , c, i) + ψi(g, c)−r·ϕ(t, g , c, i) = 0,
ϕ(t, g, c, i)>maxj6=i−Ki,j +ϕ(t, g , c, j),
ϕ(T , g, c, i) = −¯
K1·(C0−c)+−¯
K2·(C0−c)−−¯
K3i6=0,
(11)
by replacing derivatives with explicit finite differences in the first equation and directly enforcing
the barrier condition at each time-step. Using standard properties of the infinitesimal generator
LG, one obtains the convergence ϕ(0, g, c, i)−→V(0, g, c, i) as step sizes ∆t→0,∆g→0,∆c→0.
The FD method is straightforward to implement but will be slow since even in the easiest case,
where (Gt) is one-dimensional and has smooth dynamics, the pde (11) is two-dimensional in space.
Furthermore, the degenerate Ct-dynamics cause numerical instability as the pde is convection-
dominated (due to absence of ϕcc term). The algorithm is also not robust: for instance, adding
jumps to (2) produces a partial integro-differential equation which is non-local and requires special
numerical tools, see Thompson et al. (2003) for details. Similarly, pde solvers suffer from the curse
of dimensionality —making (Gt) two-dimensional is still beyond today’s computational power (in
the sense of a business-time system). On the other hand, the error analysis of FD algorithms is
well-studied and many improvements are possible, including adaptive solution grids, alternating
direction implicit schemes, relaxation methods, etc.
4. Numerical Method.
The benefit of the recursive formulation in (8)-(10) is its suitability for an efficient and scalable
numerical implementation. In this section we describe in detail the resulting algorithms.
To begin, we discretize time, setting S∆,{m∆t,m= 0,1,...,M}, ∆t=T
Mas our discrete time
grid. Managerial decisions are now allowed only at τk∈ S∆. This restriction is similar to looking
at Bermudan options as approximation to American exercise rights. Denote by
ψ∆
i(t, Gt, Ct)M
=Zt+∆t
t
e−r(s−t)·ψi(Gs, Cs)ds
the total cashflows during one time-step and let t1=m∆t, t2= (m+ 1)∆tbe two generic consecutive
time steps. In discrete time, the representation of V(t1, g, c, i) in (10) reduces to deciding between
Carmona and Ludkovski: Optimal Switching for Energy Storage
12
immediate switch at t1to some other regime j, which must then be maintained until t2(i.e. τ=t1
in (10)), versus no switching and therefore maintaining regime iuntil t2(τ > t1⇔τ≥t2). In other
words, one chooses the best (in terms of continuation value) regime jat t1, pays the corresponding
switching costs, and then waits until t2. Thus, using the notation of (4), (8) reduces to
V(t1, Gt1, Ct1, i) = max
j−Ki,j +Eψ∆
j(t1, Gt1, Ct1)+e−r∆t·V(t2, Gt2,¯
C∆t(Ct1, j), j )|Ft1.(12)
The dynamic programming method can now be applied to recursively evaluate (12) backwards in
time to obtain the discretized Snell envelopes (Dynkin 1963) of the optimal stopping problem (10).
Hence, for a numeric evaluation of Vit is sufficient to construct an algorithm for evaluating the
conditional expectations appearing in (12).
Let (Bj(g;t1, c, i))∞
j=1 be a given orthonormal basis of L2(Ft1) (selection of (Bj) is discussed
below). Recall that (Gt) is Markov, while (Ct) is determined by u. Thus, we may view the condi-
tional expectation in (12) as a map
g7→E(g;t1, c, i)M
=Ehψ∆
i(t1, Gt1, c) + e−r∆t·V(t2, Gt2,¯
C∆t(c, i), i)Gt1=gi.(13)
The latter may be approximated with a projection on the truncated basis (Bj)Nb1
j=1 :
E(g;t1, c, i) =
∞
X
j=1
αjBj(g;t1, c, i)'ˆ
E(g;t1, c, i) =
Nb1
X
j=1
αjBj(g;t1, c, i),(14)
where αjare the R-valued projection coefficients. The right hand side of (14) is a finite-dimensional
projection of the continuation values onto the basis functions and can be replaced with an empirical
regression based on a Monte Carlo simulation. This then gives a method for implementing (12) on
a computer.
Begin by generating Nsample paths (gn
m∆t), n= 1,...,N of the discretized (Gt) process with a
fixed initial condition G0=g=gn
0. As mentioned before, the inventory Ctdepends on the policy
choice, so it cannot be directly simulated. To overcome this problem, we shall construct a grid in
C-variable and compute V(t, g, c, i) only for c∈{c0=cmin, c1,...,cNC=cmax}.
