Optimal installation of renewable electricity sources: the case of Italy
OOptimal installation of renewable electricity sources: the case of Italy
Almendra Awerkin ∗ and Tiziano Vargiolu † Department of Mathematics ”Tullio Levi-Civita”University of PaduaFebruary 9, 2021
Abstract
Starting from the model in [23], we test the real impact of current renewable installed power inthe electricity price in Italy, and assess how much the renewable installation strategy which was putin place in Italy deviated from the optimal one obtained from the model in the period 2012–2018. Todo so, we consider the Ornstein-Uhlenbeck (O-U) process, including an exogenous increasing processinfluencing the mean reverting term, which is interpreted as the current renewable installed power.Using real data of electricity price, photovoltaic and wind energy production from the six mainItalian price zones, we estimate the parameters of the model and obtain quantitative results, such asthe production of photovoltaic energy impacts the North zone, while wind is significant for Sardiniaand the Central North zone does not present electricity price impact. Then we implement thesolution of the singular optimal control problem of installing renewable power production devices,in order to maximize the profit of selling the produced energy in the market net of installationcosts. We extend the results of [23] to the case when no impact on power price is presented, andto the case when N players can produce electricity by installing renewable power plants. We arethus able to describe the optimal strategy and compare it with the real installation strategy thatwas put in place in Italy. Keywords : singular stochastic control; irreversible investment; variational inequality; Ornstein-Uhlenbeck process; market impact; ARX model; Pareto optimality.
The paper [23] describes the irreversible installation problem of photovoltaic panels for an infinitely-lived profit maximizing power-producing company, willing to maximize the profits from selling elec-tricity in the market. The power price model used in that paper assumes that the company is a largemarket player, so its installation has a negative impact on power price. More in detail, the powerprice is assumed to follow an additive mean-reverting process (so that power price could possibly benegative, as it happens in reality), where the long-term mean decreases as the cumulative installationincreases. The resulting optimal strategy is to install the minimal capacity so that the power price isalways lower than a given nonlinear function of the capacity, which is characterized by solving an ordi-nary differential equation deriving from a free-boundary problem. The aim of this paper is to validateempirically that model, extending it also to wind power plants’ installation, by using time series of the ∗ [email protected] † [email protected] a r X i v : . [ m a t h . O C ] F e b ain Italian zonal prices and power production, and to assess how much the renewable installationstrategy in Italy deviated from the optimal one obtained by [23] in the period 2012–2018. In doingso, we also extend the theoretical results of [23] to the (easier) case when the amount of installedrenewable capacity has no impact on power prices, and to the case when several power producers arepresent in the market and their cumulative presence has a price impact.It is common in literature to model electricity prices via a mean-reverting behaviour, and toinclude (jump) terms representing the seasonal fluctuations and daily spikes, cf. [6, 9, 20, 28] amongothers. Here, in analogy with [23], we do not represent the spikes and seasonal fluctuations with thefollowing argument: the installation time of solar panels or wind turbines usually takes several days orweeks, which makes the power producers indifferent of daily or weekly spikes. Also, the high lifespan ofrenewable power plants and the underlying infinite time horizon setting allow us to neglect the seasonalpatterns. We therefore assume that the electricity’s fundamental price has solely a mean-revertingbehavior, and evolves according to an Ornstein-Uhlenbeck (O-U) process . We are also neglectingthe stochastic and seasonal effects of renewable power production. In fact, photovoltaic productionhas obvious seasonal patterns (solar panels do not produce power during the night and produce lessin winter than in summer), and both solar and wind power plants are subject to the randomnessaffecting weather conditions. However, since here we are interested to a long-term optimal behaviour,we interpret the average electricity produced in a generic unit of time as proportional to the installedpower. All of this can be mathematically justified if we interpret our fundamental price to be, forexample, a weekly average price as e.g. in [7, 19, 21], who used this representation exactly to get ridof daily and weekly seasonalities.In order to represent price impact of renewables in power prices, which is more and more observedin several national power markets, we follow the common stream in literature (also in analogy with[23]) and represent renewable capacity installation as a non-decreasing process, thus resulting in asingular control problem. This is also analogous to other papers modelling price impact: for example,in problems of optimal execution, [2] and [3] take into account a multiplicative and transient priceimpact, whereas [22] considers an exponential parametrization in a geometric Brownian motion settingallowing for a permanent price impact. Also, a price impact model has been studied by [1], motivatedby an irreversible capital accumulation problem with permanent price impact, and by [18], in whichthe authors consider an extraction problem with Ornstein-Uhlenbeck dynamics and transient priceimpact. In all of the aforementioned papers on price impact models dealing with singular stochasticcontrols [1, 2, 3, 18, 22], the agents’ actions can lead to an immediate jump in the underlying priceprocess, whereas in our setting, it cannot. Our model is instead analogous to [10, 11], which show howto incorporate a market impact due to cross-border trading in electricity markets, and to [26], whichmodels the price impact of wind electricity production on power prices. In these latter models, priceimpact is localized on the drift of the power price.In order to validate our model, we use a dataset of weekly Italian prices, together with photovoltaicand wind power production, of the six main Italian price zones (North, Central North, Central South,South, Sicily and Sardinia), covering the period 2012–2018. In principle, both photovoltaic and windpower production could have an impact on power prices, so we start by estimating parameters of anARX model where both