SMARTInvest: a stochastic, dynamic planning for optimizing investments in wind, solar, and storage in the presence of fossil fuels. The case of the PJM electricity market
 120 Downloads
 1 Citations
Abstract
In this paper, we present a stochastic dynamic planning model called SMARTInvest, which is capable of optimizing investment decisions in different electricity generation technologies. SMARTInvest consists of two layers: an optimization outer layer and an operational core layer. The operational model captures hourly variations of wind and solar over an entire year, with detailed modeling of dayahead commitments, forecast uncertainties and ramping constraints. The outer layer requires optimizing an unknown, nonconvex, nonsmooth, and expensivetoevaluate function. We present a stochastic search algorithm with an adaptive stepsize rule that can find the optimal investment decisions quickly and reliably. By properly capturing the marginal cost of investments in wind, solar and storage, we feel that SMARTInvest produces a more realistic picture of an optimal mix of wind, solar and storage, resulting in a tool that can provide more accurate guidance for policy makers.
1 Introduction
There has been a flurry of interest in determining whether we can meet the energy needs of our electrical grid purely through renewable sources (see [5, 12, 13]). Budischak et al. [7], for example, argues that renewables can provide as much as 99.9% of the energy required by a grid designed to serve 10% of the US population using nothing but wind, solar and a very large battery (with occasional hiccups that need to be covered by fossil generators).
We appreciate the value of this research which challenges us to think about renewables at dramatically higher levels than are present today. At the same time, we feel that it is valid to ask whether these levels are actually achievable if we challenge some of the assumptions made in their model of the problem. We applaud the study in [7] for using an adaptive simulation of wind, solar and loads, which is a significant improvement over the steady state analyses which simply use average production from solar and wind (typically obtained by multiplying the capacity times a “capacity factor”). For example, the Budischak simulator requires the use of storage, and provides an estimate of how much storage would have to be available to handle the variability of an allrenewable system serving virtually the entire load.
 1.
The Budischak model does not consider the marginal cost of wind, solar and storage investments. For example, a sufficiently large storage device would be at capacity only a tiny fraction of the time. Similarly, wind and solar generators run at their capacity only a small amount of time.
 2.
Given the high marginal cost of a pure renewable portfolio, we believe that fossil generation would retain a significant share of the generation stack even if incentives such as carbon taxes and lower utilization pushed the price of fossilgenerated electricity to very high levels. The problem is that fossil generation, in particular steam (which is the most economical, from both cost and pollution perspectives), requires some level of advance planning which introduces the issue of forecasting uncertainties when planning in the presence of wind and solar.
SMARTInvest then uses a stochastic search algorithm to optimize over different combinations of investments. Each evaluation of a set of investments uses a full simulation over the entire year, capturing all forms of variability as well as forecasting uncertainty. Parallel computation dramatically accelerates these intensive computations, allowing the model to search over different combinations of investments using a specially designed stochastic search algorithm.
Different ways to use SMARTInvest for a wide range of experiments
Generation  Complete  Aggregated  Single  Optimize  Fixed 

technology  stack  stack  price  capacity  capacity 
Wind  NA  NA  Yes  Yes  Yes 
Solar  NA  NA  Yes  Yes  Yes 
Battery  NA  NA  Yes  Yes  Yes 
Slow fossil (steam)  Yes  Yes  Yes  Yes  Yes 
Fast fossil (gas)  Yes  Yes  Yes  Yes  Yes 
Nuclear and other slow  Yes  Yes  Yes  No  Yes 
Other fast  Yes  Yes  Yes  No  Yes 
SMARTInvest is a contribution to the literature that optimizes renewable energy portfolios, a topic that is attracting growing attention. Prior to the [7] study, one of the most comprehensive studies was performed in [8, 11], which considers a portfolio focusing on wind, water and sunlight (WWS). A major problem faced by all portfolios involved high penetrations of renewables is handling the variability. These studies assume a conversion of a large portion of the transportation system to hydrogen, a fuel source that requires considerable energy to produce. Hydrogen (liquid or compressed gas) can be viewed as a form of storage, taking advantage of periods when wind and solar exceed the load. But this assumes adoption of hydrogen as a major fuel source, and as of this writing this seems unlikely. Ekren and Ekren [9] also considers the optimization of a solarwindstorage system. Other authors try to mitigate the variability of renewables (especially wind) by combining wind farms from wide areas—see [2, 14] and the references cited there. A more careful model of offshore wind integration using wind with fossil backup [23] showed that significant reserves are needed to handle the uncertainty of wind. This paper studied high levels of offshore wind using SMRTISO, and found that grid constraints limited integration at around 3GW of offshore generation.
Nahmmacher et al. [15] provide a method to select representative days to avoid the need to simulate an entire year, but this approach (which is quite popular) ignores the sequencing of decisions, where steam generation decisions are made 24 h in advance. To properly handle uncertainty, it is necessary to properly capture the lag between when decisions are made and when they impact the system (e.g. 24 h out for steam, 1 h out for gas turbines, and realtime ramping). Wogrin et al. [25] provide a more detailed unit commitment model, but make no attempt to handle the uncertainty in dayahead and hourahead forecasts. The most detailed unit commitment model we are aware of is reported in [23] called SMARTISO which carefully handles the scheduling of generators, and the uncertainty of dayahead and hourahead forecasts, but SMARTISO requires several hours to simulate a single week, which makes it impractical for doing portfolio optimization, where it is critical to handle seasonality.
SMARTInvest can be viewed as a more sophisticated version of the Budischak model, or a streamlined version of SMARTISO. It can be used to simulate an entire year in hourly increments, optimizing commitments to slow and fast fossil generators, and optimizing the use of storage over a 36 h horizon, while exploiting wind and solar energy as it becomes available. The model correctly handles advance commitments based on dayahead forecasts of wind and solar (for steam commitments) and hourahead forecasts for scheduling fast generators, and it handles ramping constraints at an aggregate level. These decisions require solving a linear program on a rolling basis to make commitments to steam generators 24 h in advance. This process is then embedded in a stochastic search algorithm for optimizing investments in wind, solar and storage, where we take advantage of the approximate convexity of the problem. We demonstrate that our search algorithm consistently finds nearoptimal mixes of wind, solar and storage given the capital costs of each of these resources, along with the cost of slow and fast fossil generators.