We will approximate the value function by the empirical average of the pathwise quasi-values
(from now on simply values) V(0, g , c, i)'1
NPN
n=1 v(0, gn
0, c, i). The values v(t, gn
t, c, i) are computed
recursively in a backward fashion, starting with the terminal condition of (7): v(T , gn
T, c, i) = −¯
K1·
(C0−c)+−¯
K2·(C0−c)−−¯
K3i6=0. Consider again two consecutive time steps t1, t2and suppose
inductively that we know v(t2, g n
t2, c, i) along the paths (gn
t2)N
n=1 and for c=c`,`= 1,...,NC. Our
goal is to compute v(t1, gn
t1, c, i). To obtain the prediction ˆ
E(gn
m∆t;t1, c, i) of the continuation value,
Carmona and Ludkovski: Optimal Switching for Energy Storage 13
one first computes v(t2, gn
t2,¯
C∆t(c, i), i). Note that in general ¯
C∆t(c, i) does not belong to the grid
{c`}, and interpolation is needed. Then one regresses v(t2, gn
t2,¯
C∆t(c, i), i) against the basis functions
(Bj(gn
t1;t1, c, i))Nb1
j=1 to find the corresponding αj≡αj(t1, c, i) and applies (14). By analogue of (12),
the estimate for v(t1, gn
t1, c, i) is then
v(t1, gn
t1, c, i) = ˆ
E(gn
t1;t1, c, i)∨max
j6=i−Ki,j +ˆ
E(gn
t1;t1, c, j).(15)
Observe that (15) performs pathwise computations, while using across-the-paths projection ˆ
E. The
scheme (15) first appeared in Tsitsiklis and van Roy (2001) in the context of American option
pricing. In our setting we call it a mixed-interpolation Tsitsiklis-van Roy scheme (MITvR).
It is also useful to think in terms of the optimal storage strategy. Let ˆn(t1;i)∈{−1,0,1}represent
the optimal decision on the n-th path at time t=t1and current regime i. The analogue of (12)
implies that (recall Ki,i ≡0)
ˆn(t1;i) = arg max
j−Ki,j +ˆ
E(gn
t1;t1, c, j).(16)
Thus, the set of paths on which it is optimal to switch at time t=m∆tis given by {n: ˆn(m∆t;i)6=
i}. This can be used to construct the switching boundaries, which partition [0,∞]×[cmin, cmax ]
into regions of optimal injection, etc., and characterize the optimal strategy at date t.
The efficiency of (15) is enhanced by using the same set of paths to compute all the conditional
expectations. Nevertheless, because of the capacity variable Cthe above approach is still time-
intensive. Indeed, at every time step m∆tand regime i, we must run a separate regression for
each inventory grid point c`. Hence, in terms of computational complexity, the above method is
equivalent to solving Ncoptimal switching problems.
The choice of appropriate basis functions (Bj(·;t, c, i)) in (14) is user-defined. A detailed analysis
of different orthogonal families is available in Stentoft (2004). Empirically, basis choice has only
a mild effect on numerical precision, but strongly affects the variance of the algorithm. Thus,
customization is desirable and it helps to use basis functions that resemble the expected shape of
the value function. In practice, Nb1as small as five or six normally suffices, and having more bases
can often lead to worse numerical results due to overfitting. Let us mention that the requirement of
an orthonormal basis is purely theoretical and any set of linearly independent functions will suffice.
Some of our favorite choices are exponential functions eαg and the polynomials gm; this choice is
essentially heuristic. Also, we typically select the bases independent of parameters (t, c, i) though
the latter offer a wide scope for additional finetuning.
Carmona and Ludkovski: Optimal Switching for Energy Storage
14
4.1. Quasi-Simulation of Inventory Levels.
To maintain numerical efficiency it is desirable to avoid the fixed discretization in the C-variable
that resembles the slow lattice schemes. Accordingly, we propose the following alternative that
uses pathwise and regime-dependent inventory levels (cn
m∆t(i)). The idea is to perform a bivariate
regression in (14) of tomorrow’s value against the (price, inventory) pair. The paths (cn
m∆t(i))M
m=1 are
generated backwards during the dynamic programming procedure by combining randomization and
guesses of today’s optimal strategy. Besides added efficiency, we are also guided by considerations of
accuracy. Quasi-simulation of inventory allows us to use the Longstaff and Schwartz (2001) scheme
of computing pathwise value functions of optimal stopping problems. From simpler problems of
American option pricing and plain optimal switching we know that the LSM scheme typically has
less bias (though more variance) than the TvR scheme (15) (Ludkovski 2005).
We inductively assume again that we are given the 3Nvalues v(t2, gn
t2, cn
t2(i), i), i∈{−1,0,1}, n =
1,...,N, as well as bivariate basis functions ( ¯
Bj(g, c;t1, i))Nb2
j=1 . For a given path n, regime iand a
given inventory cn
t1(i) (see below about obtaining cn
t1(i)) we make the optimal switching decision
as follows (compare with (15)):
1. For each k∈ {−1,0,1}, regress {e−r∆t·v(t2, gn
t2, cn
t2(k), k)}N
n=1 against the basis functions
(¯
Bj(gn
t1, cn
t2(k); t1, k), j= 1,...,Nb2). This gives a prediction
˜
E: (g, c, k)7→
Nb2
X
j=1
¯αj¯
Bj(g, c;t1, k)'Ehe−r∆t·v(t2, Gt2, c, k )Gm∆t=gi(17)
of the value tomorrow given today’s prices and tomorrow’s inventory.
2. Similarly, regress ψ∆
k(t1, gn
t1, cn
t1(k))against basis functions (Bj(gn
t1;t1, k), j= 1,...,Nb1) to
find ˆ
Eof (14).
