Optimal sizing and dispatch of solar power with storage

Abstract

Designers of utility-scale solar plants with storage, seeking to maximize some aspect of plant performance, face multiple challenges. In many geographic locations, there is significant penetration of photovoltaic generation, which depresses energy prices during the hours of solar availability. An energy storage system affords the opportunity to dispatch during higher-priced time periods, but complicates plant design and dispatch decisions. Solar resource variability compounds these challenges, because determining optimal system sizes requires simultaneously considering how the plant will be operated under the imposed market and weather conditions. We develop an approach to analyze the economic performance of hybrid and single-technology solar power plants, which incorporates optimal dispatch, and considers the expected electricity market and weather conditions. We utilize the System Advisor Model software package to simulate the operation of multiple renewable generation and energy storage technologies, in conjunction with hourly-fidelity generation decisions determined by a revenue-maximizing, mixed-integer linear program. We show that, under our assumed market and weather conditions, the lifetime benefit-to-cost ratio can be improved by 6 to 19 percent, relative to a baseline design without optimizing, and that a concentrating solar power with thermal energy storage design produces significantly more energy per year, but is less profitable under our cost assumptions.

Appendix

1.1 Plant evaluation assumptions and case study inputs

Table 11 Molten salt power tower single-owner installation cost parameters used in the CSP system of the plant evaluation procedure

The cost assumptions used in this work are the default values in the corresponding PySAM technology model or PySSC simulation. These defaults are intended to represent conservative estimates for the current state of technology and market conditions (Gilman et al. 2020; NREL 2021).

Table 11 presents the molten salt power tower single-owner installation cost parameters used. All parameters are the defaults used in the PySSC molten salt power tower model.

The PV system uses SAM’s PVWatts model. We fix the array type to single axis tracking for all plants, and utilize the default cost assumptions in SAM version 2021.12.2 for all other parameters. HOPP utilizes a simplified model of PV field costs by default, which is based on the $51.38M installation cost of a benchmark 50$\hbox {MW}_\text {dc}$ field. The PV system cost is linearly scaled by the capacity multiplier $973/$\hbox {kW}_\text {dc}$, according to the cost of the benchmark system. For the battery system, we use the SAM default values of $233.17/kW and $241.79/kWh for the installation costs. Fixed maintenance costs are $15/kW-yr, and we assume a 10-year replacement period for the battery system at the original capacity-based installation cost (Smith et al. 2017).

1.2 Dispatch optimization model formulation

We provide here the complete problem formulation ($\mathcal {H}$), which is taken from Hamilton et al. (2022), and mimics the model given in Hamilton et al. (2019). We first define notation in Table 12:

Table 12 Hybrid dispatch model, ($\mathcal {H}$), notation

The following formulation, ($\mathcal {H}$), requires the initial operational state of the system, PV field and receiver energy generation forecasts, the expected cycle conversion efficiency profile as a function of ambient temperature and thermal input, and the energy price or desired load profile depending on the system analysis. Initialization parameters used to set variable values at $t=0$ follow variable notation and are not included here. Variables and parameters describe energy (thermal MWh$_\textsf {t}$ or electric MWh$_\textsf {e}$) states and power flows (thermal MW$_\textsf {t}$ or electric MW$_\textsf {e}$) in the system. We use lowercase letters to represent variables and capital letters for parameters. All binary variables are represented with some variant of the letter y.

1.2.1 Objective function and constraints

We present two possible objective functions. The first maximizes revenue less operations and maintenance costs resulting from the dispatch solution, which is appropriate in an independent-system-operator market. The second objective minimizes the cost of following a pre-determined load profile.

Maximize revenue

(9)

Minimize cost of a load profile

(10)

Constraints having terms indexed in period t-1, but applied to all periods $t \in \mathcal {T}$, use the corresponding initial condition parameter for the decision variable when $t=1$. All time-indexed decision variables requiring an initial value use the same notation for the parameter as the decision variable indexed in period $t=0$, e.g., the initial condition for the power cycle operating binary variable $y_t$ is the parameter $y_0$.

