Abstract

Solar tower with thermal energy storage (ST-TES) represents a promising technology for large-scale exploitation of solar irradiation for electricity generation. A ST-TES has the potential to extend electricity generation to more favorable conditions, such as high electricity prices. The size of TES, however, constrains the flexibility of dispatching, especially when there is significant uncertainty in forecasts of solar irradiation and electricity prices. This study explores the impact of TES size when the plant uses model-predictive control (MPC) for dispatch planning. The performance of MPC is benchmarked against one perfect knowledge (PK) and two day-ahead strategies. The optimal achievable profit for each TES size is determined using the PK strategy. An analysis is conducted to evaluate the relative profit losses for all the other simulated strategies compared to the PK strategy. A case study is conducted for a hypothetical 115 MWe ST-TES in South Australia. For January and August, 100 tests are performed for each dispatch policy, with the TES size varying from 6 to 14 h. The revenue evaluation is conducted with both fixed and wholesale spot prices. The analysis shows that MPC-aided dispatching enables the adoption of a smaller TES compared to day-ahead policies while maintaining the same expected profit. The resulting TES size reduction from 14 to 10 h translates into approximately up to $45.4 million in capital cost savings. The findings of this study can inform the ST-TES plant’s design procedures and facilitate negotiations for electricity sales contracts.

Graphical Abstract Figure
Graphical Abstract Figure
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1 Introduction

Solar tower (ST) is a promising technology for large-scale exploitation of solar irradiation for electricity generation. Differently from other renewable technologies, the ST plant benefits from a high capacity factor as it integrates with thermal energy storage (TES) [1]. An ST with TES (ST-TES) has also the potential to decouple electricity generation from solar irradiation availability during daylight hours and allows for the extension of the dispatch profile to high electricity price periods. This justifies the development of an optimized dispatch plan (DP) to increase revenue generation from electricity sales compared to analogous systems not equipped with TES.

However, ST plants must always cope with uncertainties associated with exogenous inputs, such as weather variables, particularly solar irradiation, and volatile electricity prices. Previous studies usually neglect the influence of uncertainty in sizing TES. Madaeni et al. [2] conducted a sensitivity analysis under perfect knowledge (PK) forecasts and evaluated energy revenue, ancillary service revenue, break-even cost and percentage of capacity value for two ST plants in the United States while TES size ranges from 0 to 12 h and solar multiple from 1.5 to 2.7. The authors concluded that larger TES is more profitable but under the assumption of PK forecasts.

Another adopted approach was the development of a sophisticated optimization program for dispatching ST plants. Wagner et al. [3] developed a deterministic model that assumed a PK forecast for uncertain parameters. The objective function was the maximization of the profit discounted over the optimization horizon while limiting the degradation of critical subsystems via controlling shutdown events. This model has since been widely used in the literature. Hamilton et al. [4] extended this model for a ST-TES plant collocated with a utility-scale photovoltaic plant. Cox et al. [5] incorporated transient operations in a sub-hourly extension of Wagner’s model to allow for more frequent optimizations as weather forecasts update in real time, i.e., every 5 min their test cases. Kahvecioglu et al. [6] also expanded Wagner’s model and proposed a stochastic framework that incorporated the uncertainty associated with direct normal irradiation (DNI) and electricity, and evaluated an analogous policy that adds value to end-of-horizon TES. Wales et al. [7] integrated a maintenance and failure simulation with the optimization model suggested in Ref. [3] to account for plant downtime caused by normal wear and tear and cycling of the plant. In addition, Petrollese et al. [8] generated several scenarios for uncertain parameters (e.g., DNI) and solved the stochastic optimization problem for optimal dispatch scheduling of the ST system in the Spanish day-ahead market.

These existing studies provided a dispatch optimization tailored to the specific electricity markets adopted in those countries, such as the day-ahead market in Spain, [9,10] and certain regions in the United States [11]. However, other countries, like Australia, have different market structures allowing to dispatch of electricity to a wholesale market, which has an inherently stochastic and more volatile dynamic. Mohammadzadeh et al. [12] applied model-predictive control (MPC) to a hypothetical ST plant in South Australia and evaluated the contribution of uncertainty in the weather and electricity price forecasts to the profit of electricity sales. The authors showed that MPC is a candidate control paradigm to mitigate the influence of uncertainty on dispatch planning, particularly in the wholesale market. However, the influence of optimized DP in the assessment of TES size has not been studied yet.

The contribution of this study is to conduct a sensitivity analysis to examine the potential reduction in TES size for a ST plant when employing MPC for dispatching. To achieve this, the MPC paradigm proposed in a previous authors” study [12] is employed and benchmarked against three other dispatching strategies: one PK and two day-ahead strategies. Monte Carlo simulation is used to conduct 100 tests for each dispatch strategy in January and August, examining the profit of electricity sales under two market scenarios: (i) the contracted-revenue scenario and (ii) the wholesale market. The simulation results including profit distribution, the ability to capture high prices, the power block average monthly shutdown, relative profit loss, and dispatch weighted average (DWA) [13] are statistically analyzed and reported in detail for different TES sizes and various DPs. The results suggest that ST plants with MPC require a smaller TES to achieve the same economic performance compared to when using day-ahead strategies.

The remainder of this article is organized as follows. A simplified configuration for the ST-TES plant is outlined in Sec. 2, followed by a brief discussion of the MPC paradigm in Sec. 3. The methodology for the sensitivity analysis is outlined in Sec. 4, and the assumptions adopted in the case study are stated in Sec. 5, followed by results and discussion in Sec. 6. The conclusion and future works are discussed in Sec. 7.

2 Solar Tower With Thermal Energy Storage

A ST-TES plant consists of four main subsystems: heliostat field, receiver, TES, and power block. The heliostat field concentrates solar irradiation onto the receiver. The receiver collects the concentrated irradiation through a heat transfer fluid (e.g., molten salt). The hot heat transfer fluid stores in the storage for later use for various purposes (e.g., electricity generation and maintaining the receiver or power block at high temperature during standby). The primary objective of such systems is steam (or equivalent high-temperature fluid) generation, which is then expanded in the turbine for electricity generation. The operating principles of each subsystem in ST-TES plants were extensively discussed in previous publications [12,14].

3 Model-Predictive Control

As demonstrated in Ref. [12], the MPC paradigm enables frequent re-optimization of DP using the latest state of the system and the most recent forecasts of uncertain inputs. The MPC paradigm consists of several key modules (Fig. 1). The prediction module provides the forecasts for two exogenous inputs from the current time index, “now”, n to K in future, where K is the prediction horizon. This module generates two forecast vectors: v^n+k|n for weather variables and p^n+k|n for electricity price, where k = 1, …, K. These forecasts together with the latest state of the system (e.g., state of charge of TES at time n, i.e., SOCn) are provided to the optimization module (e.g., Mixed-Integer Linear Program (MILP)). The optimization program is solved for an optimized dispatch control sequence u^n+k|n. The first input from the optimized sequence u^n+k|n is applied to the system. The dynamic of the plant is simulated for the given control input against actual weather conditions (i.e., vn) allowing for estimation of the new state of the system, i.e., SOCn+1. The electricity is sold at the rate of w^ns generating revenues at the actual market price pn. Subsequently, the current time index n increments to n + 1, and the whole process is repeated.