3. Compute ¯
C∆t(cn
t1(i), j), the inventory tomorrow given today’s inventory cn
t1(i) and the decision
to switch to j.
4. The optimal decision is the regime ˆn(t1, i) maximizing the approximate continuation value.
cf. (12)
ˆn(t1, i) = arg max
jn˜
E(gn
t1,¯
C∆t(cn
t1(i), j), j ) + ˆ
E(gn
t1l;t1, cn
t1(i), i)−Ki,j o.(18)
5. If ¯
C∆t(cn
t1(i),ˆn) = cn
t2(ˆn) then the Longstaff-Schwartz update is used:
v(t,gn
t1, cn
t1(i), i) = Zt2
t1
e−r(t−t1)ψˆn(Gt,¯
Ct−t1(cn
t1(i),ˆn)) dt + e−r∆t·v(t2, g n
t2, cn
t2(ˆ
n),ˆn)−Ki,ˆ.
(LSM)
Else, one updates via
v(t1, gn
t1, cn
t1(i), i) = ˜
E(gn
t1,¯
C∆t(cn
t1(i),ˆ),ˆ) + ˆ
E(gn
t1;t1, cn
t1(i), i)−Ki,ˆ.(TvR)
Carmona and Ludkovski: Optimal Switching for Energy Storage 15
The first case (LSM) stands for Least Squares Monte Carlo or Longstaff Schwartz Method. Observe
that in that version the across-the-paths regression is used primarily to make the optimal switching
decision, but is not necessarily fed into the pathwise values. This helps to eliminate potential biases
from the regression step by preventing error accumulation across time-steps. In order to preserve
this beneficial look-ahead property of the Longstaff and Schwartz (2001) algorithm, we therefore
attempt to speculatively pick cn
m∆t(i) such that the first case (LSM) occurs as much as possible.
In other words, as we move back in time we try to select inventory levels that form an optimal
(price, inventory) path on the remaining time interval. When this is not possible (due to capacity
or other constraints, or if our guess of ˆis incorrect), we fall back onto the basic (TvR) scheme.
Accurate guessing of ˆmeans that we correctly select the optimal strategy (up to the errors resulting
from the projection). In such a case, we have v(t1, gn
t1, cn
t1(i), i) = RT
t1e−r(s−t)ψu∗
s(Gt,¯
Cs(u∗)) ds −
Pτ∗
k<T e−r(τk−t1)Ku∗
τ∗
k−,u∗
τ∗
k
exactly along the price path. Observe also that the method can be used
iteratively over several simulation runs, improving the guesses of ˆover time.
The terminal inventory levels cn
Tare randomized and obtained by independent and uniform
samples from [cmin, cmax ]. At each step t=m∆t, some randomization in (cn
t1(i)) is also desirable in
order to avoid clustering and allow for good fit during the regression step. Of course, randomization
reduces the number of paths satisfying (LSM), and balancing the two objectives is a detail that we
leave to implementation. We christen this scheme Bivariate Least Squares Monte Carlo (BLSM).
4.2. Algorithm Summary
1. Select a set of univariate basis functions (Bj), bivariate basis functions ( ¯
Bj) and algorithm
parameters ∆t, M, N , Nb1, Nb2.
2. Generate Npaths of the price process: {gn
m∆t,m= 0,1,...,M, n = 1,2,...,N}with fixed
initial condition gn
0=g0. Generate a random terminal inventory level cn
T(i) for each path and each
regime i.
3. Initialize the pathwise values v(T , gn
T, cn
T(i), i) from (7).
4. Moving backward in time with t=m∆t,m=M,...,0 repeat the Loop, where the computa-
tions are based on (10):
i) Guess Current Inventory: generate (cn
m∆t(i)) by guessing the optimal decision ˆn(m∆t, i) and
solving ¯
C∆t((cn
m∆t(i),ˆn(m∆t, i)) = cn
(m+1)∆t(ˆn(m∆t, i)).
ii) Regression Step: do the univariate and bivariate regressions of (17).
iii) Optimal Decision Step: find the optimal decision using (18).
iv) Update Step: compute v(m∆t, gn
m∆t, cn
m∆t(i), i) via (LSM) and (TvR).
Carmona and Ludkovski: Optimal Switching for Energy Storage
16
v) Switching Sets: the points
Cm∆t(i, j),{(gn
m∆t, cn
m∆t): nis such that ˆn(m∆t, i) = i}
define the empirical region in the (G, C)-space where switching from regime ito regime jis optimal.
This defines the optimal strategy at t=m∆t.
5. end Loop
6. Interpolate V(0, g0, c, i) from the Nvalues v(0, g0
n, c0
n(i), i) for the desired inventory level c
(using splines, kernel regression, etc.).
Remark 4. As mentioned before, in the discrete-time version we allow switching costs to be
zero, Ki,j ≡0. In that case V(t, g, c, i) does not depend on the current regime iand so one can save
on the corresponding computations.
