Off-Design Performances of Subcritical and Supercritical Organic Rankine Cycles in Geothermal Power Systems under an Optimal Control Strategy

Abstract

The conditions of heat source and heat sink in a geothermal ORC system may frequently vary due to variations in geological conditions, ambient temperature and actual operation. In this study, an off-design performance prediction model for geothermal ORC systems is developed according to special designs of critical components, and an optimal control strategy which regards the turbine guide vane angle, the refrigerant pump rotational speed and the cooling water mass flow rate as control variables is proposed to maximize the net power output. Off-design performances of both subcritical and supercritical ORCs are analyzed. The results indicate that, under the optimal control strategy, the net power output of both ORCs increase with greater geothermal water mass flow rate, higher geothermal water inlet temperature and lower cooling water inlet temperature, which is mainly due to a greater working fluid mass flow rate, higher turbine inlet pressure and lower condensing pressure, respectively. The net power output of supercritical ORC is always greater than that of subcritical ORC within the range of this study, but the difference tends to decrease when supercritical ORC activates the geothermal water reinjection temperature restriction.

1. Introduction

Due to the rapid growth of the world economy, the global energy consumption continues to increase significantly, which leads to excessive consumption of fossil fuels and serious environmental deterioration. In this instance, the utilization of low-grade thermal energy, including geothermal energy, solar energy, biomass energy and industrial waste heat, is currently drawing great attention. Geothermal energy presents favorable application prospects owing to its abundant storage capacity and convenient utilization. Among all the available technologies, the Organic Rankine Cycle (ORC) has been regarded as an efficient approach to convert low-grade heat into electricity. A great amount of research concentrated on working fluid selection and system design has been conducted to enhance the performance of ORC systems. Chen et al. [1] screened 35 kinds of working fluids for the subcritical and supercritical ORCs and proposed the selection criteria of working fluids. Zhang et al. [2] compared various working fluids for a low temperature ORC system with five different indicators as criteria. Both subcritical and supercritical ORCs were examined, and it turned out that supercritical ORC using R125 had good economics and high geothermal utilization rates. Guo et al. [3] reported that the choice of optimal working fluid was determined by the profiles of the heat source and heat sink, and the introduction of a regenerator did not always enhance the system efficiency. Liu et al. [4] pointed out that supercritical ORC could yield good performances in both thermodynamic and economic aspects, and R134a was the optimal working fluid.

These previous ORC studies were mainly focused on design conditions, however, the heat source and sink conditions probably vary during the actual operation of an ORC, which makes the operating condition deviate from the design point. Consequently, the off-design performance of ORC is of great importance and should be carefully investigated. The critical components of an ORC system, mainly including the expander and heat exchangers, significantly influence the system performance. For expanders, many studies have been carried out on the radial inflow turbine which is regarded as a suitable candidate for ORC. Sauret and Rowlands [5] preliminarily designed radial inflow turbines for a geothermal ORC system with five different working fluids, and discovered that the turbines had similar efficiencies but great distinctions in dimensions. Pan and Wang [6] pointed out the disadvantage of a given isentropic efficiency of a turbine in ORC analysis, then they computed the optimal internal efficiency of radial inflow turbines under different conditions to improve the reliability of the analysis. Baines [7] revised the loss models and proposed a mean-line model of radial inflow turbine based on NASA studies. The model could predict the turbine performance under various conditions, and the dependability of this model had been verified by a database of turbine experiments. Li and Ren [8] designed a radial inflow turbine using R123 based on one-dimensional method and performed numerical simulation of the designed turbine. The numerical results agreed well with the designed values, indicating that the one-dimensional method was reliable. Zheng et al. [9] presented a full design process of a radial inflow turbine used in ORC. Both one-dimensional calculation and three-dimensional Computational Fluid Dynamics (CFD) simulations were conducted, and the results showed a good agreement. Then off-design performance analysis was performed with the one-dimensional method which was considered to be convenient and accurate. For heat exchangers, Wang et al. [10] employed the log mean temperature difference method to compute the heat transfer area of plate type condenser and conducted a multi-objective optimization to obtain the optimal condenser geometry design. Calise et al. [11] investigated the influences of geometry parameters of heat exchangers on ORC performance and found that evaporator displayed a very different trend. Song et al. [12] established a prediction model of heat exchangers according to the efficiency-NTU (number of transfer unit) method, then the model was applied to analyze the off-design performance of ORC.