A true cost minimized solution (where fossil is part of the mix) that achieves 96% from renewables requires marginal fossil costs over $2000/MWh, and results in a levelized cost of $255/MWh.

Renewables do not enter the portfolio until the cost of energy from fossils hits $60/MWh, at which point it rises quickly to 30% of the portfolio using just wind generation.

Investments in solar do not enter the portfolio until fossil costs reach more than $300/MWh. Battery storage at $500/kWh does not enter the portfolio until fossil costs hit more than $500/MWh (these analyses ignore the use of batteries for other purposes such as frequency regulation, backup power and peak shifting to avoid transmission investment).

The investment in wind begins to increase when a CO\(_{2}\) tax of $50/ton is imposed, growing quickly and then leveling off for a tax above $70/ton.
A major contribution of this paper is the proper modeling of the sequencing of information and decisions, including the lag between when commitments are made and the information available when these decisions are made. We handle the need to make commitments for slow energy generation 24 h in advance, while fast generation is planned 1 h in advance. We model the evolution of forecasts, and propose a robust policy for handling the uncertainty. Considerable care has been invested in the notation, including the modeling of the stochastic base model and the lookahead model that forms the basis of our policy (a major point of confusion in the literature on stochastic unit commitment). We hope that our model may serve as a template for other researchers.
We begin in Sect. 2 by presenting the operating model which determines the mix of wind, solar, storage and fossil generation (fast and slow), given a fixed level of investment. We distinguish between the stochastic base model which spans an entire year, and the operating policy based on a cost function approximation in the form of a parametrically modified deterministic lookahead model. Then, Sect. 3 describes the search algorithm that optimizes the investment decisions. Section 4 presents a series of experiments to investigate the optimal generation mix under different market conditions. Section 5 concludes the paper.
2 The operating model
The operating model steps forward in hourly increments, optimizing over a 36 h horizon. The model determines how much energy to use from each of our seven generation technologies in \({\mathcal G}\) (wind, solar, battery storage, slow and fast fossil generation, and slow and fast nonfossil generation). Decisions for the available slow generation capacity are fixed 24 h out, which means that the amount of slow generation for hours 0–23 are capped by the fixed values from prior hours. We use actual wind forecasts from PJM’s forecasting vendor, as well as dayahead and shortterm solar forecasts. We use normalized historical wind and solar generation numbers for simulating wind and solar. The difference between the forecast and actual generation is the noise parameter. A tunable reserve parameter is used to produce robust policies in the face of forecasting uncertainties.
2.1 The stochastic base model
Throughout, we let w, s, and b be indices to represent wind, solar, and battery generation. Let \(\mathcal {G}^{slow}\) and \(\mathcal {G}^{fast}\) represent the mutually exclusive sets of slow generators (e.g. steam and nuclear) and fast generators (e.g. gas turbines), respectively. Also, let \({\mathcal G}=\{w,s,b\}\cup \mathcal {G}^{slow}\cup \mathcal {G}^{fast}\) be the set of sources of energy generation. New investment decisions can be made for up to (but not necessarily all) five generation technologies including wind, solar, battery, slow fossil, and fast fossil. Note that \(\mathcal {G}^{slow}\) and \(\mathcal {G}^{fast}\) can include the existing fossil and nonfossil slow and fast generators, as well as any new investment in slow and fast fossil generation. We represent the set of generation types for which we make an investment decision by \(\mathcal {I}\subseteq \mathcal {G}\).
We define the following notation.

\(g = \)The generic index for generator set \(\mathcal {G}\) or its subsets.

t \(=\) The hour for which we are solving the problem. This is a number in the range of 0–\(\mathcal {T}1\). Since we simulate for a whole year, \(\mathcal {T} = 365*24=8,760\).

\(n^{T}\) \(=\) The horizon of the problem, i.e. the number of hours in the future for which we tentatively plan.

\(n^{H}\), \((n^{H}\le n^{T})\) \(=\) The notification time, which is the number of hours that a slow generator has to be notified before it can be turned on.

\(x^{inv}_g\) \(=\) The capacity of generator type \(g\in {\mathcal G}\) (in MW). Although this is a decision variable of the investment problem (for \(g\in \mathcal {I})\), \(x^{inv}_{g}\) is a fixed parameter in the operating model. Wind and solar installed capacities are represented by \(x^{inv}_{w}\) and \(x^{inv}_{s}\), respectively.

\(c_{g}^{opr}\) \(=\) Operational cost of generating one MWh of electricity using generator \(g\in {\mathcal G}\) \((MWh)\).

\(r_{g}^{u}, r^{d}_{g}\) \(=\) Normalized ramp up and ramp down rates (as a percentage of the capacity) for generator \(g\in \mathcal {G}\).

\(e^{c}, e^{d}\) \(=\) Charging and discharging efficiency for the battery technology.

\(c^{cb}, c^{db}\) \(=\) Marginal cost of charging and discharging the battery (in MWh). These costs are set to zero in our experiments where we assume storage is a battery, but storage can come in many forms, some of which introduce actual storage costs.