4.3. Algorithm Complexity.
The BLSM algorithm requires O(N·M·((Nb1)3+ (Nb2)3))operations. The most computationally
intensive operation is the regression step where we face matrices of size N×Nb1and N×Nb2, and
which make the algorithm linear in the larger dimension Nand cubic in the smaller dimensions
Nb1, Nb2. In contrast, the algorithm complexity of the MITvR scheme is O(N·M·Nc·(Nb1)3)
where Ncis the grid size in the C-variable. However, these expressions hide the relationship between
Nb1, Nb2and N, because more basis functions require more paths for accurate evaluation of the
regression step. In fact, according to Glasserman and Yu (2004), Nmust be asymptotically expo-
nential in the number of basis functions. On the other hand, to perform the bivariate regression
(17), it is likely that a large number of basis functions Nb2is needed, about 12 −15 in our experi-
ence. Hence in the BLSM algorithm Nmust be taken larger than in the the MITvR case. Precise
comparison is hard because the BLSM scheme inherently generates more variance and we have no
hard benchmark to go by. For our examples we find that N= 16,000 for the MITvR scheme is
reasonable, while N= 40,000 is needed for BLSM. Practically speaking this implies that BLSM
is about twice as fast as MITvR, see Section 5. The memory requirements of both schemes are
O(N·M) corresponding to the need to store the entire sample paths (gn
m∆t)N
n=1 in memory.
4.4. Convergence.
The presented algorithms has several layers of approximations. Three major types of errors can
be identified: error due to time discretization and the corresponding restriction of strategies to
U∆, projection error and Monte Carlo sampling error. Detailed error analysis has been performed
in Carmona and Ludkovski (2005) for the case of the TvR scheme with no inventory. Taking Ct
Carmona and Ludkovski: Optimal Switching for Energy Storage 17
Table 1 Convergence of Monte Carlo error for Example 2 under the BLSM scheme. Standard deviations were
obtained by running the algorithm 50 times.
No. Paths NMean Std. Dev
8000 9.63 0.4179
16000 9.41 0.1362
24000 9.37 0.0961
32000 9.38 0.0663
40000 9.35 0.0647
to be a ‘dummy’ variable determined by the dynamics of Gtand policy uthe results carry over
without change. Analysis of the BLSM scheme is too involved, however see partial results in this
direction in Egloff (2005). This lack of provable convergence results is typical for Monte Carlo
optimal stopping methods, largely due to the nonlinearity introduced by the stopping boundaries.
Nevertheless, extensive empirical experiments (Stentoft 2004) have strongly supported the general
TvR/LSM methodology.
The error from discretizing (Gt) and simultaneously restricting the switching times to occur
only at the discrete time grid points is known to be O(√∆t). The error from approximating
the conditional expectations with a projection (14) is on the order of O(∆t−k·(kE−ˆ
Ek)) when
computing Vk(Carmona and Ludkovski 2005). This suggests that the projection errors multiply
in the number of decisions taken k. However, empirically the dependence on ∆tis much better,
especially under the BLSM scheme, so this upper bound is probably not tight. In any case for
practical examples, the typical number of switches is in single digits. Finally, the third source of
error is due to approximating the projections with an empirical regression using Nrealizations of
the paths (gn
m∆t,n= 1,...,N). This error is difficult to analyze due to interactions between the
path-by-path maximum taken in (16) and the across-the-paths regression. No convergence behavior
is known; however numerical experiments suggest that it is close to O((∆t·N)−1/2), which is the
expected rate for Monte Carlo methods. Table 1 illustrates this conjecture on Example 2 below.
We run the BLSM algorithm using 8000 −40000 Monte Carlo paths and tabulate the resulting
standard errors. At least in this case we see in fact a faster than O(N−1/2) convergence.
5. Numerical Results.
In this section we present several examples to show the structure of the storage problem and the
scalability of our algorithm.
Example 1. As a first illustration of our approach, consider a facility with a total capacity of
Carmona and Ludkovski: Optimal Switching for Energy Storage
18
Table 2 Comparison of numerical results for Example 1. Values are in MM$/MMBtu. Standard deviations were
obtained by running the Monte Carlo methods 50 times. The initial gas price is G0= 3 $/MMBtu, initial
inventory is C0= 4 Bcf and initial regime is store.
Method Mean Std. Dev Time (min)
Coarse FD 9.32 – 24
Fine FD 9.44 – 65
MITvR 9.86 0.021 47
BLSM 9.35 0.067 32
8Bcf rented out for one year, T= 1. The price process is taken from the data of de Jong and Walet
(2003),
dlog Gt= 17.1·(log 3 −log Gt)dt + 1.33 dWt.
Observe the very fast mean-reversion of the prices, with a half-life of 15 days. The initial inventory
is 4Bcf and the terminal condition is V(T , g, c, i) = −2·g·max(4 −c, 0). Thus, the manager is
penalized at double the market price for final inventory being less than 4 Bcf and receives no
compensation for any excess. The other parameters (in yearly units) in (1) are
ain(c) = 0.06 ·365, Ki(c)≡0.1c, Ki,j ≡0.25for i6=j,
aout(c) = 0.25 ·365, r = 0.06, bi(c)≡ai(c).
Thus, it takes about 8/0.06 = 133 days to fill the facility and 8/0.25 = 32 days to empty it. In
this simple example we have taken the injection/withdrawal rates to be independent of inventory
levels.
