Combining simulation and optimization to derive operating policies for a concentrating solar power plant

Abstract

Optimizing short-term decisions over a rolling horizon and/or using deterministic penalties to capture system stochasticity can lead to myopic policies that fail to consider unplanned events and their long-term adverse effects. We present a methodology that integrates an off-line optimization model with a simulation procedure to determine the profitability of different operating strategies; specifically, the latter is used to generate additional constraints for the former when failures occur according to system component operating lifetimes that (i) are subject to exogenous uncertainty, and (ii) may degrade more quickly under specific operating conditions. We use the feedback provided by the simulation model in a parametric analysis to obtain penalties that can be used in short-term operations scheduling to maximize the long-term revenues obtained by the optimization model. We apply this research to a concentrating solar power plant; our results show that the methodology can be used to choose an operating policy that balances maximizing profit while accounting for maintenance costs. Integrating the optimization model with a simulation procedure reveals that aggressive prices for cycling yield about 55% fewer startups and 30% fewer failures compared to using a more typical start-up operating strategy, and can save hundreds of thousands to millions of dollars in repair costs over the lifetime of the plant.

Appendix: Dispatch Formulation

This appendix presents, verbatim, the dispatch model formulation ($\mathcal {O}^\mathcal {R}$), as summarized in the appendix of Wagner et al. (2018). Units, where appropriate, are provided in brackets next to the corresponding parameter or variable. (Initialization parameters used to set variable values at $t=0$ subscribe to variable notation and are not included here.)

Indices and sets
$t \in \mathcal {T}$	Set of all time steps in the time horizon, $T = |\, \mathcal {T}\ |$
Parameters
$C^{chs}$	Penalty for cycle start-up (from hot idle) [$]
$C^{csu}$	Penalty for cycle start-up (from 0) [$]
$C^{\delta W}$	Penalty for a positive change in electricity production [$/kWe]
$C^{rsu}$	Penalty for receiver start-up (from 0) [$]
$E^c$	Required energy consumed to start cycle [kWht]
$E^{hs}$	Heliostat field startup parasitic loss [kWhe]
$E^r$	Required energy consumed to start receiver [kWht]
$E^{rsb}$	Tower piping heat trace parasitic loss [kWhe]
$E^u$	Energy storage capacity [kWht]
$L^c$	Cycle pumping power per unit energy consumed [kWe/kWt]
$L^r$	Receiver pumping power per unit power produced [kWe/kWt]
$\mathbb {M}$	A sufficiently large number [-]
$P_t$	Electricity sales price in time t [$/kWhe]
$\bar{P}$	Mean sales price $\sum _{t \in \mathcal {T}} P_t / T$; [$/kWhe]
$Q^b$	Standby thermal power consumption per period [kWt]
$Q^c$	Allowable power per period for cycle start-up [kWt]
$Q^{in}_t$	Energy generated by the solar field in time t [kWt]
$Q^l$	Minimum operational thermal power input to cycle [kWt]
$Q^{rl}$	Minimum operational thermal power delivered by receiver [kWt]
$Q^{ru}$	Allowable power per period for receiver start-up [kWt]
$Q^u$	Cycle thermal power capacity [kWt]
$W^b$	Power cycle standby operation parasitic load [kWe]
$W^h$	Heliostat field tracking parasitic loss [kWe]
$W^l$	Minimum electric power output from cycle [kWe]
$\dot{W}^{min}_t$	Minimum net power production in time t [kWe]
$\dot{W}^{net}_t$	Net power production upper limit in time t [kWe]
$W^u$	Cycle electric power rated capacity [kWe]
$\alpha$	Unit corrector for binary variables in objective [$]
$\gamma _t$	Exponential time weighting factor [-]; $\varGamma ^{(t)}$, where $\varGamma \approx 0.99$
$\varDelta$	Time step duration [hr]
$\varDelta ^l$	Minimum duration of receiver start-up in period [hr]
$\varDelta ^{rs}_t$	Estimated fraction of time step t used for receiver start-up [-]
$\eta ^{amb}_t$	Cycle efficiency adjustment factor in time t [-]
$\eta ^c_t$	Normalized condenser parasitic loss in time t [-]
$\eta ^{des}$	Cycle nominal efficiency [-]
$\eta ^{p}$	Slope of linear approximation of power cycle performance curve [-]