Fig. 1
MPC paradigm applied for dispatching ST-TES plant
Fig. 1
MPC paradigm applied for dispatching ST-TES plant
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One of the key distinctions between the MPC used here and those in the literature (e.g., Refs. [5,6]) is in the rolling horizon. In this study, MPC regularly updates the DP through intraday forecasting inputs, while the aforementioned studies employed daily rolling horizon control. The use of more accurate short-term forecasts enables MPC to be more effective in mitigating the impact of uncertainty in dispatching. The key components of the MPC paradigm used in this study are briefly discussed in the following.

3.1 Prediction Models.

Two forecast vectors are given to the optimization program. The first vector includes the forecast of weather variables. This study adopts the similarity-based forecast model developed in Ref. [12]. This model utilizes historical weather records to generate an intraday forecast for integration into the MILP model. The underlying principle of this approach assumes that two randomly selected weather profiles, denoted as A and B, will exhibit identical future behavior if their past observations are identical. The model assumes that the historical weather profiles in each month are statistically identical. The degree of similarity of each observed day is determined through weighted distance compared to other historical days. The weights are a decreasing function of the distance between the profile and historical trajectories. Further details of this model were presented in Ref. [12].

The second forecast is related to electricity prices. Australian Energy Market Operator (AEMO) provides the market participants with pre-dispatch prices with a 5-min resolution, extending until the end of the following day. However, this study focuses on optimizing DP with half-hourly resolution. This is because of the weather data availability resolution constraints, not due to any limitation in the ability of the optimization model to handle higher-resolution forecasts, as shown in Ref. [5]. Further details about how to employ pre-dispatch prices in MPC were detailed in Ref. [12].

3.2 Dispatch Optimization Model.

This is an extension of the optimization model suggested by Wagner et al. [3]. This optimization program aims to maximize the objective function, where the operating expenditures (e.g., the cost of purchasing electricity from the grid to run auxiliaries) and degradation costs (e.g., due to startup events) are deducted from revenues generated from electricity sales. The mathematical expression of the objective function is expressed as follows:
max:kK=λk[Δtp^k(w˙^ksw˙^ks)Δt(Crecq^krec+Cpcw˙^k+CcsbQcsby^kcsb)CδWw˙^kδ(Crsuy^krsup+Crhspy^krhsp+Crsdy^krsd)(Ccsuy^kcsup+Cchspy^kchsp+Ccsdy^kcsd)
(1)
where K is the optimization horizon with k intervals, Δt is the duration of each interval, λ = 0.98 is the discount factor, p^k is the electricity price forecast, w˙ks is the rate of generation dispatched to the grid, w˙kp is the power purchased from the grid to run auxiliaries, Crec is the degradation cost for each unit of thermal energy generation in the receiver, q^krec is the rate of generation in receiver delivered to the TES, Cpc is degradation cost associated with each unit of electricity generation in the power block, w˙k is the power block’s gross electricity generation, and Ccsb is the degradation cost to maintain the power block in standby, where thermal energy at the rate of Qcsb discharges from TES to maintain the subsystem at the high temperature. The degradation of critical subsystems increases as operating conditions vary. CδW represents the cost for each unit of ramping up/down and w˙kδ is the change in the power block output; Crsu and Crhsp is cost of the cold startup (restart from off-mode) and the hot startup (restart from standby) in receiver, respectively; Ccsu and Cchsp are the counterparts in the power block, respectively; Crsd and Ccsd is the shutdown cost in receiver and the power block, respectively; y^kcsb=1 indicates the power block's standby, zero otherwise; y^krsup=1 and y^krhsp=1 represent the receiver's cold and hot startup, respectively, zero otherwise; y^krsd=1 indicates the receiver shutdown, zero otherwise; similarly, y^kcsup=1 and y^kchsp=1 denote the power block's cold and hot startup, respectively, y^kcsd=1 indicates the power block's shutdown, zero otherwise. The optimization model is subjected to a set of linear constraints detailed in Ref. [12].

3.3 Monte Carlo Simulation.

The previously developed Monte Carlo simulation model [12] consists of a sampling procedure and a fail-safe strategy. The sampling procedure randomly selects days from a part of historical weather and electricity prices. The performance of DPs is tested when the plant interacts with novel weather and electricity price conditions. The fail-safe strategy is a crucial part of the simulation model preventing the implementation of nonphysical conditions. For instance, the fail-safe strategy forces a receiver to unplanned shutdown if DNI is insufficient and does not respect the minimum receiver’s generation limit. The sampling procedure and fail-safe strategy were presented in detail in Ref. [12].

4 Sensitivity Analysis

A sensitivity analysis is conducted to examine the potential reduction in TES size for ST plants when employing MPC for dispatching. Different TES sizes are considered, and DP is optimized accordingly. The optimized DP is then simulated to account for uncertainty, and the actual profit from electricity sales is evaluated. The MPC strategy is benchmarked against the following:

  1. PK forecasts, where the actual realization of uncertain parameters is used in the dispatch optimization,

  2. Day-ahead with prototypical weather profile (DAPW), where day-ahead price utility forecasts and a prototypical weather profile are used to determine the optimized DP.

  3. Persistent day-ahead (PER), where the price prediction is based on day-ahead price utility forecasts, and the weather forecast in day D is repeated from the weather conditions experienced in day D-1.

The key difference between MPC and the benchmarks is in the timing of DP optimization. MPC allows for intraday re-optimization, using the most up-to-date forecasts and the latest state of the system, whereas the benchmarks optimize the DP only once at the beginning of each operating day, without any updates throughout the day.

The DP obtained from the PK neglects the influence of uncertainty and is therefore not implementable. However, the results obtained from the PK can serve as an ideal metric to evaluate the performance of actually implementable DPs obtained from MPC, DAPW, and PER, referred to as imperfect DPs hereafter. A metric used in the sensitivity analysis is the relative reduction (i.e., loss) of profit in imperfect DPs compared to the optimum profit obtained from PK. Let Li,j represent the relative profit loss, which can be calculated as follows:
Li,j=1πjimpπipk
(2)
where i is the index of TES size in PK, j is the counterpart for TES sizes in imperfect DPs, πipk is the profit achieved from implementing PK to control TES i, and πjimp is the profit obtained from the imperfect DPs for TES j. Notably, i and j can be equal when both dispatch policies are applied to the same TES size. Another metric used in this study is DWA [13], which measures the value of 1 kWh of electricity dispatched into the wholesale market, which can be calculated as follows:
DWA=h=1HphΔtw˙hsh=1Hw˙hsΔt
(3)
where h is the time index of each half-hour, H = 48 · dm is the total number of half-hourly intervals in a month with dm days, ph is the electricity price in each time interval h, w˙hs is the average rate of electricity generation selling to the grid at time interval h, and Δt is the duration of each interval used to convert power to energy.