We solve this storage problem using three different solvers: an explicit finite-difference pde solver
discretizing (11), the MITvR scheme of (15) and the BLSM scheme of (LSM). The results are
summarized in Table 2. As an extra check we used two different grid sizes for the pde solver: a
coarse 250 ×250 (G, C)-grid with 10000 time steps and a finer 500 ×500 (G, C )-grid with 20000
time steps. The MITvR scheme used 200 time steps, 10000 paths with six basis functions and 80
grid points in the C-variable. The quasi-simulation BLSM scheme used 200 time steps and 40000
paths with fifteen basis functions.
Taking the fine pde solver as the benchmark value, we see that the simulation methods are within
5% of the optimal value and seem to have an upper bias. The computational challenges involved
are indicated by the long running times of the algorithms.4. In this light, the 45% time savings
obtained by the joint (G, C)-regression become crucial from a practical point of view.
4The simulation methods were run in Matlab on a 1.6GHz desktop. The pde solver was written in C++ and run on
the same machine.
Carmona and Ludkovski: Optimal Switching for Energy Storage 19
Figure 1 Value function surface for Example 1 showing V(0.5,g , c, store;T= 1) as a function of current gas price
Gt=gand current inventory Ct=c.
Figure 1 shows the value function V(t, g , c, i) as a function of current price and inventory for
an intermediate time t= 0.5 and store regime. Not surprisingly, higher inventory increases the
value function since one has the opportunity to simply sell the excess gas on the market. In the
Gt-variable we observe a parabolic shape with a minimum around the long-term mean 3$/MMBtu.
Thus, deviations of Gtfrom its mean imply higher future profits, confirming our intuition about
storage acting as a financial straddle.
Table 3 shows the effect of storage flexibility on the value function. Higher transmission rates
increase the extrinsic value of storage, since the manager can move more gas in and out of the
facility under “favorable” circumstances. In the example considered, the smaller injection rate acts
as a bottleneck on the manager’s flexibility, so the derived extrinsic value is more sensitive to ainj
than to aout. Table 4 studies the effect of other parameters on the extrinsic value. We find that
switching costs Ki,j have a major impact on the extrinsic value. High Ki,j makes the manager
risk-averse and unwilling to change the pumping regime until a very good opportunity comes along
(since each switch has a high upfront fixed cost, while the benefit is always risky). We also find that
because of the limited transmission rates and the mean-reverting nature of the prices, there are
dis-economies of scale with respect to facility size. Thus, cutting the facility size to 6Bcf reduces
value by nearly 16%, but an increase from 10Bcf to 12Bcf produces a benefit of just 3%.
Example 2. Our second example is based on the situation presented in Thompson et al. (2003).
Carmona and Ludkovski: Optimal Switching for Energy Storage
20
Table 3 Effect of Storage Flexibility on the Value Function. Extrinsic value corresponds to V(0,3,4,0). Results
obtained using the BLSM algorithm with 40,000 paths.
ain Bcf/day aout Bcf/day Extrinsic Value
0.06 0.25 9.35
0.03 0.125 4.75
0.12 0.50 14.50
0.18 0.75 17.28
0.12 0.25 12.33
Table 4 Effect of Engineering Characteristics on the Value Function. Results obtained using the BLSM algorithm
with 40,000 paths.
Effect of Facility Size
Capacity (Bcf) V(0,3,4,0)
6 7.78
8 9.35
10 10.26
12 10.58
Effect of Switching Costs
Ki,j ,i6=j V (0,3,4,0)
0.01 13.25
0.1 11.40
0.25 9.35
0.5 6.73
Effect of Storage Cost
KiV(0,3,4,0)
0 9.77
0.05 ·c9.56
0.1·c9.35
A mean-reverting model is taken for gas prices, with a seasonally-adjusted mean-reverting level.
The gas prices satisfy
dGt= 4 ·(6 + sin(4πt)−Gt)dt + 0.5GtdWt.
Thus, the mean-reversion level has a seasonal component representing the summer/winter price
increases in the North American markets. This seasonality implies an approximate trough-to-peak
calendar spread of $1 for each half year.
Secondly, the injection and withdrawal rates are ratcheted in terms of current inventory. Thus,
injection rate decreases as the amount of gas in the facility grows and conversely withdrawal rate
decreases as less gas is on inventory. More precisely, the facility capacity is cmax = 2 Bcf and the
yearly rates are aout(c) = 0.177 ·365√c,ain(c) = 0.0632 ·365q1
c+0.5−1
2.5. These rate functions are
Carmona and Ludkovski: Optimal Switching for Energy Storage 21
related to the ideal gas law which states that gas transmission rate is proportional to pressure
in the reservoir, which in turn is inversely quadratically related to gas volume. Thus, maximum
injectivity at c= 0 is 0.08Bcf/day, and maximum withdrawal at c= 2 is 0.25Bcf/day.
