Contents

This practical work is aimed at presenting the theory of the SDDP algorithm and apply it on a simple dam management problem.

The Stochastic Dual Dynamic Programming (SDDP) algorithm is close to the Dynamic Programming method. The main difference being that, instead of computing the Bellman function at every point, the algorithm construct a polyhedral approximation of the Bellman function. More precisely the algorithm construct a piecewise linear function that is a lower approximation of the exact Bellman function.

This document is given in three parts:

• In §1, we present a few results about piecewise linear functions. We also recall the cutting plane algorithm, known as Kelley’s algorithm.
• In §2, we present the SDDP algorithm.
• In §3, we present the dam management problem we want to tackle.
• Finally, in §4, we apply the SDDP algorithm on the dam management problem.

Before going any further you should download the following code:

1. Kelley,
2. DamData,
3. sdp,
4. CutPerAlea
5. sddp.

1 Preliminary

1.1 Piecewise linear functions

Recall that the supremum of a finite number of hyperplane is a convex piecewise linear function. We consider the function

(1)

where b_i ∈ ℝ and a_i ∈ ℝⁿ for i ∈{1,,k}.

1.2 The cutting plane method: Kelley algorithm

The Kelley algorithm enables someone to approximate the minimum of a convex function f on a compact C only by having an oracle able to compute at each point its value and derivative. Recall that a convex function is always above its tangents. The cutting plane method consists in constructing a lower approximation of the objective function f as the maximum of tangents. This lower approximation is a piecewise-linear function, and is improved throughout the algorithm.

Let describe more precisely the algorithm. We start at a given point x₀. The first lower approximation is the tangent at this point,

Then, at each step, we computes the equation of the tangent of f at the minimum of the approximate function ,

(2)

The new lower approximation ^k+1 is given by the maximum of the old one and the tangent,

(3)

2 The Structure of SDDP

In this section we present the Stochastic Dual Dynamic Programming (SDDP) algorithm. More precisely we present the so called DOASA implementation. For clarity reason we present it in the deterministic case in a first place, and then go to the stochastic case.

2.1 The Deterministic case

The SDDP algorithm in a deterministic setting is really close to the Kelley’s algorithm on a multistage problem instead of a static problem.

2.1.1 Problem statement

We consider a controled dynamic system in discrete time that follows the equation

where x_t ∈ 𝕏 is the state at time t of the system and u_t ∈ 𝕌 the control applied at time t; f_t is the dynamics of the system. We assume that the sets 𝕌 and 𝕏 are compact subset of a finite dimensionnal vector space. We also assume that the functions f_t are linear. At each time t there is a convex cost L_t(x_t,u_t) that depends on the current state and the control choosen. Moreover there is a final cost K(x_T ) that depends on the final state of the system. A policy is a sequence of functions π = (π₁,…,π_T−1) giving for any state x a control u.

We consider the following problem

(4)

This problem can be solved by dynamic programming.

2.1.2 Bellman operator

In order to understand the SDDP algorithm it is useful to introduce the Bellman operator 𝒯_t of the problem.

For any time t, and any function A mapping the set of states into ℝ we define:

(5)

Let study a few properties of the Bellman operator.

Monotonicity For any couple of functions (V,V ), we have

(6)

Convexity For any function V , if L_t is jointly convex in (x,u), V is convex, and f_t is affine then

(7)

Linearity For any piecewise linear function , if is also piecewise linear, and f_t affine, then

(8)

2.1.3 Duality property

Let J be a function mapping 𝕏 × 𝕌 into ℝ. Consider the function φ defined as

where J is jointly convex in (x,u). Then the interpretation of the multiplier as a marginal cost implies that a subgradient λ ∈ ∂φ(x₀) of φ at x₀ is the dual multiplier of constraint x₀ − x = 0 in the following problem

In particular it means that, as φ is convex

(9)

2.1.4 SDDP algorithm (deterministic case)

As in the Kelley’s algorithm, the SDDP algorithm recursively constructs a lower-approximation _t^(k) of each Bellman function V _t as the supremum of a number of affine functions. Each of this affine functions is called a cut.