5 Case Study

A case study is conducted for a hypothetical ST plant located in South Australia, where the electricity price is subjected to significant uncertainty due to a high level of renewable energy penetration and grid (e.g., interstate interconnectors) constraints. The abundant solar resource also makes this location a suitable candidate for the deployment of a large-scale solar plant [15]. The system advisor model (SAM) [16] is used to draw the key design characteristics for the hypothetical ST plant. The plant features a power block with a steam turbine of 115 MWe capacity and a thermal-to-electrical conversion efficiency of 35%. The solar multiple is set at 3.1, and the solar field configuration and size of heliostats are optimized in SAM. Table 1 reports the remaining design specifications, obtained from Ref. [12].

Table 1

Design characteristics of hypothetical ST plant

SubsystemParameter (unit)Value
Power blockDesign gross power (MWe)115
Maximum input (MWt)329
Minimum input (MWt)82.1
Startup consumption (MWht)164.3
Standby power (MWt)65.7
ReceiverMaximum generation (MWt)1000
Minimum generation (MWt)250
Startup consumption (MWht)250
Minimum startup time (h)1
Standby power (MWt)100
Height × Diameter (m2)22.15 × 21.5
Tower height (m)239.5
HeliostatWidth × Length (m2)11.3 × 10.4
Reflectivity0.95
Numbers17066
TESCapacity (h)6, 8, …, 14
SubsystemParameter (unit)Value
Power blockDesign gross power (MWe)115
Maximum input (MWt)329
Minimum input (MWt)82.1
Startup consumption (MWht)164.3
Standby power (MWt)65.7
ReceiverMaximum generation (MWt)1000
Minimum generation (MWt)250
Startup consumption (MWht)250
Minimum startup time (h)1
Standby power (MWt)100
Height × Diameter (m2)22.15 × 21.5
Tower height (m)239.5
HeliostatWidth × Length (m2)11.3 × 10.4
Reflectivity0.95
Numbers17066
TESCapacity (h)6, 8, …, 14

5.1 Operation and Degradation Costs.

The relatively short operating history of the ST technology has limited the public availability of failure data and detailed system information due to their sensitive nature. The true degradation costs for key subsystems in ST plants are not yet well understood [17]. Here, these costs are adapted from existing dispatch optimization studies [12]. Table 2 reports the cost associated with the receiver and the power block.

Table 2

Operation and degradation costs [12]

SubsystemCost (unit)Value
Power blockPower generation ($/MWh)1.7
Cold startup ($/start)5450
Hot startup ($/start)545
Ramp up/down ($/ Δ MWe)0.59
Standby cost ($/MWht)0.67
ReceiverPower generation ($/MWh)5.3
Cold startup ($/start)10000
Hot startup ($/start)1000
SubsystemCost (unit)Value
Power blockPower generation ($/MWh)1.7
Cold startup ($/start)5450
Hot startup ($/start)545
Ramp up/down ($/ Δ MWe)0.59
Standby cost ($/MWht)0.67
ReceiverPower generation ($/MWh)5.3
Cold startup ($/start)10000
Hot startup ($/start)1000

5.2 Weather Historical Records.

The historical weather data for South Australia are collected from the Solcast database [18], which includes time series of solar irradiation, used to assess the total output of the solar field, as well as ambient temperature and wind speed, used to determine receiver thermal losses based on correlations presented in Ref. [3]. The weather data are divided into training data (2007–2016) and testing data (2017–2022), with a half-hour resolution. The training set is used in the prediction model, while the testing set is used for sampling in the Monte Carlo simulation.

Figure 2 shows a set of DNI profiles in January (summer) and August (winter) in South Australia. As seen, days in January are longer and have stronger DNI, whereas days in August are characterized by shorter daylight hours and lower solar irradiation. The actual samples show the stochastic nature of solar irradiation that must be carefully considered in the DP optimization. The clear sky profile neglects the risk of cloudy days and may drive too optimistic DPs, while the average profile underestimates sunny days as cloudy days greatly contribute to averaging. The prototypical profile is derived from partitional clustering, i.e., k-Medoid algorithm [19]. Due to seasonality and auto- and cross-correlated properties of weather variables, it is assumed that historical profiles in each month have statistically identical characteristics, allowing the preservation of these properties in the sampling strategy. MPC adapts prototypical profiles wherever DNI prediction is unavailable, e.g., in the early hours of each day.

Fig. 2
DNI profiles in (a) January and (b) August
Fig. 2
DNI profiles in (a) January and (b) August
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5.3 Electricity Price Historical Records.

The publicly available AEMO database is used for trading and pre-dispatch electricity prices (2019–2022) [20]. The pre-dispatch prices are used in the DP optimization, while the trading prices are used to examine DPs under novel conditions. AEMO began publishing trading prices with a 5-min resolution after Oct. 1, 2021, while before that the settlements were reported half-hourly after being averaged over six 5-min dispatch prices. Electricity prices with a half-hour resolution are considered here to maintain consistency with the weather database.

Figure 3 shows an electricity price in August. As seen, the prices are very volatile and sometimes drop below zero. There are many factors contributing to negative prices in the market, e.g., the bidding behavior of generators to prevent high shutdown costs. In addition, the supply and demand imbalance sometimes leads the prices to reach market cap at $16,200/MWhe [20], at the time of articulating this article. Figure 4 presents the frequency distribution of half-hourly electricity prices (used in the 100 simulation tests in this study) in various price intervals for January and August. As seen, the majority of high-price events occur in January, while August sees a higher absolute frequency of negative prices. With further penetration of renewables and the retirement of thermal generators, peak prices are expected to rise, leading to more frequent spikes in wholesale prices. Therefore, it is important to evaluate DP optimization and TES sizing in the presence of high and negative price events.

Fig. 3
Electricity price from August historical records
Fig. 3
Electricity price from August historical records
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Fig. 4
Absolute and relative frequency of half-hourly electricity prices within different price intervals used in 100 monthly simulation tests in this study
Fig. 4
Absolute and relative frequency of half-hourly electricity prices within different price intervals used in 100 monthly simulation tests in this study
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5.4 Energy Market Scenario.

There are two market scenarios considered in this study. The first is the contracted-revenue scenario where the revenue generation is based on the power purchase agreement (PPA) [13]. This study assumes that PPA is constant, set at the monthly average of trading prices, i.e., $69.1/MWhe and $55.9/MWhe for January and August, respectively, and identical for all DPs in each of the 2 months. The second scenario is the wholesale price scenario where the revenue generation relies on the wholesale market electricity price.

5.5 Capital Cost of Thermal Energy Storage.

There are currently no existing ST plants in Australia. This limits the availability of detailed project cost information. In this study, the capital cost of TES is taken from the most recent version of SAM. The default value is AU$33/kWht (i.e., US$22/kWht, with 1 USD = 1.5 AUD). The conventional unit used to specify the amount of storage is kWh. However, the TES size can also be represented in terms of the number of hours that the power block can be provided by the store energy for operation at the rated capacity. Table 3 presents the capital costs for different sizes of TES considered in this case study.