The monetary reward functions are given by
Inject: ψ−1(g, c) = −(ain (c) + 0.0017 ·365)g, dCt=ain(Ct)dt,
Store: ψ0= 0, dCt= 0 dt,
Withdraw: ψ1(g, c) = aout(c)g, dCt=−aout(Ct)dt,
(19)
Note that there is gas loss during injection, represented by the constant term 0.0017 ·365. We again
take a horizon of one year T= 1 with no terminal penalty, V(T , g, c, i) = 0. There are no switching
costs and r= 0.1.
Figure 2 presents the optimal control for Example 2 at different times to maturity. The three shades
indicate the regions of injection (on the left), storage (dark region in the middle) and withdrawal
(on the right) respectively.
Switching boundary from regime 0 at 3 months to
maturity
Switching boundary from regime 1 at 6 months to
maturity
Figure 2 Optimal Controls for Example 2 using the BLSM algorithm with 32,000 paths. Each point corresponds
to a simulated (gn
t, cn
t(i)) pair, and the color indicates the optimal ˆof (18) (red for inject, black for
store, blue for withdraw).
Example 3. Finally, in our third example we illustrate the flexibility of the simulation approach
with regards to more complex price processes. It is well known that a one-factor diffusive model
does not provide a good fit to gas prices. Accordingly, let us consider a two-factor model with
jumps; namely a log-mean-reverting diffusive factor and a second mean-reverting pure jump factor.
The second factor captures spikes in natural gas prices without making the mean-reversion rate
unnecessarily high Kluge (2004).
(dG1
t= 4(log 6 −log G1
t)G1
tdt + 0.5G1
tdWt,
dG2
t= 26(0 −G2
t)dt +ξtdNt−λE[ξt]dt, (20)
Carmona and Ludkovski: Optimal Switching for Energy Storage
22
where (Nt) is an independent Poisson process with intensity λ, and the jump size ξt∼N(µJ, σJ)
has normal distribution. The total gas price is the product Gt,G1
t·exp(G2
t), and G2can be
interpreted as the multiplicative jump factor that causes price spikes on the scale of σJ%. The
possibility of price spikes makes storage much more valuable since it increases the volatility of
inter-temporal spreads. We pick λ= 12, µJ= 0.02, σJ= 0.1 for the jump component, as well as
T= 2, r = 0.06, Ki,j =Ki≡0, V (T , ·) = 0.
Implementing Example 3 requires only minor modifications to the implementation of Example
2, which essentially reduce to simulation of the bivariate price process (G1
t, G2
t) and selection of
bivariate basis functions ¯
Bj(g1, g2). This only takes a few minutes, and the resulting algorithm
will take only a little longer to run (depending on how many basis functions are added to deal
with (G2
t)). In contrast, with a pde approach, the new state dimension would require an extensive
rewrite of the code, and would slow the performance by an order of magnitude.
Since the value function V(t, g1, g2, c, i) now has three space variables, in Figure 3 we visualize
the dependence of Vjust on the two price factors (g1, g2) for different inventory levels c. Since
each factor is mean-reverting and the total price is a product of the two, the value function will
exhibit a parabolic straddle shape in each factor. Thus, in Figure 3, when g2<0, one can expect
prices to rise back to their “normal” level, and so this presents an opportunity for injection, at
least when inventory is low. Conversely, when g2>0 and inventory is high, we are in an upward
spike with prices expected to fall and an attractive withdrawal opportunity. The dependence of V
on g1is similar to that of Figure 1. Overall, as a function of the price and the spike factor, the
value function exhibits an asymmetric “bowl” shape, which in turn is dependent on the current
level of inventory.
c= 0.4Bcf .c= 1Bcf .c= 1.6B cf.
Figure 3 Value function V(1, g 1, g2, c)for Example 3 for different inventory levels c.
Carmona and Ludkovski: Optimal Switching for Energy Storage 23
6. Hydroelectric Pumped Storage.
Another important practical application of our model is hydroelectric pumped storage. Pumped
storage consists of a large reservoir of water held by a hydroelectric dam at a higher elevation.
When desired, the dam can be opened which activates the turbines and moves the water to another,
lower reservoir. The generated electricity is sold to the power grid. As the water flows, the upper
reservoir is depleted. Conversely, in times of low electricity demand (weekends, etc.), the water
can be pumped back into the upper reservoir using special, electricity-operated pumps (with the
required energy purchased from the grid). The efficiency of the system is about 80%, and the
capacity of such pumped storage facilities is typically on the order of several hundred megawatt-
hours (MWh). Currently pumped storage is the dominant type of electricity storage with more
than a hundred facilities around the world (ASCE 1996).
Beyond direct losses from upstream pumping, stored water is subject to evaporation. At the same
time, precipitation and/or upper river run-off provide reservoir replenishment. Realistic modeling
is complicated by the need to compute the potential energies of the reservoirs which depend on
the relative levels of the water and in turn modify generation rates ai(Ct). We abstract from these
concerns and treat the problem in our framework of commodity storage (5), with an addition of
another variable modeling weather. Let Ltrepresent the Markovian weather state at time t(e.g. Lt
can be a humidity index or river flow rate). (Lt) controls reservoir gains/losses, so that inventory
depletes at rate d(Lt, Ct) irrespective of the storage regime. The inventory Ctrepresents water level
in the upper reservoir5. The pumping inefficiency is represented by a multiplier ¯
K > 1, bin =¯
Kain,
bout =aout that affects the cost of pumping. The overall model is thus:
Pump: ψ−1(g, c) = −¯
K·g·ain −K−1(c), dCt= [ain −d(Lt, Ct)] dt,
Store: ψ0(g, c) = −K0(c), dCt= [a0−d(Lt, Ct)] dt,
Generate: ψ1(g, c) = +g·aout −K1(c), dCt= [−aout −d(Lt, Ct)] dt.