photovoltaic and wind power production are present as exogenous variables:the parameters of this discrete time model will then be transformed in parameters for the continuoustime O-U model by standard techniques, see e.g. [8]. Unfortunately, for three price zones we find outthat our O-U model, even after correcting for price impacts, produce non-independent residuals. Thisis an obvious indication that the O-U model is too simple for these zones, and one should instead use We allow for negative prices by modelling the electricity price via an Ornstein-Uhlenbeck process. Indeed, negativeelectricity prices can be observed in some markets, for example in Germany, cf. [25]. ad-hoc derivation, which however results in a much more elementary derivation thanthe general result in [23]. More in detail, we obtain that the function of the capacity which shouldbe hit by the power price in order to make additional installation is in this case equal to a constant,obtained by solving a nonlinear equation. The corresponding optimal strategy should thus be to notinstall anything until the price threshold is hit, and then to install the maximum possible capacity.The second aim of our paper is to check the effective installation strategy, in the different pricezones, against the optimal one obtained theoretically. In doing so, we must take into account thefact that the Italian market is liberalized since about two decades, thus there is not a single producerwhich can impact prices by him/herself, but rather prices are impacted by the cumulative installationof all the power producers in the market. We thus extend our model by formulating it for N playerswho can install, in the different price zones, the corresponding impacting renewable power source,monotonically and independently of each other: the resulting power price will be impacted by thesum of all these installations, while each producer will be rewarded by a payoff corresponding to theirinstallation. The resulting N -player nonzero-sum game can be solved with different approaches. Aformulation requiring a Nash equilibrium would result in a system of N variational inequalities with N +1 variables (see e.g. [14] and references therein), which would be quite difficult to treat analytically.We choose instead to seek for Pareto optima. One easy way to achieve this is to assume, in analogywith [13], the existence of a ”social planner” which maximizes the sum of all the N players’ payoffs,under the constraint that the sum of their installed capacity cannot be greater than a given threshold(which obviously represents the physical finite capacity of a territory to support power plants of a giventype). We prove that, in our framework, this produces Pareto optima. More in detail, by summingtogether all the N players’ installations in the social planner problem, one obtains the same problemof a single producer, which has a unique solution that represents the optimal cumulative installationof all the combined producers. Though with this approach it is not possible to distinguish the singleoptimal installations of each producer, we can assess how much the effective cumulative installationstrategy which was carried out in Italy during the dataset’s period differs from the optimal one whichwe obtained theoretically.The paper is organized as follows. Section 2 presents the continuous time model used to char-acterize the evolution of the electricity price influenced by the current installed power and presentsthe procedure for parameter estimation. In Section 3 the model is estimated using real Italian dataand the pertinent statistical tests are applied for the validation of the model. Section 4 presents theset up for the singular control problem and its analytical solution, in both the cases with impact andwith no impact, for a single producer. Section 5 extends these results to the case when N players caninstall renewable capacity and derives corresponding Pareto optima. Section 6 compares the analyticaloptimal installation strategy obtained in Section 4 with the real installation strategy applied in Italy.Finally, Section 7 presents the conclusions of the work.3 The model
We start by presenting the model introduced in [23], which we here extend to more than onerenewable electricity source.We assume that the fundamental electricity price S x ( s ), in absence of increments on the level ofrenewable installed power, evolves accordingly to an Ornstein-Uhlenbeck (O-U) process (cid:40) dS x ( s ) = κ ( ζ − S x ( s )) ds + σdW ( s ) s > S x (0) = x , (1)for some constants κ, σ, x > ζ ∈ R , where ( W ( s )) s ≥ is a standard Brownian motion defined ona filtered probability space (Ω , F , P ), more rigorous definition and detailed assumptions will be givenin the next section.We represent the increment on the current installed power level with the sum of increasing processes Y y i i , where y i is the initial installed power and the index i stands for the renewable power source type,which in our case are sun and wind. We relate Y y with solar energy and Y y with wind energy.We assume that the increment in the current renewable installed power affects the electricity priceby reducing the mean level instantaneously at time s by (cid:80) i =1 β i Y y i i ( s ) for some β i > i ∈ { , } . Therefore the spot price S x,I ( s ) evolves according to (cid:40) dS x,I ( s ) = κ ( ζ − (cid:80) i =1 , β i Y y i i ( s ) − S x,I ( s )) ds + σdW ( s ) s > S x,I (0) = x. (2)The explicit solution of (2) between two times τ and t , with 0 ≤ τ < t is given by S x,I ( t ) = e κ ( τ − t ) S x,I ( τ ) + κ (cid:90) tτ e κ ( s − t ) ζ − (cid:88) i =1 , β i Y y i i ( s ) ds + (cid:90) tτ e κ ( s − t ) σdW ( s ) (3)= e κ ( τ − t ) S x,I ( τ ) + ζ (1 − e κ ( τ − t ) ) − κ (cid:90) tτ e κ ( s − t ) (cid:88) i =1 , β i Y y i i ( s ) ds + (cid:90) tτ e κ ( s − t ) σdW ( s ) . (4)The discrete time version of (4), on a time grid 0 = t < t < . . . , with constant time step ∆ t = t n +1 − t n results in the ARX(1) model X ( t n +1 ) = a + bX ( t n ) + (cid:88) i =1 , u i Z i ( t n ) + δ(cid:15) ( t n ) . (5)where X ( t ) , X ( t ) , X ( t ) , . . . and Z i ( t ) , Z i ( t ) , Z i ( t ) , . . . are the observation on the time grid, ofprocess S x,I and Y y i i respectively. The random variables ( (cid:15) ( t n )) n = { ,...,N } ∼ N (0 ,
1) are iid and thecoefficients a , b , u , u and δ are related with κ , ζ , β , β and σ by a = ζ (1 − e − κ ∆ t ) b = e − κ ∆ t u = − β (1 − e − κ ∆ t ) u = − β (1 − e − κ ∆ t ) δ = σ √ − e − κ ∆ t √ κ . (6)4he estimation of the discrete time parameters a , b , δ and u i , i = 1 , κ , ζ , σ and β i with i = 1 , In this section we estimate the parameters of the model in Equation (5) using real Italian data ofenergy price and current installed power.