Stage k of the SDDP algorithm goes as follows.

Suppose that we have a collection of T lower approximation of the Bellman functions denoted _t^(k) that is a maximum of affine functions. In order to improve our approximation we follow an optimal trajectory (x_t^(k)) _t of the approximated problem and add a cut computed along the optimal trajectory for each Bellman function.

Thus, we begin with a loop forward in time by setting t = 0 and x_t^(k) = x ₀, and solve

We call β_t^(k+1) the value of the problem, λ _t^(k+1) a multiplier of the constraint x = x _t^(k), and u _t^(k) an optimal solution. This can also be written as

Thus we have by (9), and by monotonicity of 𝒯_t,

(11)

This ensures that the function xβ_t^(k+1) + is indeed a cut, i.e an affine function below V _t. Consequently we update our approximate of V _t by defining

Furthermore we can see that _t^(k+1) is convex and lower than V _t. We go to the next time step by setting

Upon reaching time t = T we have completed iteration k of the algorithm.

2.1.5 Initialization and stopping rule

In order to initialize the algorithm it seems that we need a lower bound to all value function. According to our assumptions this bound always exist but is not necessary in order to implement the algorithm. Indeed we can choose V _t⁽⁰⁾ = 0 in order to compute the trajectories, and simply set V _t⁽¹⁾ equal to the first cut, which means that we “forget” V ⁽⁰⁾ in the maximum that determine V _t⁽¹⁾.

Note that at any step k of the algorithm we have a (non optimal) solution (u^(k)) _t of Problem (4), and we can estimate the quality of the solution. Indeed V ₀^(k)(x ₀) is a lower bound to the optimal value of Problem (4). Moreover the simulated cost following the solution given by

is an upper bound to the value of Problem (4). A reasonable stopping rule for the algorithm is given by checking that the (relative) difference of the upper and lower bound is smaller than some constant as they are both asymptotically exact bounds, i.e. both converges toward the value of Problem (4).

2.2 The Stochastic case

Now we introduce some random variables in our problem. This complexify the algorithm in different ways:

• we need some probabilistic assumptions;
• for each stage k we need to do a forward phase that yields a trajectory (x_t^(k)) _t, and a backward phase that gives a new cut;
• we cannot compute an exact upper bound for the problem’s value.

2.2.1 Problem statement

As in the deterministic case we consider a controled stochastic dynamic system in discrete time that follows the equation

where x_t ∈ 𝕏 is the state at time t of the system, u_t the control applied at time t, and W_t an exogeneous discrete noise ; f_t is the dynamic of the system.

We assume that the stochastic process (W_t)_{t=1,…,T−1} is independent in time. Moreover we assume that at any time t we know the whole past of the noise process up to, and including, time t. The independence assumption ensures that we can only consider control u_t measurable with respect to (x_t,W_t). This is the so called hazard-decision information structure where we know at time t the realization of the noise before choosing our control.

At each time t there is a jointly convex in (x_t,u_t) cost, L_t(x_t,u_t,W_t) that depends on the current state and the control choosen ; and there is a final cost K(x_T ) the depends on the final state of the system. A policy is a sequence of measurable functions π = (π₁,…,π_T−1) giving for any state x, and realization of the noise w a control u.

We consider the following problem

(12)

2.2.2 Bellman operator

As in the deterministic case we also introduce the stochastic Bellman operator 𝒯_t of the problem. For any time t, and any function A mapping the set of states and noises 𝕏 × 𝕎 into ℝ we define:

The stochastic Bellman operator 𝒯_t, is given by

2.2.3 Duality theory

Suppose that for an iteration k and a time step t + 1 we have a lower estimation V _t+1^k+1 of V _t+1. We consider, for any w, the problem

(13)

and call _t^(k+1)(w) the optimal value, and _t^(k+1)(w) an optimal multiplier of the constraint. As in the deterministic case we have

(14)

2.2.4 SDDP algorithm (stochastic case)

At the beginning of step k we suppose that we have, for each time step t an approximation _t^(k) of V _t verifying

• _t^(k) ≤ V _t,
• V _T ^k = K,
• _t^(k) is convex.