Receiver operations

Receiver Start-up

$$\begin{aligned} u^{rsu}_t&\le u^{rsu}_{t-1} + \varDelta x^{rsu}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(11a)

$$\begin{aligned} u^{rsu}_t&\le E^r y^{rsu}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(11b)

$$\begin{aligned} y_t^r&\le \frac{u^{rsu}_t}{E^r} + y^r_{t-1}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(11c)

$$\begin{aligned} y^{rsu}_t + y^r_{t-1}&\le 1&\forall&\ t \in \mathcal {T} \end{aligned}$$

(11d)

$$\begin{aligned} x^{rsu}_t&\le Q^{ru} y^{rsu}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(11e)

$$\begin{aligned} y^{rsu}_t&\le \frac{Q^{in}_t}{Q^{rl}}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(11f)

$$\begin{aligned} y^{rsup}_t&\ge y^{rsu}_t - y^{rsu}_{t-1}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(11g)

Receiver supply and demand

$$\begin{aligned} x^{r}_t + x^{rsu}_t&\le Q^{in}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(12a)

$$\begin{aligned} x^{r}_t&\le Q^{in}_t y^r_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(12b)

$$\begin{aligned} x^{r}_t&\ge Q^{rl} y^r_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(12c)

$$\begin{aligned} y^r_t&\le \frac{Q^{in}_t}{Q^{rl}}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(12d)

Power cycle operations

Cycle Start-up

$$\begin{aligned} u^{csu}_t&\le u^{csu}_{t-1} + \varDelta Q^c y^{csu}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(13a)

$$\begin{aligned} u^{csu}_t&\le E^{c} y^{csu}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(13b)

$$\begin{aligned} y_t&\le \frac{u^{csu}_{t}}{E^c} + y_{t-1}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(13c)

$$\begin{aligned} y^{csu}_t + y_{t-1}&\le 1&\forall&\ t \in \mathcal {T} \end{aligned}$$

(13d)

$$\begin{aligned} y^{csup}_t&\ge y^{csu}_t - y^{csu}_{t-1}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(13e)

Power supply and demand

$$\begin{aligned} x_t + \frac{E^c}{\varDelta } y^{csu}_t&\le Q^u&\forall&\ t \in \mathcal {T} \end{aligned}$$

(14a)

$$\begin{aligned} x_t&\le Q^u y_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(14b)

$$\begin{aligned} x_t&\ge Q^l y_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(14c)

$$\begin{aligned} \dot{w}_t&= \frac{\eta ^{amb}_t}{\eta ^{des}}[\eta ^p x_t + (W^u - \eta ^p Q^u)y_t]&\forall&\ t \in \mathcal {T} \end{aligned}$$

(14d)

$$\begin{aligned} x^{\delta }_t&\ge x_t - x_{t-1}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(14e)

$$\begin{aligned} \dot{w}^{l}_t&= \eta ^{c}_t \dot{w}_t \nonumber \\&\quad + L^r (x^{r}_t + x^{rsu}_t) + L^c (x_t + Q^c y^{csu}_t) \nonumber \\&\quad + W^h y^{r}_t + \frac{E^{hs}}{\varDelta } y^{rsu}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(14f)

TES energy balance constraints

$$\begin{aligned} s_t - s_{t-1}&= \varDelta [x^{r}_t - (Q^c y^{csu}_t + x_t)]&\forall&\ t \in \mathcal {T} \end{aligned}$$

(15a)

$$\begin{aligned} s_t&\le E^u&\forall&\ t \in \mathcal {T} \end{aligned}$$

(15b)

$$\begin{aligned} s_{t-1}&\ge \varDelta \cdot \varDelta ^{rs}_t [Q^u (-3 + y^{rsu}_t + y_{t-1} + y_t) + x_t]&\forall&\ t \in \mathcal {T} \end{aligned}$$

(15c)

PV constraints

$$\begin{aligned} \dot{w}^{pv}_t&\le W^{pv}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(16a)

$$\begin{aligned} \dot{w}^{pv}_t&\ge 0&\forall&\ t \in \mathcal {T} \end{aligned}$$