Table 3

Capital cost of TES in $ million

TES6 h8 h10 h12 h14 h
Cost68.190.8113.5136.2158.9
TES6 h8 h10 h12 h14 h
Cost68.190.8113.5136.2158.9

5.6 Initial Conditions.

The optimized DPs are simulated for the entire months of January and August. At the start of the simulation, the TES is assumed to be set at the lower bound, meaning soc0=socmin=10%. This assumption eliminates the initial economic value of a partially filled TES that could enable electricity generation without a full system startup. The receiver and power block are initially in off-mode, requiring a cold startup to initiate their operation. Additionally, it is assumed that the thermal energy from a shutdown event is considered a part of the subsequent startup event, resulting in Qrsd = Qcsd = 0.

6 Results and Discussion

In this section, the results of the sensitivity analysis are presented and discussed in detail. One hundred simulation tests are conducted, and the DPs obtained from MPC and the benchmarks are applied to the system with different TES sizes (i.e., ranging from 6 to 14 h). The simulation results are discussed considering plant operations in January (summer) and August (winter). Each simulation comprises 31 operating days, which are randomly sampled from the testing set. The key profit components, including the revenue and the costs, are calculated for applying every DP and every TES size for dispatching at two different market scenarios, focusing on the wholesale market.

6.1 Contracted-Revenue Scenario.

Figures 5(a) and 5(b) show the monthly profit distributions for different DPs in January and August. In January, the median monthly profit is $1.851 m for the ideal PK, reaching $2.716 m with 14 h of TES. The profit fall in imperfect DPs due to inherent uncertainty, primarily from weather forecasts. With 6 h TES, MPC sees a 32.4% drop (approximately $600 k) compared to PK. This gap narrows to 19.9% (approximately $539 k) with a 14 h TES. This is while DAPW results in profit losses of 39.0% (approximately $723 k) and 23.5% (approximately. $637 k) for the smallest and largest TES, respectively. Hence, the uncertainty prevents the imperfect DPs from matching ideal PK profits unless TES size is doubled under identical conditions.

Fig. 5
Profit distributions from employing various DPs and different sizes of TES in the contracted-revenue scenario: (a) January, PPA = $69.1/MWhe and (b) August, PPA = $55.9/MWhe
Fig. 5
Profit distributions from employing various DPs and different sizes of TES in the contracted-revenue scenario: (a) January, PPA = $69.1/MWhe and (b) August, PPA = $55.9/MWhe
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In August, a similar trend is observed, albeit with reduced dispatching profits (see Fig. 5(b)). The decline in profit generation can be attributed to shorter daylight hours, lower DNI, and a smaller PPA compared to January. Simulation results indicate that the PK strategy yields 39.4% and 37.6% less profit in plants with 6 and 14 h of TES, respectively, in August compared to January. Like in summer, MPC, optimizing with recent forecasts, outperforms day-ahead DPs in winter, with more significant absolute and relative gains. The profit gap between MPC and DAPW was 6.7% and 3.6% for the smallest and largest TES sizes in summer, respectively. This gap widens to 14.7% and 20.6% in winter, emphasizing the importance of not relying on the previous day’s weather profile for DP optimization (i.e., PER), as it results in a significant profit drop compared to other benchmarks.

6.2 Wholesale Market Scenario.

The characteristics of the wholesale electricity prices allow the plant to generate significant revenue through high-price events, but the highly volatile price regime poses a risk of substantial losses. This section presents profit distributions from electricity sales using different DPs in the wholesale market, followed by an examination of DP’s ability to capture high-price events and an analysis of degradation costs’ contribution to profit reduction for each TES size.

Figure 6(a) shows monthly profit distributions for each DP and TES size during January dispatching. The price fluctuations drive the standard deviation in profit distribution. As with the contracted-revenue scenario, increasing TES reduces the gap between the median profits of imperfect DPs and the PK strategy. MPC outperforms day-ahead DPs, with a median profit drop in the range of 26.9–31.3% (or $2.402 m to $2.618 m) compared to PK, but 31.8–45.0% (or $2.841 m to $3.762 m) compared to DAPW. Combining uncertainties in weather and price forecasts exacerbates absolute profit differences from optimum profit in the wholesale market scenario. To mitigate the profit gap between MPC and PK, one potential solution is providing MPC with more accurate forecasts, as discussed in Ref. [12].

Fig. 6
Profit distributions from employing various DPs and different sizes of TES in the wholesale market scenario: (a) January and (b) August
Fig. 6
Profit distributions from employing various DPs and different sizes of TES in the wholesale market scenario: (a) January and (b) August
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In Fig. 6(b), the profit distributions for August dispatching show a similar trend, but the profits decrease for each TES size compared to January. The absolute profit loss between MPC and PK ranges from $656 k to $707 k, with the relative differences from 24.7% (for the largest TES) to 40.3% (for the smallest TES). The profit reduction is influenced by shorter daylight hours, lower DNI, and differences in the electricity price characteristics. Notably, standard deviation differs between profit distributions in January and August due to frequent large short-term spikes in intraday electricity prices, resulting in wider distributions for January (see Figs. 6(a) and 6(b)).

As part of sensitivity analysis, assessing TES size’s influence on profit-driving factors, including revenue from high-price event capture, the receiver degradation costs, and the impact of unplanned shutdown on power block degradation costs, is crucial. Table 4 reports the monthly revenue, the degradation costs, and the profit averaged across all 100 tests for PK strategy in the wholesale market. Tables 7 and 8 report relative changes in these profit drivers for imperfect DPs compared to PK for January and August, respectively.

Table 4

Average of monthly simulation results in PK strategy in the wholesale market, 000s

Profit componentMonth6 h8 h10 h12 h14 h
Total revenueJanuary$8,805.3$9,133.3$9,392.5$9,576.7$9,704.9
August$2,653.9$3,048.6$3,351.0$3,598.2$3,771.2
Receiver operation costJanuary($487.2)($565.5)($634.4)($687.6)($725.3)
August($494.6)($576.3)($651.5)($713.3)($755.7)
Receiver shutdown costJanuary($249.7)($257.2)($259.9)($260.5)($260.5)
August($270.7)($274.2)($276.2)($277.4)($277.5)
Power block operation costJanuary($57.7)($67.2)($75.1)($80.6)($84.2)
August($60.1)($69.5)($78.7)($85.6)($89.8)
Power block shutdown costJanuary($95.1)($76.6)($66.0)($61.3)($58.4)
August($122.6)($103.0)($69.0)($57.6)($56.7)
Total profitJanuary$7,887.4$8,148.2$8,344.5$8,476.7$8,568.2
August$1,753.3$2,079.2$2,343.2$2,537.6$2,665.3
Profit componentMonth6 h8 h10 h12 h14 h
Total revenueJanuary$8,805.3$9,133.3$9,392.5$9,576.7$9,704.9
August$2,653.9$3,048.6$3,351.0$3,598.2$3,771.2
Receiver operation costJanuary($487.2)($565.5)($634.4)($687.6)($725.3)
August($494.6)($576.3)($651.5)($713.3)($755.7)
Receiver shutdown costJanuary($249.7)($257.2)($259.9)($260.5)($260.5)
August($270.7)($274.2)($276.2)($277.4)($277.5)
Power block operation costJanuary($57.7)($67.2)($75.1)($80.6)($84.2)
August($60.1)($69.5)($78.7)($85.6)($89.8)
Power block shutdown costJanuary($95.1)($76.6)($66.0)($61.3)($58.4)
August($122.6)($103.0)($69.0)($57.6)($56.7)
Total profitJanuary$7,887.4$8,148.2$8,344.5$8,476.7$8,568.2
August$1,753.3$2,079.2$2,343.2$2,537.6$2,665.3

Note: ( · ) denotes negative values.