Forward path We randomly select a scenario (w₀,…,w_T−1) ∈ 𝕎^T . We recursively, by a forward loop in time, define a trajectory (x_t^(k)) _t=0,…,T by x_t+1^(k) = f _t(x_t^(k),u _t^(k),w _t), where u_t^(k) is an optimal solution of

(17)

This trajectory is a trajectory given by the optimal policy of Problem (12) where V _t is replaced by its current approximation _t^(k).

Backward path We reader improve the approximation of V _t by adding a cut at V _t(x_t^(k)). We procede backward in time by setting t = T − 1, and solve Problem (13) for every w in the support of W_t. We obtain _t^(k+1)(w) and _t^(k+1)(w). As W _t is discrete we can compute exactely the expectation λ_t^(k+1) and β_t^(k+1), and define the new lower approximation

Go one step back in time: t ← t − 1. Upon reaching t = 0 we have completed step k of the algorithm.

2.2.5 Initialization and stopping rule

We can intialize as suggested in the deterministic case. However in order to accelerate the convergence it can be useful to bypass a few forward paths by abritrarily choosing some trajectories (x_t^(k)) _t.

As in the deterministic case we have an exact lower bound of Problem (12) given by V ₀^(k)(x ₀). However, determining an upper bound is more difficult. Indeed, the upper bound is given as the expectation of the cost induced by the optimal strategy of the approximated problem. This expectation is to be taken on the whole process (W₀,…,W_T−1) and cannot be computed exactly. Consequently this expectation is estimated by Monte-Carlo methods, and we have no control over the error of our solution.

A heuristic stopping rule consist in stopping the algorithm if the lower bound is in the confidence interval of the upper bound for a determined number of Monte-Carlo simulation.

2.2.6 A few other possibilities

In the algorithm presented here, also called DOASA, we select one scenario (one realization of (W₁,…,W_T−1)) to do a forward and backward path. It is also possible to select a number N of scenarios to do the forward path (computation can be parallelized). Then during the backward path we add N cuts to V _t before computing the cuts on V _t−1. This is the original implementation of the algorithm.

Otherwise the CUPPS algorithm suggest to use _t+1^(k) instead of V _t+1^(k+1) in the computation of the cuts (14). In practice it means that we do only a forward path at step k:

• select randomly a scenario (w_t)_{t=0,…,T−1};
• at time t we have a state x_t^(k), we solve problem (13) for every possible realization of W_t in order to compute the new cut for V _t;
• choose the optimal control corresponding to the realization W_t = w_t in order to compute the state x_t+1^(k) where the cut for V _t+1 will be computed, and goes to the next step.

Moreover it is possible to select more than one scenario, and everything can be parralelized in this implementation.

Finally it is always possible, and most of the time fruitful to compute some cuts before starting the algorithm. One way of doing this is to bypass the forward phase by choosing the trajectory (x_t^(k)) _t=0,…,T instead of computing it as some optimal trajectory.

3 Problem data

We consider a dam manager intenting to minimize the intertemporal cost obtained by turbining the water in a dam, when the water inflow (rain, outflows from upper dams) is supposed to be stochastic.

3.1 Dam dynamics

with

• time t ∈𝕋 := {t₀,…,T} is discrete (days), and t denotes the beginning of the period [t,t + 1[,
• S_t volume (stock) of water at the beginning of period [t,t + 1[, belonging to the set , made of water volumes (h m³), where is the maximum dam volume,
• a_t inflow water volume (rain, etc.) during [t,t + 1[, belonging to , and supposed to be known at the beginning of the period (Hazard-Decision),
• q_t turbined outflow volume during [t,t+1[, decided at the beginning of period [t,t+1[, belonging to the set , where q^♯ is the maximum which can be turbined by time unit (and produce electricity),
• δ_t spilled outflow volume during [t,t + 1[, decided at the beginning of period [t,t + 1[, belonging to the set , and such that

  (18)

The dam manager is supposed to make a decision, here turbining q_t and spilling δ_t, at the beginning of the period [t,t + 1], after knowing the water inflow a_t.