(16b)

Battery constraints

$$\begin{aligned} b^{soc}_{t}&= b^{soc}_{t-1} + \varDelta \left( \frac{\eta ^{+} \cdot \dot{w}^{+}_t - \frac{\dot{w}^{-}_t }{\eta ^{-}}}{C^{B}} \right)&\forall&\ t \in \mathcal {T} \end{aligned}$$

(17a)

$$\begin{aligned} \underline{S}^{B}&\le b^{soc}_{t} \le \bar{S}^{B}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(17b)

$$\begin{aligned} \underline{P}^{B} y_{t}^{-}&\le \dot{w}^{-}_{t} \le \bar{P}^{B} y_{t}^{-}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(17c)

$$\begin{aligned} \underline{P}^{B} y_{t}^{+}&\le \dot{w}^{+}_{t} \le \bar{P}^{B} y_{t}^{+}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(17d)

$$\begin{aligned} y_{t}^{+} + y_{t}^{-}&\le 1&\forall&\ t \in \mathcal {T} \end{aligned}$$

(17e)

$$\begin{aligned} b^{c}&\ge \frac{\varDelta }{C^B} \sum _{t \in \mathcal {T}}\dot{w}^-_t{} & {} \end{aligned}$$

(17f)

Grid constraints

$$\begin{aligned} \dot{e}^s_t - \dot{e}^p_t&= \dot{w}^{sg}_t - \dot{w}^{sl}_t&\forall&\ t \in \mathcal {T} \end{aligned}$$

(18a)

$$\begin{aligned} \dot{e}^s_t&\le W^{g}_t y^{g}_{t}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(18b)

$$\begin{aligned} \dot{e}^p_t&\le W^{l}_t (1 - y^{g}_{t})&\forall&\ t \in \mathcal {T} \end{aligned}$$

(18c)

System connection constraints

(19a)

(19b)

(19c)

(19d)

Decision variable bounds

$$\begin{aligned} s_t, \; u^{csu}_t, \; u^{rsu}_t, \; \dot{w}_{t}, \; \dot{w}^l_{t}, \; x_t, \; x^{\delta }_t, \; x^{r}_t, \; x^{rsu}_t&\ge 0&\forall&\ t \in \mathcal {T} \end{aligned}$$

(20a)

$$\begin{aligned} y_t, \; y^{csu}_t, \; y^{csup}_t, \; y^r_t, \; y^{rsu}_t, \; y^{rsup}_t&\in \{ 0,1\}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(20b)

$$\begin{aligned} b^{c}, \; b^{soc}_{t}, \; \dot{w}^{+}_{t}, \; \dot{w}^{-}_{t}&\ge 0&\forall&\ t \in \mathcal {T} \end{aligned}$$

(20c)

$$\begin{aligned} y^{+}_{t}, \; y^{-}_{t}&\in \{ 0,1\}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(20d)

$$\begin{aligned} \dot{w}^g_t, \; \dot{w}^l_t, \; \dot{e}^s_t, \; \dot{e}^p_t&\ge 0&\forall&\ t \in \mathcal {T} \end{aligned}$$

(20e)

$$\begin{aligned} y^{g}_{t}&\in \{ 0,1\}&\forall&\ t \in \mathcal {T} \end{aligned}$$

(20f)

1.2.2 Receiver operations

Constraint (11a) accounts for receiver start-up energy “inventory,” which can assume a positive value during time periods of receiver start-up (Constraint (11b)). Power production assumes a positive value only upon completion of a start-up or if the receiver also operates in the time period prior (Constraint (11c)). In the latter case, the receiver cannot be starting up in the next time period (Constraint (11d)). Ramp-rate limits hold during the start-up procedure (Constraint (11e)). The presence of trivial solar resource prevents receiver start-up (Constraint (11f)). Constraints (11g) ensure that penalties for receiver start-up are incurred.