We have data from six main price zones of Italy, which are North, Central North, Central South,South, Sicily and Sardinia. For every zone we have weekly measurements of average energy price in e /MWh, together with photovoltaic and wind energy production in MWh. The time series goes from07/05/2012 to 25/06/2018, week 19/2012 to 26/2018, corresponding to N = 321 observations. Thetime series of current photovoltaic and wind installed power is instead available with a much lowerfrequency (i.e. year by year). In order to obtain a time series consistent with the weekly granularityof price and production, we estimate the installed power to be proportional to the running maximumof the photovoltaic and wind energy production of whole Italy, respectively. Summarizing, we use forestimation of the model in Equation (5), for every particular zone, the data summarized in Table 1.VariableType Nomenclature DescriptionTime stepobservation t , . . . , t N Weeks when the quantities are observed, N = 321.Responsevariable X ( t ) , . . . , X ( t N ) Electricity price in e /MWh relative to an Italian pricezone.Explanatoryvariable Z ( t ) , . . . , Z ( t N ) Current installed photovoltaic power in MW, estimatedas Z ( t i ) = max( E ( t ) , . . . , E ( t i )), i ∈ { , . . . , N } ,where E ( t i ) is the sum of the produced energy on thesix zones at the observation time t i . Z ( t ) , . . . , Z ( t N ) Current installed wind power in MW, estimated as Z ( t i ) = max { E ( t ) , . . . , E ( t i ) } , i ∈ { , . . . , N } , where E ( t i ) is the sum of the produced energy on the six zonesat the observation time t i .Table 1: The data used for parameter estimation of Equation (5). Using ordinary least squares considering the data described above and then setting ∆ t = t i +1 − t i = for all i = 0 , . . . , one parameters Box Pierce test κ ζ β β σ p -valueNorth Value *** 10.6056 *** 133.0670 * 0.0148 0.0012 *** 47.7527 0.6101s.e. 2.1437 32.2392 0.0082 0.0031 2.3741Central North Value *** 10.9960 *** 120.4933 0.0112 0.0027 *** 45.5106 0.2702s.e. 2.1599 30.1593 0.0076 0.0029 2.1413Central South Value *** 13.2276 *** 100.3647 0.0052 ** 0.0056 *** 45.4237 0.0093s.e. 2.3958 27.3713 0.0069 0.0026 2.05040South Value *** 11.4996 *** 98.5810 0.0059 * 0.0047 *** 41.5805 0.0086s.e. 2.2004 26.9193 0.0068 0.0026 1.7715Sicily Value *** 14.1614 ** 173.0264 0.0124 *** 0.0107 *** 81.4377 0.0132s.e. 2.5146 46.9427 0.0120 0.0044 6.4833Sardinia Value *** 18.4580 *** 94.7809 0.0020 ** 0.0129 *** 68.2290 0.1216s.e. 2.9547 33.1946 0.0085 0.0031 4.2260 Table 2: Estimated parameters for the Ornstein Uhlenbeck. Significance code: *** = p < .
01, **= p < .
05, * = p < . p -values less than0 .
05. According to the results in Table 2, the Central South, South and Sicily zones present correlationin the residuals, therefore the proposed O-U model for electricity price is not the right choice. On theother hand the North, Central North and Sardinia zones have independent residuals implying that themodel is able to explain the behavior of the electricity price. Regarding the parameters significancefor this latter three zones, only the North and Sardinia zones present price impact: in the Norththere is only photovoltaic impact while in Sardinia only wind impact. We re-estimate the parametersconsidering only the zones which pass the Box-Pierce test and with only the significant price impactparameters. Table 3 summarizes the obtained results.
Zone parameters Box Pierce tests κ ζ β β σ p -valueNorth Value *** 10.3702 *** 140.5894 ** 0.0172 0 *** 47.6586 0.6206s.e. 2.0514 26.4732 0.0054 2.3747Central North Value *** 9.2648 *** 55.6085 0 0 *** 65.9346 0.2771s.e. 1.9273 2.8265 4.6367Sardinia Value *** 18.5248 *** 102.4620 0 *** 0.0123 *** 68.2889 0.1296s.e. 2.9510 6.6813 0.0017 4.2260 Table 3: Significant estimated parameters for Ornstein Uhlenbeck . Significance code: ∗∗∗ = p < . ∗∗ = p < . ∗ = p < . In this section we give the general set up and description for the singular control problem ofoptimally increasing the current installed power in order to maximize the profit of selling the producedenergy in the market net of the installation cost. This problem is completely described and solved in[23] when β >