Continuous variables
$x^r_t$	Thermal power delivered by the receiver at time t [kWt]
$x^{rsu}_t$	Receiver start-up power consumption at time t [kWt]
$u^{rsu}_t$	Receiver start-up energy inventory at time t [kWht]
$x_t$	Cycle thermal power consumption at time t [kWt]
$u^{csu}_t$	Cycle start-up energy inventory at time t [kWht]
$\dot{w}_t$	Electrical power generation at time t [kWe]
$\dot{w}^{\delta }_t$	Positive change in electricity production at time t [kWe]
$s_t$	Thermal energy storage reserve quantity at time t (auxiliary variable) [kWht]
Binary variables
$y^r_t$	1 if receiver is generating “usable” thermal power at time t; 0 otherwise
$y^{rsu}_t$	1 if receiver is starting up at time t; 0 otherwise
$y^{rsup}_t$	1 if receiver incurs a penalty for start-up at time t; 0 otherwise
$y_t$	1 if cycle is generating electric power at time t; 0 otherwise
$y^{csu}_t$	1 if cycle is starting up at time t; 0 otherwise
$y^{csb}_t$	1 if cycle is in standby mode at time t; 0 otherwise
$y^{csup}_t$	1 if cycle is starting up at time t from off state; 0 otherwise
$y^{chsp}_t$	1 if cycle is starting up at time t from standby mode; 0 otherwise

The objective function and constraints follow:

$$ \begin{gathered} ({\mathcal{O}}^{{\mathcal{R}}} )\,\,\,\,{\text{maximize}} \hfill \\\qquad\qquad\qquad \sum\limits_{{t \in {\mathcal{T}}}} [ \Delta \cdot P_{t} \left( {(1 - \eta _{t}^{c} )\dot{w}_{t} - L^{r} (x_{t}^{r} + x_{t}^{{rsu}} ) - L^{c} x_{t} } \right. \hfill \\ \qquad\qquad\qquad \left. { - W^{h} y_{t}^{r} - W^{b} y_{t}^{{csb}} - (E^{{rsb}} /\Delta + E^{{hs}} /\Delta )y_{t}^{{rsu}} } \right) \hfill \\\qquad\qquad\qquad - (C^{{rsu}} y_{t}^{{rsup }} + C^{{csu}} y_{t}^{{csup }} + C^{{chs}} y_{t}^{{chsp}} + C^{{\delta W}} \dot{w}_{t}^{\delta } ) \hfill \\\qquad\qquad\qquad + \gamma _{t} (\bar{P}\Delta x_{t}^{r} + \alpha y_{t}^{r} )] \hfill \\ \end{gathered} $$

(A.1)

subject to

(Receiver start-up)

$$\begin{aligned}&u^{rsu}_t \le u^{rsu}_{t-1} + \varDelta \cdot x^{rsu}_t\quad \forall t \in \mathcal {T} : t \ge 2 \end{aligned}$$

(A.2a)

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

(A.2b)

$$\begin{aligned}&y_t^r \le \frac{u^{rsu}_t}{E^r} + y^r_{t-1} \ \ \forall t \in \mathcal {T} : t \ge 2 \end{aligned}$$

(A.2c)

$$\begin{aligned}&y^{rsu}_t + y^r_{t-1} \le 1 \ \ \forall t \in \mathcal {T} : t \ge 2 \end{aligned}$$

(A.2d)

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

(A.2e)

$$\begin{aligned}\textsf {If} \quad Q^{in}_t = 0 \, \textsf {then:} \nonumber \\\qquad y^{rsu}_t = 0 \ \ \forall t \in \mathcal {T} \end{aligned}$$

(A.2f)

(Receiver supply and demand)

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

(A.3a)

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

(A.3b)

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

(A.3c)

$$\begin{aligned}&\textsf {If} \quad Q^{in}_t = 0 \, \textsf { then:} \nonumber \\&\qquad y^r_t = 0 \ \ \forall t \in \mathcal {T} \end{aligned}$$

(A.3d)

$$\begin{aligned}&y^{rsup}_t \ge y^{rsu}_t - y^{rsu}_{t-1} \ \ \forall t \in \mathcal {T} : t \ge 2 \end{aligned}$$

(A.3e)