\(n^{B}\) \(=\) The number of hours required to fully charge or discharge the battery. We use this rate along with the battery energy capacity to calculate power capacity of the battery (in MW).

\(R^{b}_{t}\) \(=\) State of charge of the storage device at time t (in MWh).

\(R^{slow}_{t,t',g}\) \(=\) The available capacity of slow generator \(g\in \mathcal {G}^{slow}\) for time \(t'\), as it is known at time t (in MW). If \(t't >24\), then this is equal to \(x^{inv}_{g}\). The available slow generation is fixed when \(t'=t+24\), and then remains the same for \(t't < 24\) due to notification time requirements.

\(q^{0}_{t,g}\) \(=\) The starting generation level for generator g at time t (in MW). This parameter is decided at time \(t1\).

\(f^{w}_{t,t'}\) \(=\) The normalized (i.e. as a percentage of capacity) wind forecast of hour \(t'\) measured at hour t (in MW). These data points are obtained from the official wind forecast figures of the PJM. In our model, we assume that, at each hour t, a wind forecast is available for 48 h in the future.

\(f^{s}_{t,t'}\) \(=\) Solar forecast of hour \(t'\) made at hour t, expressed as a percentage of the maximum solar output (in MW).

\(f^{d}_{t,t'}\) \(=\) Demand forecast of hour \(t'\) made at hour t.

\(q^{w}_{t}, q^{s}_{t}\) \(=\) The actual wind and solar generation at time t (in MW).

\(d_{t}\) \(=\) Demand in hour t (in MW).

\(x^{gen}_{t, g}\) \(=\) The amount of generation from g at time t (in MW).

\(x^{bc}_{t}, x^{bd}_{t}\) \(=\) The amount that the battery is charged from (discharged into) the market at time t (in MW).

\(x^{gen}_{t,b} \)=\( x^{bc}_{t}x^{bd}_{t}\).

\(R^{slow}_{t,t+n^{H},g}\) \(=\) The available slow generator g for time \(t+n^{H}\) , which is determined at time t (in MW).
 The battery status \(R^b_t\) evolves according to:$$\begin{aligned} R^{b}_{t+1} =R^{b}_{t}+e^{c}x^{bc}_{t}\frac{1}{e^{d}}x^{ bd}_{t} . \end{aligned}$$

Slow ramping generators require advance notification to prepare for generating the requested amount. Therefore, energy markets also run longerterm problems, for example a day ahead of actual generation (a dayahead problem). We plan generation over a 36 h horizon, but make commitments to slow units \(n^H = 24\) h into the future, meaning that slow generation may not be fully available for times less than 24 h in the future. The decision function \(X^{\pi }(S_{t}(x^{inv}_{g})_{g\in \mathcal {G}})\) determines the value of \(R^{slow}_{t,t+n^{H},g}\).
 Starting generation from g for hour \(t+1\) is equal to the generation level of g at hour t:$$\begin{aligned} q^{0}_{t+1,g} = x^{gen}_{t,b}. \end{aligned}$$
 The wind forecast, solar forecast, and generation noise parameters determine how wind and solar forecasts and generation processes update through time:$$\begin{aligned} f^{w}_{t+1,t'}= & {} f^{w}_{t,t'}+\varepsilon ^{wf}_{t+1,t'},\\ f^{s}_{t+1,t'}= & {} f^{s}_{t,t'}+\varepsilon ^{sf}_{t+1,t'},\\ f^{d}_{t+1,t'}= & {} f^{d}_{t,t'}+\varepsilon ^{df}_{t+1,t'},\\ q^{w}_{t+1}= & {} f^{w}_{t,t+1}+\varepsilon ^{w}_{t+1},\\ q^{s}_{t+1}= & {} f^{s}_{t,t+1}+\varepsilon ^{s}_{t+1}.\\ d^{}_{t+1}= & {} f^{d}_{t,t+1}+\varepsilon ^{d}_{t+1}. \end{aligned}$$
2.2 The hourly optimization model \(X^{\pi }_{t}(S_{t}\theta ,(x^{inv}_{g})_{g\in \mathcal {G}})\)
The lookahead model is formulated over a horizon \(n^T\) spanning \(t' = t, t+1, \ldots , t+n^T\). The model uses point forecasts of wind, solar and loads, but achieves robust behavior using a modified formulation which factors the forecast of energy from wind and solar to account for uncertainty. This adjustment, known as a parametric cost function approximation, has to be tuned using the stochastic base model represented by Eq. (1). Let us first define the following notation.

\(\theta \) \(=\) A parameter that factors the forecast of wind and solar to handle uncertainty (\(\theta \) \(=\) 1 implies no adjustment for uncertainty, \(\theta >1\) factors up the forecast to introduce a buffer to handle uncertainty).

\(\tilde{x}^{gen}_{t, t', g}\) \(=\) The decision variable of the hourly problem indicating the amount of generation from g at time \(t'\) (\(t\le t'< t+n^{T}\)) according to the hourly problem solved at hour t,

\(\tilde{x}^{bc}_{t,t'}, \tilde{x}^{bd}_{t,t'}\) \(=\) The amount that the battery is charged from (discharged into) the market at time \(t'\) according to the hourly problem solved at hour t,

\(\tilde{x}^{gen}_{t,t',b}\) \(=\) \(\tilde{x}^{bc}_{t,t'}\tilde{x}^{bd}_{t,t'}\),

\(\tilde{x}^{bs}_{t,t'}\) \(=\) The battery status (energy stored in the battery) at hour \(t'\) according to the hourly problem solved at hour t.