0. However, the case when β = 0 can be obtained using the same procedure, whichwe describe in this section. Also we include a brief description and practical results of the case when β > .1 General set up and description of the problem Let (Ω , F , ( F t ) t ≥ , P ) be a complete filtered probability space where a one-dimensional Brownianmotion W is defined and ( F t ) t ≥ is the natural filtration generated by W , augmented by the P -nullsets.As we have already seen, only one type of energy influences the energy price in each price zone,therefore in the sequel we use the model in Equation (2) with only one single process influencing theprice dynamics. Therefore, let us define a stochastic process Y = ( Y y ( s )) s ≥ , with initial condition y ∈ [0 , θ ], representing the current renewable installed power of a company, which can be increasedirreversibly by installing more renewable energy generation devices, starting from an initial installedpower y ≥
0, until a maximum θ . This strategy is described by the control process I = ( I ( s )) s ≥ andtakes values on the set I [0 , ∞ ) of admissible strategies, defined by I [0 , ∞ ) (cid:44) { I : [0 , ∞ ) × Ω → [0 , ∞ ) : I is ( F t ) t ≥ - adapted , t → I ( t ) is increasing, cadlag,with I (0 − ) = 0 ≤ I ( t ) ≤ θ − y , ∀ t ≥ } . Hence the process Y is written as Y y ( t ) = y + I ( t ) . (7)As we already said, the aim of the company is to maximize the expected profits from selling theproduced energy in the market, net of the total expected cost of installing a generation device, whichfor an admissible strategy I is described by the following utility functional J ( x, y, I ) = E (cid:20)(cid:90) ∞ e − ρτ S x,I ( τ ) aY y ( τ ) dτ − (cid:90) ∞ ce − ρτ dI ( τ ) (cid:21) , (8)where ρ > c is the installation cost of 1 MW of technology, a > S x ( s )is the electricity price, with x as initial condition. The objective of the company is to maximize thefunctional in Equation (8) by finding an optimal strategy ˆ I ∈ I [0 , ∞ ) such that V ( x, y ) = J ( x, y, ˆ I ) = sup I ∈I [0 , ∞ ) J ( x, y, I ) . (9) β = 0 In this section we present the systematic procedure to construction the optimal solution andcharacterize the value function (9) when there is no impact price, i.e. when β = 0. Recall that in thiscase the electricity price evolves accordingly to the O-U process in Equation (1).Notice that, for a non-installation strategy I ( s ) ≡ ∀ s ≥
0, we have J ( x, y,
0) = E (cid:20)(cid:90) ∞ e − ρs S x ( s ) ayds (cid:21) = axyρ + κ + aζκyρ ( ρ + κ ) =: R ( x, y ) . (10)The possible strategies that the company can follows at time zero are: do not install during a timeperiod ∆ t and earn money selling the energy already installed, or immediately install more power.7he first strategy carries one equation which is obtained applying the dynamic programming principleand the second one carries another equation, obtained by perturbing the value function (9) in thecontrol. As a result we arrive to a variational inequality that the candidate value function w shouldsatisfy max (cid:18) L w ( x, y ) − ρw + axy, ∂w∂y − c (cid:19) = 0 , (11)with boundary condition w ( x, θ ) = R ( x, θ ) and the differential operator L is defined as L u ( x, y ) = σ ∂ u∂x + κ ( ζ − x ) ∂u∂x . (12)Equation (11) defines two regions: a waiting region W and an installation region I , given by W = (cid:40) ( x, y ) ∈ R × [0 , θ ) : L w ( x, y ) − ρw + axy = 0 , ∂w∂y − c < (cid:41) , (13) I = (cid:40) ( x, y ) ∈ R × [0 , θ ) : L w ( x, y ) − ρw + axy < , ∂w∂y − c = 0 (cid:41) . (14) Theorem 4.1 (Verification theorem) Suppose there exists a function w ∈ C , which solves Equation (11) with boundary condition w ( x, θ ) = R ( x, θ ) and satisfies the growth condition | w ( x, y ) | ≤ K (1 + | x | ) (15) for some K > : then w ≥ V on R × [0 , θ ] . Moreover, suppose that for every initial values ( x, y ) ∈ R × [0 , θ ) there exists a process ˆ I ∈ I , with associated trajectories ˆ Y y , such that (cid:16) S x ( s ) , ˆ Y y ( s ) (cid:17) ∈ ¯ W , ∀ s ≥ , P -a.s., (16)ˆ I ( s ) = (cid:90) s (cid:8) ( S x ( τ ) , ˆ Y y ( τ )) ∈ I (cid:9) d ˆ I ( τ ) , ∀ s ≥ , P -a.s. (17) Then V ( x, y ) = w ( x, y ) , ( x, y ) ∈ R × [0 , θ ] (18) and ˆ I is optimal. This theorem is a particular case of the Theorem 3.2 in [23], thus we invite the interested reader tothat reference for the proof. 8 .2.1 Constructing an optimal solution