Generally, under off-design conditions, a specific control strategy ought to be adopted to maintain the steady operation of an ORC system. In view of the thermal degradation of geothermal resources, Gabbrielli [13] developed a detailed supercritical ORC off-design performance model to determine the optimal design parameters. The expander operated in sliding pressure mode with constant nozzle area to satisfy various operation conditions. It turned out that the ORC system designed based on the lowest geothermal temperature displayed the best economic performance. Fu et al. [14,15] conducted off-design performance analysis of a 250-kW ORC system with variable heat source temperatures and flow rates. A pressure control method was applied to ensure R245fa was in liquid and vapor saturation states at the exits of the preheater and evaporator, respectively. It turned out that higher temperature and greater flow rate of heat source led to increases in both net power output and thermal efficiency. Nevertheless, they only considered the off-design performance of the preheater. Ibarra et al. [16] researched the off-design performance of ORCs from a thermodynamic perspective and discovered that the isentropic efficiency of a scroll expander significantly influenced the cycle performance. In this work, both expander rotation speed and working pressure were regulated to satisfy different load demands. Nusiaputra et al. [17] focused on the thermal-economic performance of ORCs utilized in a mid-enthalpy geothermal power system. In order to obtain optimal operating points under off-design conditions, evaporation pressure, turbine inlet temperature and condensation temperature were respectively controlled by the turbine nozzle opening, pump speed and fan speed. Mazzi et al. [18] established a comprehensive dynamic model of a regenerative ORC to explore the influences of heat source flow rate and heat sink temperature on the cycle performance. Three PID (Proportion Integration Differentiation) controllers were employed to keep the system stable and meet actual operating constraints under off-design conditions. To develop a more accurate and reliable off-design analysis method of ORC, Hu et al. [19,20] specially designed the critical components including radial inflow turbine and plate heat exchangers for a subcritical ORC. Under different heat sources conditions, sliding pressure operation, constant pressure operation, and an optimal control strategy combining the former two methods were studied, respectively. The results showed that the optimal control strategy was superior to the other two methods. However, the variation of heat sink condition was not considered in their works. Usman et al. [21] compared the part-load operation of air cooled and cooling water tower-based geothermal ORC systems installed at different geographical locations. A sink auxiliary control optimization was performed to provide maximum power output at off-design conditions by changing the fan speed of cooling tower, air cooled condenser and mass flow rate of sink side. Rajabloo et al. [22] investigated the behaviors of low temperature and high temperature ORCs with varying conditions of the heat source and heat sink, respectively. The turbine inlet pressure, the condensing pressure and the mass flow rate of working fluid following were regulated to meet cycle requirements at off-design conditions. They considered the thermal decomposition of the working fluid and found that the properties of decomposed products could significantly affect cycle efficiency. To study the off-design performance of a small scale subcritical ORC, Liu et al. [23] developed a working fluid charge oriented model based on the component models and discovered that the optimal working fluid charge volume would differ depending on the expander’s output work. In this study, the solution of off-design model was based on the optimization of pump rotation speed to keep the system output to a stable value.

Most investigations focused on the off-design analysis of ORC were conducted based on the performance curves or simple formulas of critical components which were not specially obtained from ORC systems. Hence, predicting the off-design performance on the basis of specially designed components would be more dependable. In addition, few studies have compared the off-design performances of subcritical and supercritical ORCs with a specific control strategy. In this paper, to analyze and compare off-design performances of subcritical and supercritical geothermal ORCs with a water cooling system, an off-design performance prediction model is developed based on special designs of the critical components. Then a specific optimal control strategy which maximizes the net power output is proposed to regulate the operating parameters of both ORCs with varying geothermal water mass flow rate, geothermal water temperature and cooling water temperature. The control variables of the control strategy are turbine guide vane angle, rotational speed of refrigerant pump and cooling water mass flow rate. The findings can provide references and guidelines for the off-design operation regulation of a geothermal ORC system.

2. ORC System

In this work, both subcritical and supercritical ORCs are adopted to utilize the geothermal energy. The layout of geothermal ORC system and T-S diagrams are displayed in Figure 1 and Figure 2, respectively. The following processes are included in ORC:

• 1–2: The high pressure working fluid vapor expands to do work in the turbine.
• 2–5: The exhausted vapor is condensed by cooling water in the condenser.
• 5–6: The subcooled liquid is pumped to a high pressure in the refrigerant pump.
• 6–1: The high pressure liquid is heated into vapor by geothermal water in the evaporator (vapor generator).

2.1. Thermodynamic Modeling

The power yielded by turbine, consumed by the refrigerant pump and cooling water pump are respectively expressed as:

$$P_{Turbine}=m_{wf}(h_1-h_{2,s}){\eta }_{Turbine}=m_{wf}(h_1-h_2)$$

(1)

$$P_{RP}=\frac{m_{wf}(h_{6,s}-h_5)}{{\eta }_{RP}}=m_{wf}(h_6-h_5)$$

(2)

$$P_{WP}=\frac{m_{cw}gH}{{\eta }_{WP}}$$

(3)

Then the net power output of ORC is calculated as:

$$P_{net}=P_{Turbine}-P_{RP}-P_{WP}$$

(4)

2.2. Thermodynamic Parameter Optimization on the Design Condition

The design condition of the geothermal system is given as

Table 1. Design operating conditions of the geothermal ORC system.