The parameter $Q^{in}_t$ serves as an upper bound on the thermal power produced by the receiver, from which any energy used for start-up detracts (Constraint (12a)). Constraint (12b) permits the receiver to generate thermal power only while in power-producing mode. Receiver thermal power generation is subject to a lower bound by Constraint (12c). The receiver cannot operate (Constraint (12d)) in the absence of thermal power.

1.2.3 Power cycle operations

Constraint (13a) accounts for start-up energy “inventory,” which can only be positive during time periods in which the cycle is starting up (Constraint (13b)). Normal cycle operation can occur upon completion of start-up energy requirements or if the cycle is operating normally (Constraint (13c)). In the latter case, the cycle cannot start up in the time period directly following operation (Constraint (13d)). Cycle start up penalties are incurred via Constraint (13e).

Constraint (14a) limits the cycle input thermal power during periods when the cycle is starting up. This is a model approximation to derate power cycle output during startup periods. In reality, the cycle power output is not derated but the total energy production during the time period is reduced due to the time the cycle is starting up during that period. Constraint (14b) and Constraint (14c) form the upper and lower bounds on the heat input to the power cycle, respectively. The relationship between electrical power and cycle heat input is modeled as a linear function with corrections for ambient temperature effects (Constraint (14d)). Constraint (14e) measures the positive change in cycle thermal input, i.e., ramping, over time. Constraint (14f) calculates the CSP system load depending operational decisions.

1.2.4 Thermal energy storage balance

Constraints (15a)-(15c) measure TES state of charge via energy flow. Constraint (15a) balances energy to and from TES with the charge; a time-scaling parameter $\varDelta$ reconciles power and energy. Constraint (15b) imposes the upper bound to TES charge state. If the power cycle is operating in time periods $t-1$ and t, and if the receiver is starting up in time t, then there must be a sufficient charge level in the TES in time $t-1$ to ensure that the power cycle can operate through its start-up period (Constraint (15c)). Constraint (15c) uses $Q^u$ as a ‘big M’ value to make the constraint non-binding when the specific condition is not occurring, i.e., the cycle operating and receiver starting up in the same period. The expected fraction of a time period used for receiver start-up is given by (21), if applicable.

$$\begin{aligned} \varDelta ^{rs}_t = \min \left\{ 1, \max \left\{ \varDelta ^l, \frac{E^r}{ \max \left\{ \epsilon , Q^{in}_{t} \varDelta \right\} } \right\} \right\} \end{aligned}$$

(21)

1.2.5 PV field operations

Within the hybrid framework, we assume that the PV system has a take it or leave it policy in which the hybrid system can take up to the available generation at any time (Constraint (16a)) or curtail part or all of the available generation depending other system constraints. Non-negativity is enforced by Constraint (16b).

1.2.6 Battery operations

Battery state-of-charge must be updated (Constraint (17a)), and this quantity is bounded both below and above (Constraint (17b)). Power flow into and out of the battery is bounded by Constraints (17c) and (17d). The battery cannot be charging and discharging simultaneously (Constraint (17e)) while Constraint (17f) measures battery cycle count similar to what is done in Scioletti et al. (2017).

Constraint (18a) provides an energy balance at the transmission interconnect of the hybrid system. Electricity sales are limited by the transmission limit during periods in which the system is generating net power (Constraint (18b)). During periods in which the hybrid system net generation is negative, Constraint (18c) limits the load that the system can pull from the grid.

1.2.7 Grid operations

Constraints (19a) and (19b) enforce energy balance for the grid module on system generation and load, respectively. Constraints (19c) and (19d) are conditional constraints depending on specific requirements imposed on the hybrid system’s battery charging. Constraint (19c) limits battery charging to only electricity produced by the hybrid system locally, i.e., the battery cannot be charged by the electric grid. Constraint (19d) restricts battery charging to only electricity generated by the PV system. Constraint (19d) is more restrictive than Constraint (19c); therefore, Constraint (19c) can be omitted if Constraint (19d) is imposed. Variable bounds are enforced in constraints (20a) through (20f).