6.2.1 Capturing High Prices Events.

Table 4 illustrates that the revenue from electricity sales increases with the TES size. However, the imperfect DPs result in reducing the revenue compared to PK, attributed to uncertainties in the wholesale price forecasts. In January, the MPC’s revenue drop from 21.5% (for the largest TES) to 26.0% (for the smallest TES) compared to the PK strategy. In August, the revenue losses are smaller, ranging from 16.1% to 24.7%, primarily due to the lower electricity prices. The closest day-ahead DP shows even larger revenue losses, ranging from 23.7% to 30.9% in January and from 21.3% to 26.8% in August. These findings highlight the importance of regularly updating the DP with recent forecasts to maintain revenue closer to the ideal profit.

The high-price event capture significantly influences revenue generation. Table 5 demonstrates DP performance in capturing 1%, 5%, and 10% of high-price events with TES sizes from 6 to 14 h in January. The results for August are reported in Table 6. The perfect forecast dispatching captures these events with percentages ranging from 98.6% to 99.3% for 1%, 94.2% to 96.7% for 5%, and slightly less for 10%. As the percentage of high-price events increases, the capturing rate decreases, a trend observed across all imperfect DPs.

Table 5

DP’s performance in capturing high prices, January

TESPKMPCDAPWPER
1% of high prices
6 h98.6%75.5%66.0%54.3%
8 h98.9%76.9%72.7%58.2%
10 h99.1%78.8%74.4%59.6%
12 h99.1%80.1%74.6%60.0%
14 h99.3%81.3%75.1%60.7%
5% of high prices
6 h94.2%67.3%63.6%51.7%
8 h95.5%70.9%68.4%54.8%
10 h96.0%74.2%71.4%57.1%
12 h96.4%76.1%72.0%57.9%
14 h96.7%77.7%72.9%58.7%
10% of high prices
6 h89.6%64.1%62.5%50.5%
8 h92.1%69.1%66.8%53.4%
10 h93.0%72.3%70.4%55.9%
12 h93.6%74.3%71.3%57.0%
14 h94.3%75.9%72.6%58.1%
TESPKMPCDAPWPER
1% of high prices
6 h98.6%75.5%66.0%54.3%
8 h98.9%76.9%72.7%58.2%
10 h99.1%78.8%74.4%59.6%
12 h99.1%80.1%74.6%60.0%
14 h99.3%81.3%75.1%60.7%
5% of high prices
6 h94.2%67.3%63.6%51.7%
8 h95.5%70.9%68.4%54.8%
10 h96.0%74.2%71.4%57.1%
12 h96.4%76.1%72.0%57.9%
14 h96.7%77.7%72.9%58.7%
10% of high prices
6 h89.6%64.1%62.5%50.5%
8 h92.1%69.1%66.8%53.4%
10 h93.0%72.3%70.4%55.9%
12 h93.6%74.3%71.3%57.0%
14 h94.3%75.9%72.6%58.1%
Table 6

DP’s performance in capturing high prices, August

TESPKMPCDAPWPER
1% of high prices
6 h94.2%75.3%65.7%60.9%
8 h95.2%81.2%70.3%63.5%
10 h96.2%83.3%75.5%66.8%
12 h96.5%84.9%76.8%69.2%
14 h97.7%85.3%77.7%71.3%
5% of high prices
6 h87.2%67.0%58.1%52.8%
8 h89.8%73.7%64.5%57.9%
10 h91.5%78.8%71.7%62.3%
12 h92.7%81.0%74.5%65.7%
14 h93.8%81.8%75.6%67.2%
10% of high prices
6 h81.5%59.8%52.6%47.0%
8 h86.4%67.9%59.9%52.8%
10 h88.7%74.0%67.3%57.3%
12 h89.8%77.6%71.2%61.6%
14 h90.8%79.2%72.6%63.8%
TESPKMPCDAPWPER
1% of high prices
6 h94.2%75.3%65.7%60.9%
8 h95.2%81.2%70.3%63.5%
10 h96.2%83.3%75.5%66.8%
12 h96.5%84.9%76.8%69.2%
14 h97.7%85.3%77.7%71.3%
5% of high prices
6 h87.2%67.0%58.1%52.8%
8 h89.8%73.7%64.5%57.9%
10 h91.5%78.8%71.7%62.3%
12 h92.7%81.0%74.5%65.7%
14 h93.8%81.8%75.6%67.2%
10% of high prices
6 h81.5%59.8%52.6%47.0%
8 h86.4%67.9%59.9%52.8%
10 h88.7%74.0%67.3%57.3%
12 h89.8%77.6%71.2%61.6%
14 h90.8%79.2%72.6%63.8%

MPC performs closer to PK in capturing high-price events than day-aheads. For instance, MPC with 6 h of TES captures 75.5% of 1% high-price events, increasing to 81.3% with 14 h of TES in January. DAPW captures 1% of high-price events in the range of 66.0–75.1% as TES increases from 6 to 14 h. Notably, PER is the worst performer, capturing only 54.3–60.7% of 1% high-price intervals with the smallest and largest TES, respectively. In summary, MPC closely aligns with PK in capturing high-price events, while relying on the previous day’s weather profile in optimizing DP reduces the chance of capturing such events.

6.2.2 Receiver Degradation Costs.

The total receiver degradation cost comprises the receiver’s operation cost, proportional to the rate of receiver generation to charge the TES and the receiver shutdown cost. In Table 4, the receiver operation cost in PK ranges from $478.2 k to $725.3 k in January and $494 k to $755 k in August, primarily due to increased receiver generation with larger TES size. Tables 7 and 8 indicate that imperfect DPs reduce the receiver operation cost, signifying less charging of the TES compared to PK.