Let us suppose that there exists a function F : [0 , θ ] → R which separates the installation andwaiting regions, called the free boundary, such that, W = (cid:8) ( x, y ) ∈ R × [0 , θ ) : x < F ( y ) (cid:9) , (19) I = (cid:8) ( x, y ) ∈ R × [0 , θ ) : x ≥ F ( y ) (cid:9) . (20)For all ( x, y ) ∈ W , the candidate value function w should satisfy L w ( x, y ) − ρw ( x, y ) + axy = 0 . (21)A general solution for (21) is w ( x, y ) = A ( y ) ψ ( x ) + B ( y ) φ ( x ) + R ( x, y ) , (22)where ψ ( x ) and φ ( x ) are the two strictly monotone and positive fundamental solutions of the homo-geneous equation L w ( x, y ) − ρw ( x, y ) = 0 (see Lemma 4.3 in [18] or Lemma A.1 in [23]). More indetail, ψ ( x ) is strictly increasing and positive and φ ( x ) strictly decreasing and positive (see Lemma4.3 in [18] or Lemma A.1 in [23]). Nevertheless, φ ( x ) growths exponentially when x → −∞ and dueto the linear growth of w and the form of the waiting region we should have B ( y ) = 0. The resultingsolution w is then given by w ( x, y ) = A ( y ) ψ ( x ) + R ( x, y ) , with ψ ( x ) given by ψ ( x ) = 1Γ( ρκ ) (cid:90) ∞ t ρκ − e − t − ( x − ζσ √ κ ) t dt (23)At the free boundary, w should satisfy ∂w ( x, y ) ∂y − c = 0 (24)and also ∂ w ( x, y ) ∂x∂y = 0 . (25)Using expression (22) in (24) and (25), and evaluating them in the free boundary x = F ( y ), we obtain A (cid:48) ( y ) ψ ( F ( y )) + R y ( F ( y ) , y ) − c = 0 , (26) A (cid:48) ( y ) ψ (cid:48) ( F ( y )) + R xy ( F ( y ) , y ) = 0 . (27)Differentiating (26) we obtain 9 = A (cid:48) ( y ) ψ (cid:48) ( F ( y )) F (cid:48) ( y ) + A (cid:48)(cid:48) ( y ) ψ ( F ( y )) + R yy ( F ( y ) , y ) + R xy ( F ( y ) , y ) F (cid:48) ( y )= F (cid:48) ( y )( A (cid:48) ( y ) ψ (cid:48) ( F ( y )) + R xy ( F ( y ) , y )) + A (cid:48)(cid:48) ( y ) ψ ( F ( y )) + R yy ( F ( y ) , y )= A (cid:48)(cid:48) ( y ) ψ ( F ( y )) , (28)where in the last equality we used Equations (28) and (10). As ψ is strictly positive for any x ∈ R ,we get A (cid:48)(cid:48) ( y ) = 0, which implies A ( y ) = ay + b . By imposing the condition A ( θ ) = 0, we get b = − aθ .From (26) we have for all y ∈ [0 , θ ) aψ ( F ( y )) + F ( y ) ρ + κ + ζκρ ( ρ + κ ) − c = 0 , (29)taking the derivative with respect to y , we get F (cid:48) ( y ) (cid:18) aψ (cid:48) ( F ( y )) + 1 ρ + κ (cid:19) = 0 . (30)Therefore from (30) it follows that ∀ y ∈ [0 , θ ) F ( y ) = C (31)or aψ (cid:48) ( F ( y )) + 1 ρ + κ = 0 . (32)From (32) we have a = − ρ + κ ) ψ (cid:48) ( F ( y )) (33)and the denominator is always non zero. Replacing (33) in (29), we get F ( y ) = c ( ρ + κ ) − ζκρ + ψ ( F ( y )) ψ (cid:48) ( F ( y )) . (34)Define now the function H ( x ) = c ( ρ + κ ) − ζκρ + ψ ( x ) ψ (cid:48) ( x ) − x , x ∈ R . (35) Proposition 4.2
There exist a unique solution ¯ x ∈ R to H ( x ) = 0 . roof. Set ˆ c = c ( ρ + κ ) − ζκρ . By Lemma 4.3 in [18], we know that for x ∈ R the derivative ψ ( k ) ( x ), forall k ∈ N ∪ { } is strictly convex and up to constant, it is the strictly increasing fundamental solutionof L u − ( ρ + kκ ) u = 0 and for any k ∈ N ∪ { } , φ k +2 ( x ) φ k ( x ) − φ k +1 ( x ) > , ∀ x ∈ R . Therefore wehave (cid:18) ψ ( x ) ψ (cid:48) ( x ) (cid:19) (cid:48) = ψ (cid:48) ( x ) − ψ (cid:48)(cid:48) ( x ) ψ ( x ) ψ (cid:48) ( x ) < , ∀ x ∈ R . (36)Hence H (cid:48) ( x ) = (cid:16) ψ ( x ) ψ (cid:48) ( x ) (cid:17) (cid:48) − < ∀ x ∈ R by (36), then H is decreasing. Observe that if x ≤ ˆ c , then H ( x ) >
0. If x > ˆ c + ψ (ˆ c ) ψ (cid:48) (ˆ c ) > ˆ c , then H ( x ) = ˆ c + ψ ( x ) ψ (cid:48) ( x ) − x < ψ ( x ) ψ (cid:48) ( x ) − ψ (ˆ c ) ψ (cid:48) (ˆ c ) < , (37)where the last inequality comes from the fact that ψ ( x ) ψ (cid:48) ( x ) is decreasing. Therefore, by continuity andmonotonicity of H we get the desired result.By Proposition 4.2 we conclude that the free boundary F ( y ) in this case is a constant function,whose value is given by F ( y ) = ¯ x , ∀ y ∈ [0 , θ ) . (38)and we have W = ( −∞ , ¯ x ) × [0 , θ ), I = [¯ x, + ∞ ) × [0 , θ ). Consider now the candidate value function w ( x, y ) = (cid:40) A ( y ) ψ ( x ) + R ( x, y ) , if ( x, y ) ∈ W ∪ ( { θ } × ( −∞ , ¯ x )) R ( x, θ ) − c ( θ − y ) , if ( x, y ) ∈ I ∪ ( { θ } × (¯ x, ∞ )) , (39)with R ( x, y ) defined in Equation (10), ψ ( x ) given by Equation (23) and A ( y ) given by A ( y ) = θ − y ( ρ + κ ) ψ (cid:48) (¯ x ) . (40)Equation (39) is in fact a solution of the variational inequality (11). We state this result in thefollowing lemma. Lemma 4.3
The function w ( x, y ) defined in (39) is a C , ( R × [0 , θ ]) solution to (11) , for all ( x, y ) ∈ R × [0 , θ ) with boundary condition w ( x, y ) = R ( x, θ ) . Proof.
Follows the arguments in [23]. 11 heorem 4.4
Recall the constant free boundary F ( y ) ≡ ¯ x and w ( x, y ) defined in (39) . The function w ( x, y ) identifies with the value function V from (9) , and the optimal installation strategy, denoted by ˆ I , is given by, ˆ I ( t ) = (cid:40) , t ∈ [0 , τ ) θ − y , t ≥ τ , (41) with τ = inf { t ≥ , X ( t ) ≥ ¯ x } . Proof.