(21)
Once a suitable model is chosen for (Lt) (see e.g. Cao et al. (2004)), implementation would be
similar to Example 3 above, and would require minimal changes to the simulation algorithm.
Note that one could mix-and-match different model types for different variables, for instance a
jump-diffusion model for gas prices, and a seasonal AR(1) model for river run-off. Such flexibility
would be hard to achieve outside of simulation paradigm (compare to the stochastic programming
approach of Nowak and R¨omisch (2000), Doege et al. (2006)).
5A full model should also specify the lower reservoir inventory since the latter also depletes over time.
Carmona and Ludkovski: Optimal Switching for Energy Storage
24
7. Extensions.
In this section we discuss various extensions and modifications that can be made to our model. First,
let us remark that many other resource management problems can be recast in our framework. Such
problems all feature fluctuating commodity prices, finite inventory constraints and a small number
of operating regimes describing the facility state. Below we elaborate on some of the possibilities.
7.1. Other Applications.
7.1.1. Mine management A producer extracts metal from a mine with initial capacity C0.
As the resource is mined, the inventory Ctdeclines. In the meantime, the producer can control
the mine operating regime to time the extraction with high commodity prices represented by
(Gt). In this situation, the remaining resource amount Ctis non-decreasing, since only extraction
is possible. Exhaustion implies that no profit is available when Ct= 0: V(t, g, 0, i) = 0. Armed
with our methodology we can e.g. redo in a more efficient manner (see Ludkovski (2005) for the
computation) the copper mine example analyzed in Brennan and Schwartz (1985). Moreover, we
can easily add further constraints to their model.
A related application is production of oil from oilfields. In the latter context, Ctcan be increased
by further exploration; at the same time fixed extraction costs increase as the field gets depleted,
so that Ki≡Ki(Ct). One may also add a termination option that allows total shutdown and avoids
the ongoing O&M costs Ki.
7.1.2. Hydroelectric Power Generation This setting is similar to the pumped storage
problem; however no pumping is available and the dammed reservoir is replenished solely with river
run-off. The latter is stochastic and is modelled by a stochastic process (Lt). When the turbines
are running the produced electricity is sold at the spot power price Gt. As before, the inventory Ct
is the current amount of water in the dammed reservoir. Reservoir management has been already
studied by ancient Egyptians and Mesopotamians; related stochastic control models have recently
been considered by Keppo (2002) and McNickle et al. (2004). Note that on a practical level a major
challenge is the long-memory hydrological features of (Lt).
7.1.3. Power Supply Guarantees Yet another possibility similar to the pumped storage
above is the case of power supply guarantees. The latter involve a hybrid energy storage/power
generation setting. By law, the North American Load-Serving Entities (LSE), i.e. the local power
utilities, are obligated to provide power irrespective of demand. The latter is stochastic so that
the LSE faces uncertain demand (volume risk) coupled with uncertain fuel prices (price risk). To
insulate against risk, the LSE might operate an energy storage facility (e.g. a natural gas aquifer)
Carmona and Ludkovski: Optimal Switching for Energy Storage 25
that acts as a buffer between risky supply costs and risky demand needs. Letting (Dt) represent
the demand at time t, one obtains a model similar to (21) where the reservoir is depleted at rate
Dtdue to the requirement of producing fuel. Note that typically (Dt) is highly correlated with the
fuel price (Gt) as high demand drives up the spot prices. Thus, the marginal cost of storage is high
precisely when prices are high. See see Deng et al. (2005) for further details.
7.1.4. Emissions Trading Another application area is emissions trading, cf. Insley (2003).
A firm running a factory is subject to emission laws and must account for its pollution by buying
publicly traded emission permits with current price Gt. The non-increasing inventory Ctin this
case corresponds to the total number of remaining factory orders that must filled in the current
quarter. Hence, the management must satisfy all the orders while minimizing emission costs. The
firm opportunistically runs its production given emission price Gtand current shipment backlog.
This setup is similar to supply guarantees, with an additional constraint of Ct≤C0−Otwhere Ot
is the (deterministic) shipping timetable supplied by the customers. Violation of this constraint
causes a severe penalty as the firm misses its shipment.
Many other situations can be imagined—forest management, oilfield development, pipeline ship-
ping, etc. From the above descriptions it should be evident that our numerical algorithm would
carry over easily to the new settings. Summarizing, optimal switching with inventory is a widespread
financial setting with many practical applications.