Table 7

Relative increase (decrease) of the profit components compared to PK in the wholesale market, January

Profit component/TESMPCDAPWPER
Revenue from electricity sales
6 h(26.0%)(30.9%)(44.6%)
8 h(25.0%)(26.7%)(42.0%)
10 h(24.1%)(24.8%)(41.1%)
12 h(22.9%)(24.4%)(40.7%)
14 h(21.5%)(23.7%)(39.9%)
Receiver operating cost
6 h(13.6%)(7.1%)(28.0%)
8 h(13.8%)(7.0%)(28.4%)
10 h(13.6%)(7.1%)(28.8%)
12 h(12.3%)(6.9%)(29.0%)
14 h(11.2%)(6.2%)(28.4%)
Receiver shutdown cost
6 h14.4%65.9%40.2%
8 h10.8%63.2%36.9%
10 h10.6%61.3%36.4%
12 h11.4%56.3%32.3%
14 h11.9%55.7%31.9%
Power block operating cost
6 h(12.9%)(8.9%)(29.9%)
8 h(13.4%)(9.0%)(30.4%)
10 h(13.2%)(9.0%)(30.7%)
12 h(11.4%)(8.6%)(30.5%)
14 h(10.3%)(7.6%)(29.7%)
Power block shutdown cost
6 h49.9%88.8%65.9%
8 h75.1%110.4%91.6%
10 h97.0%121.2%107.2%
12 h110.1%130.0%115.2%
14 h115.6%126.4%111.5%
Profit from dispatching
6 h(29.4%)(37.2%)(49.9%)
8 h(28.4%)(32.5%)(47.0%)
10 h(27.4%)(30.3%)(45.9%)
12 h(26.3%)(29.7%)(45.4%)
14 h(24.7%)(28.9%)(44.4%)
Profit component/TESMPCDAPWPER
Revenue from electricity sales
6 h(26.0%)(30.9%)(44.6%)
8 h(25.0%)(26.7%)(42.0%)
10 h(24.1%)(24.8%)(41.1%)
12 h(22.9%)(24.4%)(40.7%)
14 h(21.5%)(23.7%)(39.9%)
Receiver operating cost
6 h(13.6%)(7.1%)(28.0%)
8 h(13.8%)(7.0%)(28.4%)
10 h(13.6%)(7.1%)(28.8%)
12 h(12.3%)(6.9%)(29.0%)
14 h(11.2%)(6.2%)(28.4%)
Receiver shutdown cost
6 h14.4%65.9%40.2%
8 h10.8%63.2%36.9%
10 h10.6%61.3%36.4%
12 h11.4%56.3%32.3%
14 h11.9%55.7%31.9%
Power block operating cost
6 h(12.9%)(8.9%)(29.9%)
8 h(13.4%)(9.0%)(30.4%)
10 h(13.2%)(9.0%)(30.7%)
12 h(11.4%)(8.6%)(30.5%)
14 h(10.3%)(7.6%)(29.7%)
Power block shutdown cost
6 h49.9%88.8%65.9%
8 h75.1%110.4%91.6%
10 h97.0%121.2%107.2%
12 h110.1%130.0%115.2%
14 h115.6%126.4%111.5%
Profit from dispatching
6 h(29.4%)(37.2%)(49.9%)
8 h(28.4%)(32.5%)(47.0%)
10 h(27.4%)(30.3%)(45.9%)
12 h(26.3%)(29.7%)(45.4%)
14 h(24.7%)(28.9%)(44.4%)
Table 8

Relative increase (decrease) of the profit components compared to PK in the wholesale market, August

Profit component/TESMPCDAPWPER
Revenue from electricity sales
6 h(24.7%)(26.8%)(36.3%)
8 h(22.2%)(24.3%)(35.5%)
10 h(18.7%)(20.5%)(34.0%)
12 h(16.7%)(20.6%)(32.9%)
14 h(16.1%)(21.3%)(32.0%)
Receiver operating cost
6 h(7.7%)(8.4%)(20.6%)
8 h(7.7%)(8.6%)(22.2%)
10 h(6.9%)(7.4%)(23.2%)
12 h(6.3%)(10.3%)(23.4%)
14 h(6.3%)(12.4%)(23.3%)
Receiver shutdown cost
6 h12.0%39.1%23.8%
8 h11.6%37.8%22.6%
10 h11.8%36.9%22.2%
12 h11.6%37.3%22.3%
14 h11.5%37.2%21.1%
Power block operating cost
6 h(7.5%)(10.8%)(23.4%)
8 h(7.1%)(10.9%)(24.7%)
10 h(7.1%)(10.2%)(25.9%)
12 h(6.8%)(11.8%)(25.7%)
14 h(6.5%)(13.4%)(25.2%)
Power block shutdown cost
6 h40.2%64.3%52.2%
8 h50.4%81.3%76.9%
10 h81.4%138.9%138.3%
12 h71.8%132.0%140.2%
14 h61.6%125.5%126.7%
Profit from dispatching
6 h(40.4%)(49.3%)(56.8%)
8 h(35.1%)(43.0%)(53.1%)
10 h(29.7%)(36.7%)(49.7%)
12 h(26.0%)(34.4%)(46.6%)
14 h(24.6%)(34.0%)(44.5%)
Profit component/TESMPCDAPWPER
Revenue from electricity sales
6 h(24.7%)(26.8%)(36.3%)
8 h(22.2%)(24.3%)(35.5%)
10 h(18.7%)(20.5%)(34.0%)
12 h(16.7%)(20.6%)(32.9%)
14 h(16.1%)(21.3%)(32.0%)
Receiver operating cost
6 h(7.7%)(8.4%)(20.6%)
8 h(7.7%)(8.6%)(22.2%)
10 h(6.9%)(7.4%)(23.2%)
12 h(6.3%)(10.3%)(23.4%)
14 h(6.3%)(12.4%)(23.3%)
Receiver shutdown cost
6 h12.0%39.1%23.8%
8 h11.6%37.8%22.6%
10 h11.8%36.9%22.2%
12 h11.6%37.3%22.3%
14 h11.5%37.2%21.1%
Power block operating cost
6 h(7.5%)(10.8%)(23.4%)
8 h(7.1%)(10.9%)(24.7%)
10 h(7.1%)(10.2%)(25.9%)
12 h(6.8%)(11.8%)(25.7%)
14 h(6.5%)(13.4%)(25.2%)
Power block shutdown cost
6 h40.2%64.3%52.2%
8 h50.4%81.3%76.9%
10 h81.4%138.9%138.3%
12 h71.8%132.0%140.2%
14 h61.6%125.5%126.7%
Profit from dispatching
6 h(40.4%)(49.3%)(56.8%)
8 h(35.1%)(43.0%)(53.1%)
10 h(29.7%)(36.7%)(49.7%)
12 h(26.0%)(34.4%)(46.6%)
14 h(24.6%)(34.0%)(44.5%)

This reduction is evident in Table 9, reporting the average monthly curtailment from the receiver. In January, the abundant thermal power allows MPC to be conservative in TES charging to avoid unplanned shutdown risks, leading to higher curtailment than DAPW. Despite lower curtailment, DAPW incurs higher shutdown costs than MPC (43.8–52.4% higher). Overall, while MPC experiences higher receiver curtailment, its receiver operation and the shutdown costs are lower than PK and DAPW.