See [18, 23]. β > In this case, the candidate value function w satisfies the variational inequalitymax {L w ( x, y ) − ρw ( x, y ) + axy, w y ( x, y ) − c } = 0 (42)with boundary condition w ( x, θ ) = R ( x, θ ), where R ( x, y ) := J ( x, y,
0) = axyρ + κ + aζκyρ ( ρ + κ ) − aκβy ρ ( ρ + κ )and L is the differential operator L u ( x, y ) = κ (( ζ − βy ) − x ) ∂u ( x, y ) ∂x + σ ∂ u ( x, y ) ∂x . (43)As in the case β = 0, Equation (42) defines two regions: a waiting region W and an installation region I , such that W = { ( x, y ) ∈ R × [0 , θ ) : xy − ρw ( x, y ) + L w ( x, y ) = 0 , w y ( x, y ) − c < } (44) I = { ( x, y ) ∈ R × [0 , θ ) : xy − ρw ( x, y ) + L w ( x, y ) ≤ , w y ( x, y ) − c = 0 } (45)which define when it is optimal to install more power or not. Again, these two regions are separated bythe strictly increasing function F : [0 , θ ] → R [23, Corollary 4.5], called the free boundary. Therefore W and I are written as W = { ( x, y ) ∈ R × [0 , θ ) : x < F ( y ) } , (46) I = { ( x, y ) ∈ R × [0 , θ ) : x ≥ F ( y ) } . (47)Now we can describe the optimal strategy using (46) and (47). When the current electricity price S x ( t ) is sufficiently low, such that S x ( t ) < F ( Y y ( t )), then the optimal choice is to not increment theinstalled power until the electricity price cross F ( Y y ( t )), passing to the installation region, where theoptimal choice is to increase the installed power in order to maintain the pair price-power ( S x ( t ) , Y y ( t ))12ot below of the free boundary. We explain again this strategy in Section 6 observing the numericalsolutions and graphics obtained for the Italian case.Setting ˆ F ( y ) = F ( y ) + βy , the free boundary is characterized by the ordinary differential equation[23, Proposition 4.4 and Corollary 4.5] ˆ F (cid:48) ( y ) = β × N ( y, ¯ F ( y )) D ( y, ¯ F ( y )) , y ∈ [0 , θ )ˆ F ( θ ) = ˆ x. (48)where N ( y, z ) = (cid:16) ψ ( z ) ψ (cid:48)(cid:48) ( z ) − ψ (cid:48) ( z ) (cid:17) (cid:18) ρ + 2 κρ ψ (cid:48) ( z ) + (cid:16) ( ρ + κ ) (cid:16) c − ˆ R ( z, y ) (cid:17) ψ (cid:48)(cid:48) ( z ) + ψ (cid:48) ( z ) (cid:17)(cid:19) ,D ( y, x ) = ψ ( x ) (cid:16) ( ρ + κ )( c − ˆ R ( x, y )) (cid:16) ψ (cid:48) ( x ) ψ (cid:48)(cid:48)(cid:48) ( x ) − ψ (cid:48)(cid:48) ( x ) (cid:17) + ψ ( x ) ψ (cid:48)(cid:48)(cid:48) ( x ) − ψ (cid:48) ( x ) ψ (cid:48)(cid:48) ( x ) (cid:17) and ˆ R ( x, y ) = aζκ + aρx − aβ ( ρ + 2 κ ) yρ ( ρ + κ ) . (49)On the other hand, the boundary condition ˆ x in (48) is the unique solution of ψ (cid:48) ( x )( c − ˆ R ( x, θ )) + ( ρ + κ ) − ψ ( x ) = 0 . (50) Remark 4.5
The solution ˆ x is such that ˆ x ∈ (cid:16) ¯ c, ¯ c + ψ (¯ c ) ψ (cid:48) (¯ c ) (cid:17) , with ¯ c = c ( ρ + κ ) − ζκ − β ( ρ +2 κ ) θρ [23,Lemma 4.2]. Remark 4.6
Observe that, when β = 0 , the boundary condition ˆ x correspond with the constant freeboundary obtained in Proposition 4.2. N producers As mentioned in the Introduction, Italy has a liberalized market, thus there is not a single producerwhich can impact prices by him/herself as is assumed in Section 4. Conversely, prices are impacted bythe cumulative installation of all the power producers which are present in the market. For this reason,we now consider a market with N producers, indexed by i = 1 , . . . , N . The cumulative irreversibleinstallation strategy of the producer i up to time s , denoted by I i ( s ), is an adapted, nondecreasing,cadlag process, such that I i (0) = 0. We assume that the aggregated installation of the N firms isallowed to increase until a total maximum constant power θ , that is, N (cid:88) i =1 ( y i + I i ( s )) ≤ θ P − a.s., s ∈ [0 , ∞ ) , (51)13here y i is the initial installed power for the firm i . We denote by I the set of admissible strategiesof all the players I (cid:44) { ¯ I : [0 , ∞ ) × Ω → [0 , ∞ ) N , non decreasing, left continuous adapted processwith I i (0) = 0, P -a.s., (cid:80) Ni =1 ( y i + I i ( s )) ≤ θ } . and notice that each player is constrained, in its strategy, by the installation strategies of the otherplayers. The social planner problem consists into finding a efficient installation strategy ˆ I ∈ I whichmaximizes the aggregate expected profit, net of investment cost [13]. This is expressed as V SP = sup ¯ I ∈I J SP ( ¯ I ) , (52)where J SP ( ¯ I ) = N (cid:88) i =1 J i ( I i ) (53)and for i = 1 , , . . . , N , J i ( t, x, I i ) = E (cid:20)(cid:90) ∞ e − ρτ S x ( s )( τ ) a ( y i + I i ( τ )) dτ − c (cid:90) ∞ e − ρτ dI i ( τ ) (cid:21) , (54)where ρ , a and c are the same defined in (8). The process S x ( s ) is the electricity price affected bythe sum of the installations of all the agents which, in analogy with the one-player case, we assume tofollow an O-U process with an exogenous mean reverting term, whose dynamics is given by (cid:40) dS x,I ( s ) = κ ( ζ − β (cid:80) Ni =1 ( y i + I i ( s )) − S x,I ( s )) ds + σdW ( s ) s > S x,I (0) = x. (55)Call now ν ( t ) = (cid:80) Ni =1 I i ( t ) and γ = (cid:80) Ni =1 y i : then, by substituting on the social planner functional(53), we get J SP ( ¯ I ) = N (cid:88) i =1 E (cid:20)(cid:90) ∞ e − ρτ S x ( τ ) a ( y i + I i ( τ )) dτ − c (cid:90) ∞ e − ρτ dI i ( τ ) (cid:21) (56)= E (cid:34)(cid:90) ∞ e − ρτ S x ( τ ) a (cid:32) N (cid:88) i =1 y i + N (cid:88) i =1 I i ( τ ) (cid:33) dτ − c (cid:90) ∞ e − ρτ d (cid:32) N (cid:88) i =1 I i ( τ ) (cid:33)(cid:35) (57)= E (cid:20)(cid:90) ∞ e − ρτ S x ( τ ) a ( γ + ν ( τ )) dτ − c (cid:90) ∞ e − ρτ dν ( τ ) (cid:21) . (58)Observe that we have the same optimal control problem as in the single company case (Section 4),therefore we can guess that the optimal solution for the social planner will be equal to that for thesingle company. The aggregate optimal strategy for the N producer of a given region results to bePareto optimal (see Lemma 5.1 below). 14 emma 5.1 If ˆ I ∈ arg max J SP ( I ) , then ˆ I is Pareto optimal. Proof.