Equation (3)—Balancing constraint, i.e. total generation must be equal to demand at each point in time, where \(\theta \) is a parameter (typically greater than 1) that forces the system to schedule reserve generation to compensate for uncertainty.

Equation (4)—Battery generation can be negative (if charging), and we can represent it using two nonnegative variables \(\tilde{x}^{bd}_{t,t'}\) and \(\tilde{x}^{bc}_{t,t'}\).

Equation (5)—Battery status transition from 1 h to the next.

Equations (6)–(7)—Ramping constraints for all generators except slow generators.

Equations (8)–(9)—Ramping constraints for slow generators. These generators may not be available fully at a particular time, and thus the ramping is also proportional to their available capacity instead of \(x^{inv}_{l}\).

Equations (10)–(13)—Similar to (6), (7), (8) and (9), but for the first hour.

Equation (14)–(15)—Wind and solar generation must be less than or equal to the forecasted value.

Equation (16)—Available capacity for slow generators.

Equation (17)—Capacity constraint.

Equation (18)—Energy capacity constraint for the battery.

Equation (19)–(20)—Power capacity for charging or discharging the battery.

Equation (21)–(22)—Maximum amount of charging or discharging the battery is also constrained by the status of the battery. For example, if the battery is full, charging is not possible.

Equation (23)—Initial battery status.
3 The investment problem
We next consider the problem of finding the optimal mix of investment in generators \(\mathcal {I}\subseteq \mathcal {G}\ \) while minimizing the total yearly costs. Total cost includes prorated yearly cost of investment, operational costs (independent of generation amount), and cost of generation (e.g. fuel costs).
Before proceeding, we need to introduce some new notation.
3.1 The investment optimization algorithm
Assuming that generator installed capacities are known, we can simulate the yearly operation of the market by going forward in time and solving hourly problems for all hours \(t\in \mathcal {T}\). As we mentioned earlier, the purpose of this yearly simulation is to calculate total yearly generation cost (i.e. \(C_{}^{gen}((x^{inv}_{g})_{g\in \mathcal {I}}, (x^{inv}_{g})_{g\in \mathcal {G}\setminus \mathcal {I}})\)). Although we can calculate \(C^{gen}\) numerically, it is an unknown nonsmooth function, and in order to solve the investment problem (Eq. (26)) we need to develop an appropriate algorithm to solve this complex problem.
3.2 An adaptive stepsize rule

Superscript n: Iteration counter.

\(\alpha _{n}\): The stepsize at iteration n.

\(\delta ^{Max}\): The maximum change in each iteration (maximum step) in all directions.

\(\delta ^{Min}\): The minimum acceptable change in each iteration. If the change in all directions becomes smaller than \(\delta ^{Min}\), we accept that move regardless of whether it improves the objective value or not.
 1.
Initialization: Start with an initial capacity level (e.g. \(x^{0}_{g}=0, \forall g\in \mathcal {I}\)), and set \(n=0\).
 2.Numerically calculate \(\nabla _{} C_{}((x^{n}_{g})_{g\in I}) \) using finite differences:where, \(\epsilon \) is a small nonnegative parameter that needs to be tuned for an application (we used \(\epsilon = 1\) MW for SMARTInvest), and$$\begin{aligned} \nabla _{g'}C_{}((x^{n}_{g})_{g\in I})=\frac{C_{}(( x^{n}_{g}+\epsilon e_{g,g'})_{g\in I})C_{}((x^{n}_{g})_{g\in I})}{\epsilon }, \end{aligned}$$$$\begin{aligned} e^{}_{g,g'}= {\left\{ \begin{array}{ll} 0 &{} g\ne g',\ \\ 1\ &{} g=g'. \\ \end{array}\right. } \end{aligned}$$
 3.Set \(n=n+1\), and calculate \(\alpha ^{0}_{n}\) from (26) and (27), and set \(\alpha _{n}=\alpha ^{0}_{n}\).$$\begin{aligned} x^{n}_{g}(\alpha _{n})&=\max \left\{ 0,x_{g}^{n1}\alpha _{n}\nabla _{g}C_{}(x^{n1})\right\} \end{aligned}$$(26)$$\begin{aligned} \alpha ^{0}_{n}&=\left\{ \alpha _{n}:\max _{g}\left\{ \left x^{n}_{g} (\alpha _{n})x^{n1}_{g}\right \right\} =\delta ^{Max}\right\} \end{aligned}$$(27)
 4.
Calculate \((x_{g}^{n})_{g\in \mathcal {I}}\) from (26) and then compute \(C((x_{g}^{n})_{g\in \mathcal {I}})\).
 5.
If \(C((x_{g}^{n})_{g\in \mathcal {I}})\le C((x_{g}^{n1})_{g\in \mathcal {I}})\) or \(\max _{i}\left\{ \left x^{n}_{g} x^{n1}_{g}\right \right\} \le \delta ^{Min}\) go to step 2, otherwise, set \(\alpha _{n}=\frac{\alpha _{n}}{2}\), and go to step 4.
 6.
If the objective function has not improved in \(k^{max}\) iterations stop the algorithm.

Computing the gradient numerically is computationally expensive since it requires running the yearly simulation (which itself includes running almost 9000 hourly problems) several times (once for each decision variable). By only accepting improving steps, this algorithm reduces the number of iterations, and therefore decreases the number of times the gradient is computed. Note that finding an improving step does not require recomputing the gradient.