Table 9

Average of monthly receiver curtailment, GWht

TESPKMPCDAPWPER
January
6 h152.7165.2159.2178.4
8 h138.0152.7145.5168.2
10 h125.0141.2133.5159.4
12 h115.0130.9124.0152.5
14 h107.9123.3116.3146.7
August
6 h88.295.496.0107.4
8 h72.881.182.196.9
10 h58.767.167.887.1
12 h47.055.560.878.5
14 h39.048.056.772.2
TESPKMPCDAPWPER
January
6 h152.7165.2159.2178.4
8 h138.0152.7145.5168.2
10 h125.0141.2133.5159.4
12 h115.0130.9124.0152.5
14 h107.9123.3116.3146.7
August
6 h88.295.496.0107.4
8 h72.881.182.196.9
10 h58.767.167.887.1
12 h47.055.560.878.5
14 h39.048.056.772.2

August, with shorter daylight hours, results in a less thermal energy being available for the receiver to charge the TES, as seen in lower receiver curtailment compared to January. In August, MPC and DAPW exhibit the similar receiver operation costs until the TES size is below 10 h. Beyond 10 h, the MPC’s receiver operation cost rises as it accepts more thermal energy, resulting in less curtailment than DAPW. Therefore, the receiver’s curtailment decreases as the TES size increases for all DPs.

6.2.3 Power Cycle Shutdowns.

There are two types of shutdowns for the power block: planned shutdowns commanded by the DP to stop electricity generation, and unplanned shutdowns leading to the power block's generation curtailment due to insufficient storage. Figures 7(a) and 7(b) show the distribution of monthly power block shutdowns in January and August, respectively. Table 10 shows the average contribution of unplanned shutdowns to average monthly shutdowns for every imperfect DP.

Fig. 7
Monthly power block shutdowns from different DPs in wholesale market: (a) January and (b) August
Fig. 7
Monthly power block shutdowns from different DPs in wholesale market: (a) January and (b) August
Close modal
Table 10

Percent of power block unplanned shutdowns

TESMPCDAPWPER
January
6 h7.8%65.9%63.3%
8 h4.9%76.3%71.3%
10 h4.9%81.7%77.9%
12 h4.1%80.8%75.9%
14 h4.0%83.1%78.3%
August
6 h10.7%60.3%63.4%
8 h8.8%59.0%60.7%
10 h9.5%59.9%63.8%
12 h10.3%70.4%74.5%
14 h10.0%70.8%75.3%
TESMPCDAPWPER
January
6 h7.8%65.9%63.3%
8 h4.9%76.3%71.3%
10 h4.9%81.7%77.9%
12 h4.1%80.8%75.9%
14 h4.0%83.1%78.3%
August
6 h10.7%60.3%63.4%
8 h8.8%59.0%60.7%
10 h9.5%59.9%63.8%
12 h10.3%70.4%74.5%
14 h10.0%70.8%75.3%

Results exploration reveals that as the TES size increases, the number of monthly power block shutdowns decreases for all DPs. PK consistently results in fewer shutdowns at each TES size, while the risk of unplanned shutdowns in imperfect DPs increases with uncertainty in the weather forecast. In January, as the TES size increases from 6 to 14 h, the average monthly shutdowns decrease from 16 to 9 for PK, from 25 to 21 for MPC, and from 32 to 22 for DAPW. The difference between MPC and DAPW is more pronounced in August due to a higher risk of empty storage from shorter daylight hours and lower DNI. In winter, MPC (with 6 h of TES) leads to 30 monthly shutdowns, a 17.3% reduction compared to DAPW, and MPC (with 14 h of TES) yields 13 monthly shutdowns, a 31.8% reduction compared to DAPW.

For imperfect DPs, poor prediction of DNI drives unplanned shutdowns, with MPC having less than 7.8% and 10.7% of total shutdowns as unplanned in January and August, respectively. This indicates that MPC reduces the risk of unplanned shutdowns through conservative TES use. In contrast, the majority of shutdowns in DAPW are unplanned, ranging from 65.9% to 83.1% in January and 59.0% to 70.8% in August. Figures 7(a) and 7(b) show a minimal difference in shutdowns between 12 and 14 h of TES, particularly for PK and MPC, suggesting a 2-h increase in TES size does not significantly reduce monthly power block shutdowns.

This means regular updating DPs with recent price forecasts allows effective TES use and reduces the risk of unplanned shutdowns. Day-ahead DP increases the risk of empty TES and unplanned shutdowns, even with the largest TES. The statistics of the shutdown events for the contracted-revenue scenario can be found in the previous study [12].

6.2.4 Power Cycle Standby.

An effective approach to minimize shutdown costs is to place the power block on standby during short-term negative prices, provided the DP receives accurate forecasts of these periods. During standby, the power block consumes thermal power from the TES to maintain a high temperature but does not generate electricity, avoiding revenue loss. This study uses the number of hot startups as a metric to measure the frequency of standby events and potential savings in shutdowns.

Figure 8(a) shows the distribution of monthly hot startups for all DPs in January. With an increase in TES, the number of hot startups and the savings in shutdowns increase. For MPC, with 6 and 14 h of TES, the number of hot startups is approximately 9 and 12, respectively. This suggests that MPC can save 9 shutdowns with 6 h of TES and 12 shutdowns with 14 h, whereas DAPW results in saving 6 and 8 shutdowns for 6 and 14 h of TES, respectively.

Fig. 8
Monthly power block hot startup from different DPs in wholesale market: (a) January and (b) August
Fig. 8
Monthly power block hot startup from different DPs in wholesale market: (a) January and (b) August
Close modal

In Fig. 8(b), the occurrence of short-term negative prices leads to more hot startups and wider distributions in August than in January. In the case of PK, the number of monthly hot startups increases from 6 in January to 13 in August for 6 h of TES and from 11 in January to 21 in August for 14 h of TES. Again, MPC outperforms day-ahead benchmarks and saves 9 and 16 shutdowns in 6 and 14 h of TES, respectively. DAPW allows for saving 8 and 12 shutdowns in 6 and 14 h of TES. Therefore, considering intraday negative electricity prices allows for more savings in power block shutdowns compared to day-ahead DPs, and as with shutdowns, the increase in TES from 12 to 14 h does not significantly reduce the number of hot startups (or save shutdowns).

6.2.5 Relative Profit Losses.

Figures 9 and 10 show relative profit losses in January and August, respectively. The x-axis and y-axis represent TES size in PK and imperfect DPs, respectively, while contours denote the relative profit loss (Li,j in Eq. (2)). Three cases are considered. In the first case, profit losses are compared to PK as TES increases in imperfect DPs. Profit losses decrease with a larger TES. In January, MPC shows a 30% profit loss compared to PK with the smallest TES, decreasing to 20% with 14 h of TES. Day-ahead DPs exhibit larger profit losses, 37% and 24% for 6 and 14 h of TES, respectively. The reduction in relative loss with increased storage size is reasonable, as the larger TES allows imperfect DPs to capture higher high-price events (see Table 5) and generate more revenue (see Table 7).