Suppose ˆ I ∈ arg max J SP ( ¯ I ) and assume ˆ I is not Pareto optimal, then there exist I ∗ suchthat, J i ( I ∗ i ) ≥ J i ( ˆ I i ) , ∀ i ∈ { , . . . , N } (59)where at least one inequality is strict. Then, N (cid:88) i =1 J i ( I ∗ ) > N (cid:88) i =1 J i ( ˆ I ) , (60)contradicting the fact that ˆ I is maximizing.As already said in the Introduction, with this approach it is not possible to distinguish the singleoptimal installations of each producer, as we can only characterize the cumulative installation ν ( t ) = (cid:80) Ni =1 I i ( t ), while the single components I i ( t ) remain to be determined. However, our declared aimis about the effective cumulative installation strategy which was carried out in Italy during the timeperiod covered by the dataset. Thus, in the next section we compare this with the optimal one whichwe obtained theoretically. In this section we solve numerically Equation (48), using the parameters’ values estimated inSection 3 for the North, Central North and Sardinia zones. Recall from Table 3 that the price impactin the North zone is due to photovoltaic power production, while in Sardinia is due to wind powerproduction. Both are cases when the parameter impact is β >
0, which we describe in Section 4.1.2.On the other hand, Central North has not price impact from power production (at least from thesetwo renewable sources), so here we are in the case β = 0 described in Section 4.1.1.The parameters c and a presented in (8) should be selected according to the type of renewableenergy which has an impact on the corresponding price zone. In the case of photovoltaic power weconsider a yearly average of the installation cost of 1 MW of the prices available in the market, seee.g. [27]. On the other hand, for the wind power installation cost we consider the invested moneyon an offshore wind park that is being installed in Sardinia [24]. In both cases we adjust for thepresence of government incentives for renewable energy installation (usually under the form of taxbenefits), therefore we consider around a 40% and a 60% of the real investment cost c of photovoltaicand wind power, respectively, for our numerical simulation. The parameter a is the effective powerproduced during a representative year: as we consider a yearly scale for simulation, the parameter a will convert our weekly data of produced power into yearly effective produced power. Additionally,the a value depends on the type of produced power. This information is available e.g. in [17, Chapter4]. The parameter ρ is the discount factor for the electricity market and is the same in the threecases: no impact, photovoltaic and wind power impact. The parameter θ is the effective power thatcan be produced considering the real installed power of the respective type of energy. In the case ofthe estimated parameters κ , ζ , β and σ , we choose a value from the 95% confidence interval, based onbetter heuristic numerical performance simulation criteria.We summarize in Table 4 the parameters considered for the numerical simulations.15one Parameters’ values κ ζ β σ c a θ ρ North 6.7 124.7 0.0091 47.7 290000 1400 6500 0.1Central North 5.6029 50.2381 0 58.9796 290000 1400 6500 0.1Sardinia 13.213 115.1565 0.0091 68.2889 1944400 7508 5700 0.1Table 4: Parameter values used for the North, Central North and Sardinia zones.For the Central North case, we consider the cost of photovoltaic installation, because it is the mainrenewable energy produced in this zone.
We solved the ordinary differential equation (48) using the data in Table 4 for the North, usingthe backward Euler scheme with step h = 0 . F ( θ ) = 976 . e /MWh , which wasobtained by solving Equation (50) with the bisection method considering as initial points the extremesof the interval on Remark 4.5. The graph of the solution for the free boundary F ( y ) = x is presentedin Figure 1a, with a detail on realized power prices in Figure 1b. (a) Simulated free boundary and real data for theNorth (b) Detail of free boundary and real data for theNorth In Figure 1a, the point at zero installation level corresponds to F (0) = 64 . e /MWh. The redirregular line corresponds to the realized trajectory t → ( X ( t ) , Y ( t )), i.e. to the values of electricityprice vs effective photovoltaic installed power in the North: from it we can see that, at the beginningof the observation period (2012), the installed power was already around 3600 MW. Instead, theblue smooth line corresponds to the computed free boundary F ( y ) = x , which expresses the optimalinstallation strategy in the following sense: when the electricity price S x ( t ) is lower than F ( Y ( t )),i.e. when we are in the waiting region (see (46)), no installation should be done and it is necessaryto wait until the price S x ( t ) crosses F ( Y ( t )) to optimally increase the installed power level. Whenthe electricity price S x ( t ) is between F (0) and F ( θ ), enough power should be installed to move thepair price-installation in the up-direction until reaching the free boundary F . In the extreme case recall that Y is really just an estimation of the installed power, which is officially given with yearly granularity;moreover, Y is expressed in units of rated power, i.e. in production equivalent to a power plant always producing thepower Y S x ( t ) ≥ F ( θ ) the energy producer should install instantaneously the maximum allowed power θ . In the detailed Figure 1b we can observe the strategy followed in the North zone: the installationlevel from 3500 MW until 4500 MW was approximately optimal, in the sense that the pair price-installed power was around the free boundary F , with possibly some missed gain opportunities when,between 4300 and 4500 MW, the price was deep into the installation region; nevertheless, the rise inrenewable installation from 4500 MW to 4800 MW was at the end done with a power price whichresulted lower than what should be the optimal one. At around 4800 MW, there was an optimalno installation procedure until the price entered again the installation region: again, the consequentinstallation strategy was executed with some delay, resulting in a non-optimal strategy. At the end ofthe installation (around 5200 MW), we can see that the pair price-installed power moved again deepinto the installation region: we should then expect an increment in installation. In this case we do not have price impact, hence the constant free boundary F ( y ) ≡ ¯ x was obtainedsolving Equation (35). As before, we used the bisection method considering as initial points theextremes of the interval described in Remark 4.5. The obtained value is F ( y ) = ¯ x = 29 . e /MWh . (a) Simulated free boundary and real data for Cen-tral North (b) Detail of free boundary and real data for Cen-tral North In Figure 2a the vertical blue line corresponds to the constant free boundary ¯ x = 29 . e /MWh,while the red irregular line with the realized values of price-installation action that was put in place inthe Central North zone. In this case, the optimal strategy is described as follows: for electricity pricesless than ¯ x , no increments on the