The large steps allow the algorithm to jump over the small bumps. The minimum stepsize ensure that, when it seems that we have already obtained the optimal solution, the algorithm travels to other points and checks new directions.
3.3 Parallelization
Running time for each function evaluation with and without parallelization (yearly vs. monthly) on a computer with 16 CPUs and 3059 MHz CPU speed
Without parallelization (s)  With parallelization (s)  

Complete offer stack  8738  1786 
Aggregated offer stack  135  30 
Optimal investment with and without parallelization
Without parallelization  With parallelization  

Wind (MW)  93718.15  93699.62 
Solar (MW)  8619.51  8610.29 
Battery (MWh)  6365.69  6442.68 
3.4 Algorithm performance
Figure 2a, b show two heat maps of the cost function on a two dimensional discretized grid. Each dimension spans for about 600 GW and the smallest unit in each direction of the grid is 3 GW. Therefore, in a two dimensional problem (e.g. the one in Fig. 2a) there are 40,000 grid points. In these heatmaps the cost value ranges from blue (minimum) to red (maximum). The first figure represents a case where the solar capacity is fixed, and the second figure corresponds to a case with a fixed wind capacity. The first observation is that although the cost function is not a convex function (because of small bumps), it looks like a convex function (in largescale) with very small nonconvexities (bumps). Thus, if our algorithm can avoid the small bumps, we can be hopeful to obtain a near optimal solution.
The algorithm results from different starting points (left) vs. best results from the grid search (right)
Optimal point (stochastic stepsize rule)  Best six points (grid search)  

Initial  Initial  Opt.  Opt.  Total cost  Opt.  Opt.  Total cost 
solar  batt.  solar  batt.  ($ \(\times 10^{11}\))  solar  batt.  ($ \(\times 10^{11}\)) 
0  0  121,862  279,313  1.0594  120,000  291,000  1.0607 
400,000  550,000  121,064  283,872  1.0596  120,000  282,000  1.0609 
300,000  300,000  121,693  280,869  1.0592  117,000  291,000  1.0611 
400,000  0  121,902  273,427  1.0605  120,000  279,000  1.0612 
100,000  100,000  118,847  217,409  1.0618  117,000  300,000  1.0614 
0  550,000  118,602  283,613  1.0598  111,000  279,000  1.0619 
The algorithm results from different starting points (left) vs. best results from the grid search (right)
Optimal point (stochastic stepsize rule)  Best five points (grid search)  

Initial  Initial  Opt.  Opt.  Total cost  Opt.  Opt.  Total cost 
wind  batt.  wind  batt.  ($ \(\times 10^{11}\))  wind  batt.  ($ \(\times 10^{11}\)) 
400,000  400,000  646,497  220,771  1.0709  657,000  215,000  1.0710 
0  0  664,763  215,292  1.0714  651,000  224,000  1.0716 
0  550,000  624,963  240,337  1.0713  654,000  209,000  1.0717 
600,000  200,000  656,181  215,311  1.0709  660,000  215,000  1.0717 
800,000  0  657,941  216,443  1.0706  654,000  218,000  1.0719 
To obtain a more precise understanding of the algorithm performance, we can compare the stopping points of our algorithm with the most costeffective solutions from the grid. Tables 4 and 5 compare the results of different runs of our algorithm with the best grid points (i.e. the least cost solutions from the heat map). According to these tables, our algorithm outperforms the best solution over the grid most of the time. When wind capacity is assumed to be a fixed number (Table 4), the algorithm provides better results than the best solution from the grid search for five out of six starting points. The remaining starting point which provides the worst algorithm performance among these six starting points yields a better objective function than the sixth best point on the grid. Table 5 presents similar results for a case with a fixed solar capacity. In this case as well, three out of five starting points lead to better solutions than the best solution over the grid.
These results show that our algorithm not only is effective and fast in solving this optimization problem, it can also produce better solutions than a coarse grid search.
3.5 The optimal reserve parameter (\(\theta \))
Optimal \(\theta \) for different levels of renewable integration and CO\(_{2}\) tax
CO\(_2\) tax  Wind  Solar  Batt.  Wind gen.  Opt. 

$/ton  MW  MW  MWh  (%)  \(\theta \) 
0  0  0  0  0.0  1.000 
50  51284  170  19738  19.0  0.985 
80  82066  3741  4556  29.8  0.998 
100  80250  6584  6005  29.2  1.001 
150  120251  33372  30939  42.0  1.020 
0  0  0  0  0.0  1.000 
0  51284  170  19738  18.8  1.191 
0  80250  6584  6005  29.2  1.041 
0  120251  33372  30939  40.7  1.197 
0  120251  33372  30939  40.7  1.197 
50  120251  33372  30939  41.0  1.134 
80  120251  33372  30939  42.2  1.011 
100  120251  33372  30939  42.1  1.011 
Table 6 consists of three different parts, each representing a separate experiment. Results of the first part of this table are obtained from two sets of experiements. First, CO\(_{2} \) tax is changed and the optimal level of renewable investment is calculated. Then, CO\(_{2} \) tax and the optimal levels are assumed to be a fixed number, as given in the table, and SMARTInvest is used to find the optimal \(\theta \) value for each market configuration. In the second part, we investigate the effect of larger renewable integration on the value of optimal \(\theta \). There is no CO\(_{2}\) tax in this case. In the third part, only CO\(_{2} \) tax changes and the renewable capacity is fixed.
If there are no renewables in the generation mix, we expect the optimal \(\theta \) to be equal to 1.0, because there is no uncertainty in such a system. As expected, the optimal \(\theta \) obtained from SMARTInvest is 1.0 for this case, as shown in Table 6. However, with larger amounts of renewables, one may expect to observe a larger \(\theta \) value, which is equivalent to planning generation reserves as a hedge against the risk of using expensive fast fossil generators to deal with unexpected variations in energy from wind and solar sources. Table 6 (second part) shows that this may not be always true. Planning for larger demand means preplanning larger amounts of slow generation and thus the system operator may end up using the more expensive slow fossil instead of wind in some time periods.
A tax on CO\(_{2}\) is another important parameter that affects the optimal \(\theta \) value. The result from the third part of this table shows that a larger CO\(_{2}\) tax results in smaller \(\theta \) values. This can be attributed to the fact that a larger CO\(_{2}\) tax, as we show in Sect. 4.3, causes less generation from slow fossil sources and more from fast fossil sources, and this increases the ability of the system to deal with uncertainty and decreases the need for preplanning and thus results in a lower \(\theta \) value.
4 Policy studies

Is it possible to generate a large percentage of our electricity needs (e.g. 99%) from renewable sources such as wind, solar, and battery?