Fig. 9
Relative profit losses compared to PK (%), January: (a) MPC, (b) DAPW, and (c) PER
Fig. 9
Relative profit losses compared to PK (%), January: (a) MPC, (b) DAPW, and (c) PER
Close modal
Fig. 10
Relative profit losses compared to PK (%), August: (a) MPC, (b) DAPW, and (c) PER
Fig. 10
Relative profit losses compared to PK (%), August: (a) MPC, (b) DAPW, and (c) PER
Close modal

The same trend is observed for August, with MPC incurring fewer losses than day-ahead DPs. Imperfect DPs, even with the largest TES, cannot achieve zero profit difference with PK with the smallest storage, primarily due to the stochastic nature of the weather profile and electricity prices, especially when participating in the wholesale market.

In the second case, the relative profit losses are evaluated as TES increases only in the PK but not in imperfect DP. The results show that increasing TES in PK leads to the higher relative profit losses. In January, the relative profit loss is 24% with 10 h of TES, but 30% with the largest TES. The relative profit loss between DAPW with 10 h of TES against PK with the smallest TES is about 28%, but 32% against PK with the largest TES. This increase in the profit loss is reasonable, as increased TES size in PK results in a higher revenue generation gap compared to imperfect DPs.

In the third case, the profit losses between PK and imperfect DPs with the same TES are compared. As TES increases from 6 to 14 h, the profit losses decrease. Using MPC achieves the same profit as day-ahead DPs but with 2–4 h less TES, potentially saving approximately $22.7 m to $45.4 m in capital costs.

6.2.6 Dispatch Weighted Average.

Figures 11(a) and 11(b) show the distribution of monthly DWA for various TES sizes and every DP in January and August, respectively. The two primary factors influencing DWA are wholesale electricity prices and the rate of electricity generation during each price interval. Therefore, DWA is driven by electricity generation during high prices and is impacted negatively by generation at low or negative prices. Notably, the distributions of monthly DWAs are wider, and the median is higher for dispatching in January compared to August, reflecting the higher frequency of large short-term price spikes in January.

Fig. 11
Distributions of monthly DWA in wholesale market: (a) January and (b) August
Fig. 11
Distributions of monthly DWA in wholesale market: (a) January and (b) August
Close modal

Exploring January results, the median DWA for PK with 6 h of TES is $301.6/MWh, that for MPC is $273/MWh, and that for DAPW is $232.5/MWh. As TES increases to 14 h, the median DWA drops by 26% ($78.4/MWh) for PK, 27.0% ($74/MWh) for MPC, and 80.3% ($45.9/MWh) for DAPW. The increase in TES from 12 to 14 h does not significantly alter the median DWA, particularly for PK and MPC. Consequently, PK forecasting enables the capture of more high-price events, resulting in higher monthly DWAs compared to imperfect DPs across all TES. Moreover, smaller TES yields higher electricity sale values, while the larger TES allows dispatching at a wider range of wholesale prices, reducing DWA.

The simulation results also show that MPC, by actively incorporating electricity price forecast updatesin DP optimization, allows for conservative TES utilization and maintains a reserve for high-price events. In January, MPC improves the average DWAs by $33/MWh (14.5%) and $16/MWh (8.8%) compared to DAPW with 6 and 14 h of TES, respectively. It is noteworthy that the achieved DWA from the wholesale market dispatch is much higher than the assumed PPA value, which represents the average electricity price in each month under the contracted-revenue scenario. In January, DWA is about four to five times higher than PPA, while in August, it is about two times higher. This suggests that plant operators may consider DWA in negotiating the PPA to hedge revenue against wholesale price volatility.

7 Conclusion

In this study, a sensitivity analysis is conducted to explore potential reductions in thermal energy storage (TES) size for a solar tower (ST) plant using model-predictive control (MPC) for dispatching. The proposed MPC was previously developed and benchmarked against perfect knowledge (PK) and day-ahead dispatch plans (DPs), including day-ahead with prototypical weather profile (DAPW) and PER. The analysis was applied to a hypothetical 115 MW ST with TES (ST-TES) plant in South Australia, using Monte Carlo simulation to sample from historical weather and electricity price profiles, conducting 100 simulation tests for every DP while varying TES size from 6 to 14 h. The dispatch results are discussed for dispatching under a contracted-revenue scenario and wholesale market scenario.

The results indicate that achieving the same profit as PK is unlikely for imperfect DPs, even with the largest (14 h) TES. However, MPC, actively updating DP using recent forecasts, outperforms day-ahead DP, particularly with smaller TES sizes, mitigating the impact of uncertainty. Profit distributions in the wholesale market scenario have a higher standard deviation due to wholesale price fluctuations. Despite this, the wholesale price characteristics enable operators to generate higher revenue by capturing high-price events. MPC with 10 h of TES achieves the same profit as DAPW with 14 h in the wholesale market, translating to $45.4 m in plant capital cost savings. Compared to DAPW, MPC reduces power block shutdown events by up to 31.8%, indicating lower degradation and additional operation and maintenance cost savings. MPC also allows for higher DWA compared to day-ahead DPs, suggesting its consideration in negotiating the power purchase agreements (PPA).

The improved dispatch efficiency encourages investment in additional concentrating solar thermal technology, potentially reducing the wholesale prices in the energy market and benefiting end-users. Future work could explore MPC performance with shorter dispatch intervals (e.g., 5 min) and evaluate the impact of additional features, such as the size of the heliostat field, for further improvements.

Acknowledgment

The authors acknowledge the support of the Australian Government for this study, through the Australian Renewable Energy Agency (ARENA) and within the framework of the Australian Solar Thermal Research Institute (ASTRI).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

d =

number of days in a month

p =

electricity price ($ Wh −1)

q =

receiver output thermal power (W)

t =

time-step (h)

u =

dispatch control sequence

y =

1 if operation mode actives, 0 otherwise

w =

electrical power (w)

C =

degradation cost ($ Wh −1)

Q =

thermal power (W)

DWA =

dispatch weighted average ($ Wh −1)

Greek Symbols

γ =

discount factor

Δ =

difference

δ =

ramping up/down

π =

profit ($)

L =

relative profit loss (%)

Superscripts

chsp =

power block hot startup

csb =

power block standby

csd =

power block shutdown

csup =

power block cold startup

pc =

power block generation

rec =

receiver generation

rhsp =

receiver hot startup

rsb =

receiver standby

rsd =

receiver shutdown

rsup =

receiver cold startup

Subscripts

h =

time index of each half-hour data

i =

index of storage size in PK planning

j =

index of storage size in imperfect planning

k =

time index

  m =

index of month

n =

current time index, now

Abbreviation

DAPW =

day-ahead with prototypical weather profile

DNI =

direct normal irradiation

DP =

dispatch plan

MPC =

model predictive control

PER =

persistent day-ahead

PK =

perfect knowledge

PPA =

power purchase agreement

ST =

solar thermal

TES =

thermal energy storage

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