A water inflows scenario is a sequence as

(19)

Compared to other methods, we have added the spilled outflow volume δ in order to get a linear dynamics and we are placed in the Hazard-Decision case. These two points are mandatory to apply the SDDP algorithm.

3.2 Criterion: intertemporal cost

The manager original problem is one of cost minimization where turbining one unit of water has unitary cost −p_t. On the period from t₀ to T, the costs sum up to

(20)

where

• a scenario of costs

  (21)

  is given (data of the problem) and supposed to be known in advance (deterministic setting),

• the final term gives value to the water volume in the dam at the horizon T.

3.3 Probabilistic model for water inflows

We suppose that sequences of uncertainties (a_t0,…,a_T−1) are random variables with a known probability distribution ℙ on the set {0,…,a^♯}, with ℙ{a _t = a} given.

Notice that the random variables (a_t0,…,a_T−1) are independent, but that they are not necessarily identically distributed. This allows us to account for seasonal effects (more rain in autumn and winter).

3.4 Numerical data

We now perform numerical simulations, and try different policies.

Time. We consider a daily management over one year

(22)

Bounds. Concerning the dam and water inflows, we consider the following bounds:

(23)

These figures reflect the assumptions that, during one week, one can release at maximum 40% of the dam volume, and that during one week of full water inflows, an empty dam can be half-filled.

Costs scenario. The scenario of costs is known in advance. We produce it by one sample from the expression

(24)

where is drawn from a sequence of i.i.d. uniform random variables in .

Water inflow scenarios. We produce scenarios a(⋅) := a_t0,…,a_T−1 of water inflows by a sequence of independent random draws from the set {0, 1,,a^♯}.

3.5 Water turbined policy

An admissible policy π : 𝕋 × 𝕏 × 𝕎 → 𝕌 × 𝔻 assigns an amount of water turbined and spilled to any state  S of dam stock volume, to any decision period t ∈ 𝕋 and to any water inflow a ∈ 𝕎, while respecting the constraints

(25)

that can also be written

(26)

The system (26) is equivalent to

(27)

In the following, we will simply write as defined A ⋅ + S_t.E ≤ b(t).

Once the policy given, we obtain a volume trajectory S(⋅) = , a turbined trajectory q(⋅) = and a spilled trajectory δ(⋅) = produced by the ”closed-loop” dynamics

(28)

and function of the scenario a(⋅). Thus, in the end, we obtain the cost

(29)

3.6 Minimization under constraints

The aim of this part is to compute the minimum of a given convex function under linear constraints and to find a cutting plane. We will use this to find an optimal strategy.

The problem is

(30)

In order to use the linear solver, we have to transform this formula into one of the form

(31)

We remind the reader that V _t+1^k is a piecewise linear function written

which is equivalent to (as seen in § 1.1)

(32)

By writting

, , and , we obtain the wanted form for the minimization problem (see (31)).

The last step is to find the parameters of the cutting plane of the Bellman function V _t^k at a given point z. To do so, we transform the state z in the previous equation in a variable x and add the constraint x = z. The Lagrange multiplier of this equality constraint is the slope of a new cutting plane and the minimum cost is the ordinate at the origin.

The following macro returns the parameters of the plane (λ,β) and the point at which the minimum is reached u_opt given the matricial form of the cost (−p,V ), the constraints (A,b(t),E), the value of the uncertainty a, the dynamics given by (Q,R) and the state z.

4 Dam optimal management

We now solve the problem presented earlier in this practical work, by Dynamic Programming and by Stochastic Dual Dynamic Programming.

4.1 DP resolution

We solve exactly the problem using Dynamic Programming.

4.2 SDDP resolution

We now apply an SDDP approach to the same problem.

In the files that were downloaded at the beginning, the function SDDP creates an approximate of the Bellman function given the constraints and random trajectories, then the function sddp_policy defines the optimal policy given the former approximate.

4.3 Comparison of SDDP and DP