What is the sensitivity of our results to cost parameters (e.g. for a future scenario where investment cost in solar is half of today’s cost)?

How much renewable investment is optimal in today’s market, and how much renewable incentives such as CO\(_{2}\) tax can affect the results?
Parameter values for different generation technologies
Wind  Solar  Battery  Slow Fossil  Fast Fossil  

Capital cost ($/MW)  2,213,000  3,873,000  500,000  1,023,000  676,000 
Yearly oper. cost ($/MW/yr)  39,550  24,690  0  15,370  7,040 
Life time (year)  30  30  15  30  30 
Ramp up rate (frac. of cap.)  1  1  1  0.038  1 
Ramp down rate (frac. of cap.)  1  1  1  0.037  1 
Battery parameters and interest rate
\(e^{c}\)  \(e^{d}\)  \(c^{cb}\)  \(c^{db}\)  \(n^{B}\)  r 

0.90  0.90  0.00  0.00  6.00  0.02 
4.1 Experiment 1: cost of fast fossil
For our first experiment, we find the optimal investment decisions for wind, solar, and battery technologies, assuming fast fossil generation is available at different cost levels and large capacity. This experiment also provides a comparison with Budischak’s paper, since the scenario with very large costs of fast fossil generation is equivalent to the model of [7].
We also calculate the levelized cost of electricity which is the average cost of generating one MWh of electricity using different technologies. We compute the levelized cost of electricity by dividing total generation cost (including investment and operational costs) over the length of the simulation by total met demand during the simulation. Note that investment and operational costs are calculated proportional to the length of the simulation, and we include the time value of money in our computations. The levelized cost of electricity at different fossil generation costs is given in Fig. 4.
At a cost of $2000/MWh for fossil generators, 96% of generation is from renewables, and this is equivalent to a levelized cost of $255/MWh. Therefore, one may conclude that figures such as 99% generation from renewables are not viable. The reason for the big discrepancy between this conclusion and that of the Budischak paper, we believe, is because a) we evaluate the value of investments on the margin and b) we require that the entire load be covered by our system, without having to resort to asking the grid to cover rare (but significant) outages. As long as the marginal cost of fast fossil generation is smaller than the value of lost load, we will not observe uncovered load in the solution.
As discussed before, different motivations can incentivize investment in renewables and the investment cost may not be the only determining factor. For example, tax credits, carbon taxes, or SREC requirements are some of the key factors. We have exclusively included renewable incentives such as emission taxes, and SREC prices in the SMARTInvest model, and many other factors (e.g. tax credits) can be modeled as reductions in the investment cost or simply as added wind and solar incentives. Investment costs and renewable incentives can also change in the future, and one may be interested in how markets respond to these changes. Figure 5 shows the renewable penetration plot under four scenarios. The wind investment cost of $500/kW represents the base case discussed earlier in this section. We produce the same plot for wind costs of $375/kW, and $250/kW, and also for a case with SREC prices of $100/MWh. This figure shows that, even if the cost of investment in wind is half of today or if we introduce other renewable incentives, reaching renewable integration above 80% is only optimal if the alternative flexible generation technology costs more than $300/MWh.
4.2 Experiment 2: cost of investment
In this section, we investigate the effect of investment cost on the optimal technology mix. The economics of renewables is fairly complex, and the investment in wind and solar is affected by a mixture of state requirements (RPS portfolios), tax credits and feedin tariffs. There is also some evidence of companies that are simply willing to pay more as a hedge against what they project is higher gas prices. So, the wind and solar might be more expensive for investment now, but because of hedging, people might still invest in them even without subsidies. Thus, it is interesting to investigate other investment cost scenarios to account for these incentives or as a hypothetical technology change in the future resulting in a reduction in investment costs.
Figures 6 and 7 compare optimal wind, solar, and battery capacities under different levels of wind, solar and battery investment costs. The base cost values are given in Table 7, and we multiply these numbers by a factor to account for different cost levels.
Simulations testing the effect of battery costs on investment in wind and solar indicate that battery costs have very little impact on the optimal investment in renewables. However, Fig. 6 shows the impact of battery costs on the optimal investment in battery.
An interesting observation from Fig. 7 is that total generation from renewables is nearly the same under different scenarios. It only decreases slightly for higher levels of wind and solar investment costs (e.g. it has a very shallow slope as a function of wind investment cost in Fig. 7). However, a change in the investment cost can significantly impact total investment in renewables (i.e. sum of wind and solar capacities). As shown in Fig. 7, the amount of investment in wind alone is in the range of 300–1000 GW, which is several times more than total demand (which is around 100–150 GW), and the same can be said about solar and battery. So in most time periods, a change in wind or solar capacity does not change their generation (because they already generate more than load), but in some periods with a very small wind and solar availability, the amount of installed capacity can make a difference. One may conclude that most of the changes in investment (due to various levels of investment costs) are to avoid the high cost of $400/MWh only for a small percentage of time periods.