(Cycle start-up)

$$\begin{aligned}&u^{csu}_t \le u^{csu}_{t-1} + \varDelta \cdot Q^c y^{csu}_t \ \ \forall t \in \mathcal {T}: t \ge 2 \end{aligned}$$

(A.4a)

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

(A.4b)

$$\begin{aligned}&y_t \le \frac{u^{csu}_t}{E^c} + y_{t-1} + y^{csb}_{t-1} \ \ \forall t \in \mathcal {T}: t \ge 2 \end{aligned}$$

(A.4c)

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

(A.4d)

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

(A.4e)

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

(A.4f)

(Power supply and demand)

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

(A.5a)

$$\begin{aligned}&\dot{w}^{\delta }_t \ge \dot{w}_t - \dot{w}_{t-1} \ \ \forall t \in \mathcal {T} : t \ge 2 \end{aligned}$$

(A.5b)

$$\begin{aligned}&\textsf {If} \quad \dot{W}^{net}_t \ge \dot{W}^{min}_t \ \textsf {then:} \nonumber \\&\qquad \dot{W}^{net}_t \ge (1-\eta ^{c}_t)\dot{w}_t - L^r (x^r_t + x^{rsu}_t) - L^c x_t - W^h y^r_t \nonumber \\&\qquad \qquad - \left( \frac{E^{rsb}}{\varDelta } + \frac{E^{hs}}{\varDelta } \right) y^{rsu}_t - W^b y^{csb}_t \ \ \forall t \in \mathcal {T} \end{aligned}$$

(A.5c)

$$\begin{aligned}&\textsf {else:} \nonumber \\&\qquad \dot{w}_t = 0 \ \forall t \in \mathcal {T} \end{aligned}$$

(A.5d)

(Logic governing cycle modes)

$$\begin{aligned}&y^{csu}_t + y_{t-1} \le 1 \ \ \forall t \in \mathcal {T}: t \ge 2 \end{aligned}$$

(A.6a)

$$\begin{aligned}&y^{csb}_t \le y_{t-1} + y^{csb}_{t-1} \ \ \forall t \in \mathcal {T}: t \ge 2 \end{aligned}$$

(A.6b)

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

(A.6c)

$$\begin{aligned}&y_t + y^{csb}_t \le 1 \ \ \forall t \in \mathcal {T} \end{aligned}$$

(A.6d)

$$\begin{aligned}&y^{csup}_t \ge y^{csu}_t - y^{csu}_{t-1} \ \ \forall t \in \mathcal {T}: t \ge 2 \end{aligned}$$

(A.6e)

$$\begin{aligned}&y^{chsp}_t \ge y_t - (1 - y^{csb}_{t-1}) \ \ \forall t \in \mathcal {T}: t \ge 2 \end{aligned}$$

(A.6f)

(Energy balance)

$$\begin{aligned} s_t - s_{t-1}= & {} \varDelta \cdot [ x^r_t - (Q^c y^{csu}_t + Q^b y^{csb}_t + x_t ) ] \nonumber \\&\qquad \forall t \in \mathcal {T}: t \ge 2 \end{aligned}$$

(A.7a)

$$\begin{aligned} x_{t+1} + Q^b y^{csb}_{t+1}\le & {} \frac{s_t}{\varDelta ^{rs}_{t+1}} - \mathbb {M} \cdot (-3 + y^{rsu}_{t+1} + y_t + y_{t+1} + y^{csb}_t + y^{csb}_{t+1}) \nonumber \\&\qquad \forall t \in \mathcal {T} \ : \ t \le {T-1} \end{aligned}$$

(A.7b)

(Variable bounds)

$$\begin{aligned}&x^{rsu}_t, u^{rsu}_t \ge 0 \ \ \forall t \in \mathcal {T} \end{aligned}$$

(A.8a)

$$\begin{aligned}&u^{csu}_t, \dot{w}_t, \dot{w}^{\delta }_t \ge 0 \ \ \forall t \in \mathcal {T} \end{aligned}$$

(A.8b)

$$\begin{aligned}&0 \le s_t \le E^u\ \ \forall t \in \mathcal {T} \end{aligned}$$

(A.8c)

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

(A.8d)

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

(A.8e)

We maximize the product of electricity price and power generation less parasitic losses and cost penalties; this represents revenue associated with electricity sales minus total costs, and each of these differences is summed over the time horizon. The final term in the objective function incentivizes energy dumping from the solar field – if necessary – to occur later in the time horizon to improve agreement between the expected net electricity production and the actual modeled value, as pumping parasitics associated with solar field operation are significant.