7.2. Incorporating Other Features.
From a practical standpoint the presented models are gross simplifications. However, as already
advertised, the simulation framework permits great flexibility. To illustrate the possibilities, we
briefly discuss additional features that a practitioner is likely to implement. First of all, one is likely
to use a more general price model for (Gt) than (2). As already mentioned, all that is absolutely
necessary is to satisfy Assumption 1, thus extra features such as regime-switching or latent factors
are easily implementable. As already shown in Examples 2 and 3, seasonality effects, models with
jumps and multi-factor models can be implemented straightforwardly.
The dynamics of (Gt) might also be affected by the choice of strategy. Indeed, since the manager
tends to buy when prices are low and sell when prices are high, her influence is to smooth out
the price fluctuations of (Gt). This effect can be quite pronounced in segmented markets based
on supply and demand (e.g. gas markets with little outside connectivity). As long as the effect is
limited to the coefficients of (2), µ=µ(ut,·), σ =σ(ut,·) such market impact can be treated by
independently simulating price paths under each separate regime and then modifying the algorithm
Carmona and Ludkovski: Optimal Switching for Energy Storage
26
like in Carmona and Ludkovski (2005). If the transmission losses and engineering costs Kiare
nonlinear in the pumping rates, it may become optimal to inject/withdraw gas at sub-maximal
rates. In such a case, the optimal control u∗will take on a continuum of values. Our method relies
heavily on u∗belonging to a (small) finite number of regimes; however a reasonable first-order
correction would be to add a few more regimes (i.e. to discretize the range of u∗) to the original
three considered here.
Another challenge is proper modeling of borrowing constraints faced by the manager. In the
North American gas industry, the facility typically borrows money in the summer to inject gas
and then repays its loans in the winter as gas is withdrawn. In the meantime, the creditors often
impose margin requirements regarding the value of stored gas versus the original loan. Thus, if
prices drop, the manager might receive a margin call that would require him to sell off some of
the inventory (at a loss) to raise capital. To account for this, one can let Btbe the cumulative
borrowed capital taken out for inventory purchase. The margin constraints are then imposed on the
net equity Bt−Ct·Gt; alternatively some absolute borrowing limits Bt>−Kcould be required.
The option of forward sales is another crucial feature. Forward sales allow the manager to lock-in
future profits while reducing earnings volatility and form a bread-and-butter business in the gas
storage industry. From the modeling perspective, a forward contract is challenging due to its non-
Markovian nature, which necessitates complex account-keeping for gas already sold but still sitting
in the facility (or gas already bought but still not on inventory). Indeed, imagine that at time t=t0,
the manager forward-sells some quantity C0of gas at future date t=t1. This now affects her possible
future strategies: the manager must have at least C0in inventory at t=t1, and must also start
to withdraw C0after t1. Such constraints are modelled easily enough in our framework; however
because they affect the future, they are not easily implementable in the dynamic programming
method —to find the value of the forward sale at t=t0we must recompute the optimal strategy
under the new admissibility restrictions ¯
U(t, c, i), which is computationally challenging (essentially
requiring as much effort as the original computation). Hence, modeling of a forward sale would
lead to a “tree” of simulations with a separate branch for each possible forward sale or purchase.
This might still be practical to do if the forward sales occur infrequently.
Finally, an interesting research direction would be incorporation of realistic risk objectives for the
manager. In this paper we have assumed that the manager maximizes total (discounted) earnings
from the asset. In practice this would lead to overly aggressive strategies and high earnings volatility.
Thus, it is desirable to impose risk constraints that lead to more conservative speculation. One
method for doing so can be achieved by replacing the linear conditional expectations in (12) with
Carmona and Ludkovski: Optimal Switching for Energy Storage 27
non-linear expectations that take into account risk preferences Musiela and Zariphopoulou (2004).
Another perspective can be to make the value function depend on the cumulative gains/losses that
have been denoted by Btabove, and to penalize for variance in say BT.
8. Conclusion.
This paper presented a simple model for energy storage that emphasizes the intertemporal option-
ality of the asset. Assuming that the commodity is bought and sold on the spot market, we have
maximized the expected profit given operational constraints, in particular inventory limits and
switching costs. While the model sidesteps the possibilities of forward trades, it properly accounts
for the dynamic nature of the problem, which is a crucial aspect of revenue maximization.
Our approach is scalable and robust and we provide a detailed description of implementation. As
our numerical examples attest, the model is computationally efficient and we believe better than
any other proposed in the literature. We hope it can fill in the gap between current practitioner
needs and academic models. Moreover, our strategy is applicable to many related problems, such
as hydroelectric pumped storage, power supply guarantees, natural resource management and
emissions trading.
As the next step in improving our model, one should study financial hedging and more advanced
risk objectives. Financial hedging would permit further risk-management by considering the oppor-
tunity to hedge operations through trading in liquid instruments. For instance, for gas storage one
can trade in the Henry Hub contracts available on the New York Mercantile Exchange (NYMEX).
Such hedges are likely to be imperfect, because the facility buys gas based on local prices that are
different from the NYMEX index. Thus, financial hedging exposes the agent to basis risk. Con-
sequently, to study hedging one must consider the risk-preferences of the manager, an issue that
was alluded to in Section 7.2. On the theoretical level, financial hedging would require analysis of
a combined control problem, namely the mixture of optimal switching and portfolio optimization
in an incomplete market.
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