Another interesting observation from Fig. 7 is that a battery investment is more responsive to solar capacity rather than wind capacity. When solar capacity increases (and thus wind capacity decreases), investment in the battery also increases. This may indicate that the existence of some storage is more critical for solar generation. This can be attributed to the fact that solar generation is not available at all during the night time.
These results also show that an increase in wind investment cost is equivalent to lower investment in wind and higher investment in solar and vice versa. Also, battery capacity, as expected, decreases, when the cost of investment in the battery increases.
4.3 Experiment 3: carbon tax
In the previous experiments, we assumed that unlimited fossil generation is available at a fixed cost level. When concerned with large levels of renewable integration into electricity markets, this seems to be a logical assumption, since many of the conventional sources of electricity generation may not be fast and large enough to cope with huge variability and unpredictability resulting from large integration of intermittent renewables.
Figure 8 shows the results of an experiment with the current market configuration (before and after CO\(_{2}\) tax). In this experiment, we optimize investment decisions for wind, solar, and battery, while we assume the other types of generation have a constant capacity equal to their current installed capacity in PJM. According to this figure, no investment in wind, solar, and battery is economically efficient in today’s market conditions. However, when CO\(_2\) tax increases to $50/ton, about 10% generation from wind appears in the optimal solution, and at $70/ton 30% of generation is met by wind. Prices above $100/ton is required to observe some generation from solar and battery. This figure also shows that higher CO\(_2\) tax values do not change total generation from nuclear energy greatly, but has a dramatic effect on slow fossil generation. In effect, slow fossil is replaced with fast fossil and wind while CO\(_2\) tax increases from $0/ton to $150/ton.
Another observation is that the percentage of nuclear generation is decreased at the CO\(_{2}\) tax level of $150/ton. Also, slow fossil completely vanishes from the optimal mix after leveling off at lower tax levels. To explain this, we first note that the marginal cost of wind and solar is zero, while it is around $9/MWh for nuclear and is more expensive for slow fossil generators. At high levels of CO\(_{2}\) tax, investment in wind is large enough to replace a large portion of more expensive slow generation. Among slow generation types, slow fossil has become very expensive because of the CO\(_{2}\) tax, and is removed first from the energy mix. When we have a large investment in wind e.g. more than total load, less expensive (but more expensive than wind) nuclear generation can also be replaced with wind at some time periods.
5 Conclusion
In this paper, we introduced a new model (called SMARTInvest) that is able to find the optimal generation configuration under different market conditions. It is a very flexible model, and can be used for a wide range of experiments. SMARTInvest can model the generation technologies using the existing offer stack, an aggregated version of offer stacks, or a single offer band. It can also optimize for up to five technology types. SMARTInvest provides a flexible framework for a wide range of experiments.
The mathematical model of SMARTInvest carefully captures the distinction between a stochastic base model and a robust operating policy using a parametric cost function approximation in the form of a modified deterministic lookahead model. This is widely used in practice, but this appears to be the first time the entire process has been modeled formally. We then describe a stochastic optimization algorithm for the investment problem which exploits the approximate convexity of the problem while overcoming the finegrained nonconvexities using a carefully designed stepsize algorithm. The algorithm makes it possible to find nearoptimal solutions with only a few hundred function evaluations.
We also design a few experiments to gain insights into the optimal generation mix under different market conditions. Firstly, we show that reaching really large levels of renewable integration (e.g. 99% as claimed in the literature) is very expensive socially and so may not happen in a real electricity market, although it is possible to reach relatively largescale wind integration. We also evaluate the market conditions for reaching such levels. In addition to this, we investigate the level of wind and solar integration under different investment cost scenarios. The results of our experiments, for example, show that the battery investment cost is not a very important parameter in determining the optimal wind and solar capacities. Even more interesting, different levels of investment cost in battery or wind do not greatly change the percentage of generation from renewables. A higher cost of investment in, say, wind, means more generation from solar and less from wind, but the total generation percentage does not change greatly. However, this amounts to a noticeable change in total renewable installed capacity. We also analyse the current PJM electricity market and investigate the effect of CO\(_2\) cost on the amount of renewable generation. It turns out that without renewable incentives such as CO\(_2\) tax or SREC markets, no investment in renewables is economical; however, for higher levels of these economical incentives, we observe that less generation from slow fossil and more from renewables and fast fossil appears in the optimal solution.