Constraint (A.2a) accounts for start-up energy “inventory”; we use an inequality to permit a zero value in time periods following completion of a start-up. Constraint (A.2b) ensures that inventory can only be positive when the receiver starts. Constraint (A.2c) allows the receiver to produce power under only one of the following two circumstances: (i) after the completion of a start-up or (ii) if, in the previous time step, the receiver had been operating. Constraint (A.2d) ensures that a receiver cannot be starting up and operating in power-producing mode simultaneously. Constraint (A.2e) limits the power used for receiver start-up based on the ramp rate. Constraint (A.2f) precludes receiver start-up in time periods with insufficient solar resource.

There is a limit on the total power produced by the receiver, which is adjusted for start-up energy consumption by Constraint (A.3a). Constraint (A.3b) ensures that the receiver must be in power-producing mode to generate thermal power. There is also a minimum limit on receiver energy generation by Constraint (A.3c) to adhere to molten-salt pump operating limits and heat transfer requirements. Constraint (A.3d) precludes power from being produced in the absence of available energy. Constraint (A.3e) governs logic associated with incurring a receiver start-up.

Inventory associated with start-up energy is accounted for by Constraint (A.4a); when the cycle is starting up, inventory can assume a positive value by Constraint (A.4b). Typical operation of the cycle can occur only after completion of start-up, or if the cycle had been operating or had been in stand-by mode in the previous time step, by Constraint (A.4c). Use of thermal power during cycle start-ups is curtailed in Constraint (A.4d), and thermal power use by the power cycle is restricted to a maximum value by Constraint (A.4e). Constraint (A.4f) ensures a lower limit on power production when the cycle is operating.

Electrical power production is given by a constraint that relates cycle performance to thermodynamic efficiency, allowing the model to capture the relationship between efficiency and thermal input while maintaining a linear relationship between the two. Constraint (A.5a) exploits this relationship; the power cycle does not always operate at maximum efficiency owing to other cost considerations accounted for by the model. To resolve the nonlinearities present in the relationship between thermodynamic efficiency and thermal input, we approximate electrical output with a linear function of cycle thermal power consumption, as given in Constraint (A.5a) where

$$\begin{aligned} \eta ^p = \frac{W^u - W^l}{Q^u - Q^l} \end{aligned}$$

(A.9)

Constraint (A.5b) determines a positive change in electrical power production in any given time step, which is discouraged in the objective function. Electrical power production is subject to a lower bound in Constraint (A.5c) which, if it lies below the minimum cycle power value, forces the cycle to be turned off; see Constraint (A.5d). We refer the reader to §2.4.4 of Wagner et al. (2017) for more detail.

Start-up mode cannot occur if the cycle is operating by Constraint (A.6a). Constraint (A.6b) enforces the same type of restriction for standby mode. The modes of standby and start-up cannot simultaneously occur (Constraint (A.6c)), nor can the modes of standby and power production (Constraint (A.6d)). Penalties of start-up from an off or standby state are incurred based on logic implemented in Constraints (A.6e) and (A.6f), respectively.

The charge state of thermal storage ($s_t$) is equal to the difference between the corresponding positive and negative amount of power. Related constraints enforce energy balance with respect to thermal energy storage, where the parameter $\varDelta$ converts from power to energy; see Constraint (A.7a). We account for the hourly time resolution of this model and the possible sub-hourly amount of time required to start the receiver while the power cycle is operating with Constraint (A.7b). If applicable, Equation (A.10) represents the fraction of each time step used for receiver start-up.

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

(A.10)

Thermal energy storage state of charge is calculated in every time step by Constraints (A.7a)-(A.7b).

Constraint (A.8a) imposes nonnegativity on receiver start-up power consumption and receiver start-up energy inventory. Constraint (A.3c) guarantees nonnegativity for $x^r_t$. Nonnegativity for cycle start-up energy inventory, electrical power generation, and positive change in electricity production is given by Constraint (A.8b). Nonnegativity for $x_t$ is ensured via Constraint (A.4f). Constraint (A.8c) bounds thermal energy storage in each time step. Constraints (A.8d) and (A.8e) enforce binary requirements.