installation level should be done. Conversely, when the electricityprice is grater or equal to ¯ x the producer should increment the installation level up to the maximumlevel allowed for photovoltaic power (here we posed θ = 6500 MW). As we can clearly see on Figure 2a,the electricity price has always been greater than ¯ x in the observation period; however, the incrementson the installation level was not high enough to arrive to the maximum level θ = 6500 MW, thereforethe performed installation was not optimal. As in the North case, we solved the differential equation (48) using the data in Table 4 for Sardinia,using the backward Euler scheme with step h = 0 . F ( θ ) = 1453 . e /MWh , whichwas obtained by solving Equation (50) using the bisection method and considering as initial points17he extremes of the interval in Remark 4.5. The graph of the solution for the free boundary F ( y ) = x is presented in Figure 3a. (a) Simulated free boundary and real data for Sardinia (b) Detail free boundary and real data for Sardinia In Figure 3a the point at zero installation level corresponds to F (0) = 61 . e /MWh . The redirregular line corresponds with the realized values of electricity price vs effective wind installed powerin Sardinia, from which we can see that the installed wind power at the beginning of the observationperiod was already around 600 MW. The blue smooth line corresponds to the simulated free boundary F ( y ) = x , which expresses the optimal installation strategy as was already explained for the Northcase. In the detailed Figure 3b we can observe the strategy followed in the Sardinia zone: until the level1600 MW the power price was very deeply into the installation region, but the installation incrementswere not high enough to be optimal. Optimality came between the levels 1600 MW and 2400 MW,where the performed strategy was to effectively maintain the pair price-installed power around the freeboundary F . However, the subsequent increments were not optimal, in the sense that the installedpower was often increased in periods where the electricity price was too low, and in other situationsthe power price entered deeply in the installation region without the installed capacity following thattrend, or rather doing it with some delay. We must start by saying that we did not expect optimality in the installation strategy. In fact,firstly this strategy has been carried out by very diverse market operators, including hundreds of thou-sands of private citizens mounting photovoltaic panels on the roof of their houses, thus not necessarilyby rational agents which solved the procedure shown in Sections 4 and 5. Moreover, we must also saythat renewable power plants like photovoltaic panels or wind turbines often meet irrational resistancesby municipalities, especially when performed at an industrial level: more in detail, photovoltaic farmsare perceived to ”steal land” from agriculture (see e.g. [15]), while high wind turbines are genericallyperceived as ”ugly” (together with many other perceived drawbacks, see the exaustive monography[12] on this).Despite all these possible adverse effects we saw that, in the North and Sardinia price zones, partof the realized trajectory of power price and installed capacity was very near to the optimal freeboundary, while in other periods the installation was put in place in moments when power price wasnot the optimal one — possibly, the installation was planned when the power price was high anddeep into the installation region (time periods like this have been described both in the North as in18ardinia, see Sections 6.1 and 6.3) but the installation was delayed by adverse effects like e.g. the onesdescribed above. Summarizing, in these two regions the final installation level resulting at the end ofthe observation period (2018) seems consistent with the price levels reached during the period.It is instead difficult to reach such a conclusion in the Central North region: in fact, in that casethe realized trajectory of power price and installed capacity was always deeply into the installationregion, as the power price was always above the constant free boundary F ( y ) = ¯ x which resulted inthis case: the optimal strategy should then have been to install immediately the maximum possiblecapacity. We did observe a rise in installed renewable power during the period, which was obviouslynot optimal in the execution time (which spanned several years), given the peculiar nature of the freeboundary. However, in analogy to what already said for the North and Sardinia price zones, it ispossible that the performed installation, which at the end took place during the observation period,has been planned in advance but delayed by the same adverse effects cited above. We apply to real modeling and simulation the model presented in [23], which assumes that theelectricity price evolves accordingly to an O-U process and that it is affected by renewable power in-stallation on the mean-reverting term. The original model considers one big company that influencesthe electricity price with its activities. To be more realistic, we also study the case when N producershave an impact on electricity price by their aggregate installation. To solve this N -player noncooper-ative game, we use a ”social planner” approach as in [13] and maximize the aggregate utility of the N producers: this approach produces Pareto optima, and brings the problem back to the one-producercase.Using real data from the six main Italian price zones we found that, under an O-U model withan exogenous influence on the mean reverting term, there exists significant price impact of renewablepower production in the North and Sardinia zones. Also we found that for the Central North pricethere is not renewable production impact on power price, which is well described by an O-U modelwithout exogenous term.Once we solve numerically the ordinary differential equation for the free boundary or triggerfrontier, which describes when it is optimal to increment the installed power, we compare it with thereal installation strategy that was put in place in the North, Central North and Sardinia zones. Wefound that for the North the installation was optimal until the 4500 MW level, while in Sardinia theinstallation was optimal between 1600 MW and 2400 MW level. On the other hand, the capacityexpansion in Central North was executed but not in an optimal way, and the increment on theinstallation level should possibly be higher than what it was. We also present a discussion on this,stating some possible reasons why the installation has not been fully optimal. Acknowledgments.
T. Vargiolu acknowledges financial support from the research grant BIRD172407-2017 ”New perspectives in stochastic methods for finance and energy markets” and BIRD190200/19 ”Term structure dynamics in interest rates and energy markets: modelling and numerics”,both of the University of Padova. We wish to thank Giorgio Ferrari, Markus Fischer, KatarzynaMaciejowska and Rafa(cid:32)l Weron for useful comments and suggestions.
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