We would like to emphasize that this research is not about predicting the future and more about what level of renewable energy integration is socially optimal. That is why a costminimization approach is used in this paper. This model is striking a balance between the very simplistic Budischak model, and much more detailed models such as SMARTISO (which still does not model market behavior, and is not appropriate for this study). It is important to remember that all models have to make approximations to handle the questions they are trying to answer. Readers can refer to [10] for more information on the market behaviour.
The ongoing changes in the climate have elevated the risk of integrating large amounts of renewable energy into the power mix. Future extensions of SMARTInvest can include the addition of riskaverse optimization methods (see e.g. [3, 16]).
References
 1.Archer, C., Simao, H. P., Kempton, W., Powell, W., Dvorak, M. (2015). The challenge of integrating offshore wind power in the US electric grid. part I: Wind forecast error. Princeton University, Dept. of Operations Research and Financial EngineeringGoogle Scholar
 2.Archer, C. L., Jacobson, M. Z.: Supplying baseload power and reducing transmission requirements by interconnecting wind farms. J. Appl. Meteorol. Climatol 46(11), 1701–1717. ISSN:15588424 (2007)Google Scholar
 3.Asamov, T., Ruszczyski, A.: Timeconsistent approximations of riskaverse multistage stochastic optimization problems. Math. Program (2014). ISSN:00255610. doi: 10.1007/s101070140813x
 4.Bazaraa, M.S., Sherali, H.D., Shetty C.M.: Nonlinear programming: theory and algorithms. Wiley, Hoboken, NJ (2013)Google Scholar
 5.Becker, S., Frew, B., Andresen, G., Zeyer, T., Schramm, S., Greiner, M., Jacobson, M.: Features of a fully renewable US electricity system: Optimized mixes of wind and solar PV and transmission grid extensions (2014). arXiv preprint arXiv:1402.2833
 6.Bertsekas, D.P.: Nonlinear programming. Athena scientific, Belmont (1999)Google Scholar
 7.Budischak, C., Sewell, D., Thomson, H., Mach, L., Veron, D.E., Kempton, W.: Costminimized combinations of wind power, solar power and electrochemical storage, powering the grid up to 99.9% of the time. J. Power Sour. 225, 60–74 (2013)CrossRefGoogle Scholar
 8.Delucchi, M.A., Jacobson, M.Z.: Providing all global energy with wind, water, and solar power, Part II: Reliability, system and transmission costs, and policies. Energy Policy 39(3), 1170–1190 (2011). ISSN:03014215. doi: 10.1016/j.enpol.2010.11.045
 9.Ekren, O., Ekren, B.Y.: Size optimization of a PV/wind hybrid energy conversion system with battery storage using simulated annealing. Appl. Energy 87(2), 592–598 (2010). ISSN:03062619. doi: 10.1016/j.apenergy.2009.05.022
 10.Gabriel, S.A., Conejo, A.J., Fuller, J.D., Hobbs, B.F., Ruiz, C.: Complementarity modeling in energy markets, vol. 180. Springer Science & Business Media (2012)Google Scholar
 11.Jacobson, M.Z., Delucchi, M.A.: Providing all global energy with wind, water, and solar power, Part I: Technologies, energy resources, quantities and areas of infrastructure, and materials. Energy Policy 39(3), 1154–1169 (2011). ISSN:03014215. doi: 10.1016/j.enpol.2010.11.040
 12.Jacobson, M.Z., Delucchi, M.A., Ingraffea, A.R., Howarth, R.W., Bazouin, G., Bridgeland, B., Burkart, K., Chang, M., Chowdhury, N., Cook, R., et al.: A roadmap for repowering California for all purposes with wind, water, and sunlight. Energy 73, 875–889 (2014)Google Scholar
 13.Jacobson, M.Z., Howarth, R.W., Delucchi, M.A., Scobie, S.R., Barth, J.M., Dvorak, M.J., Klevze, M., Katkhuda, H., Miranda, B., Chowdhury, N.A., et al.: Examining the feasibility of converting New York state‘s allpurpose energy infrastructure to one using wind, water, and sunlight. Energy Policy 57, 585–601 (2013)Google Scholar
 14.Kempton, W., Pimenta, F.M., Veron, D.E., Colle, B.A.: Electric power from offshore wind via synopticscale interconnection. Proc. Natl. Acad. Sci. 107(16), 7240–7245 (2010)CrossRefGoogle Scholar
 15.Nahmmacher, P., Schmid, E., Hirth, L., Knopf, B.: Carpe diem: a novel approach to select representative days for longterm power system modeling. Energy 112, 430–442 (2016)CrossRefGoogle Scholar
 16.Philpott, A., de Matos, V., Finardi, E.: On solving multistage stochastic programs with coherent risk measures. Oper. Res. 61(4), 957–970 (2013)MathSciNetCrossRefMATHGoogle Scholar
 17.Powell, W.B.: Clearing the jungle of stochastic optimization. Informs Tutor. Oper. Res. 2014, 109–137 (2014)Google Scholar
 18.Powell, W.B., Meisel, S.: Tutorial on Stochastic Optimization in Energy II : an energy storage illustration. IEEE Trans. Power Syst XX(X), 1–8 (2015)Google Scholar
 19.Regis, R.G., Shoemaker, C.A.: Constrained global optimization of expensive black box functions using radial basis functions. J. Glob. Optim. 31(1), 153–171 (2005)MathSciNetCrossRefMATHGoogle Scholar
 20.Regis, R.G., Shoemaker, C.A.: Improved strategies for radial basis function methods for global optimization. J. Glob. Optim. 37(1), 113–135 (2007)Google Scholar
 21.Regis, R.G., Shoemaker, C.A.: A stochastic radial basis function method for the global optimization of expensive functions. Inf. J. Comput. 19(4), 497–509 (2007)Google Scholar
 22.Ryan, S.M., Wets, R.J.B., Woodruff, D.L., SilvaMonroy, C., Watson, J.P.: Toward scalable, parallel progressive hedging for stochastic unit commitment. In: Power and Energy Society General Meeting (PES), 2013 IEEE, pp. 1–5. IEEE (2013)Google Scholar
 23.Simao, H.P., Powell, W., Archer, C., Kempton, W.: The challenge of integrating offshore wind power in the US electric grid. part II: Simulation of electricity market operations. Renew. Energy 103, 418–431 (2017)CrossRefGoogle Scholar
 24.Takriti, S., Birge, J.R., Long, E.: A stochastic model for the unit commitment problem. Power Syst. IEEE Trans. 11(3), 1497–1508 (1996)CrossRefGoogle Scholar
 25.Wogrin, S., Galbally, D., Ramos, A.: CCGT unit commitment model with firstprinciple formulation of cycling costs due to fatigue damage. Energy 113, 227–247 (2016)CrossRefGoogle Scholar