Parameter	Value
Geothermal water inlet temperature (°C)	150
Geothermal water reinjection temperature (°C)	≥70
Geothermal water flow rate (kg/s)	10
Cooling water inlet temperature (°C)	20
Evaporator pinch temperature (°C)	10
Vapor generator pinch temperature (°C)	10
Condenser pinch temperature (°C)	5
Turbine isentropic efficiency (%)	80
Refrigerant pump isentropic efficiency (%)	70
Cooling water pump head (m)	20
Cooling water pump efficiency (%)	80

. For subcritical ORC, R600a is selected as a preferred working fluid [17,24,25]. Superheating is not advantageous for R600a which is a dry working fluid [26,27], so just a small superheating of 5 K is set to satisfy the practical operational constraints. For supercritical ORC, R134a is chosen due to its good performance [4,25]. To avoid cavitation in the refrigerant pump, a subcooling of 2 K is adopted in both ORCs. The regenerator cannot always enhance the thermodynamic performance of ORC at the expense of higher cost [25,27]. In addition, the regenerator can bring operative problems when geothermal water temperature is low under off-design conditions [13]. Therefore, the regenerator is not considered in this work. The geothermal water reinjection temperature should be not less than 70 °C to avoid silica oversaturation.

To validate the models developed for the simulation of ORCs, the available data with the same operating conditions and working fluids in the literature are utilized. The comparisons of simulation results with those reported in literature are presented in Figure 3. As can be seen, there is a good agreement between the values of parameters calculated in this paper and those published in literature [24,28].

To conduct the thermodynamic performance calculation, some other thermodynamic variables need to be determined in addition to the parameters listed in Table 1. For subcritical ORC, the variables are turbine inlet pressure and condensation pressure, while for supercritical ORC, an additional variable, turbine inlet temperature should be considered. These variables are regarded as decision variables to conduct a single-objective optimization based on a genetic algorithm (GA). The net power output is chosen as optimization objective since the maximum value is desirable for a geothermal power system [29,30]. In this work, all the programs are written in MATLAB 2011b (The MathWorks, Natick, Massachusetts, USA), and the thermodynamic properties are attained from REFPROP 9.0 (NIST, Gaithersburg, Maryland, USA). Assumptions and constraints adopted are listed as below:

(1)

The system is supposed in steady state.

(2)

The pressure drops in heat exchangers and pipelines are ignored.

(3)

The vapor quality of turbine outlet is restricted to not less than 1 to avoid the droplet wear phenomenon.

The optimization results are displayed in

Table 2. Optimization results of ORC systems on the design condition.

Parameter	Subcritical ORC	Supercritical ORC
Turbine inlet pressure (kPa)	1787.1	5106.6
Condensing pressure (kPa)	400.7	769.7
Turbine inlet temperature (°C)	99.4	132.8
Net power output (kW)	330.4	371.2
Mass flow rate of working fluid (kg/s)	8.07	15.06
Mass flow rate of cooling water (kg/s)	136.6	127.5

. The specific internal parameters of ORCs can be calculated based on the optimal decision variable values, including the mass flow rates of working fluid and cooling water, inlet and outlet thermodynamic states of each component. Then the critical components can be specially designed.

3. System Components Modeling and Design

To conduct the off-design performance analysis of ORC system, the performance prediction model and specific design of each important component should be determined.

3.1. Radial Inflow Turbine

Radial inflow turbines are characterized by easy manufacture, compact structures and high efficiency, and they are suitable for low flow rate and high pressure ratio situations. A variable inlet guide vane (VIGV) is a familiar method to adjust the radial turbine performance. The adjustable nozzles can rotate around the rotation centers, leading to the changes in nozzle outlet angle and area. Then the turbine performance can be regulated to adapt variable working conditions without throttling losses [31]. Thus, radial turbine with VIGV presents smooth off-design performance, and it is applied in this paper. A schematic of a radial inflow turbine is displayed in Figure 4.

The one-dimensional analysis method based on mass and energy conservation equations is employed to calculate the turbine performance. Compared with the computational fluid dynamics (CFD), this method is more convenient and practical, moreover, the prediction results can match well those of CFD [8,9]. In the turbine, the expansion process is preliminarily assumed to be isentropic, then relevant loss models are introduced to modify the flow process, making the prediction more accurate.

In stator vanes, the flow process can be expressed by the following equations:

$$h_{01}+\frac{c_{01}^2}{2}=h_{02}+\frac{c_{02}^2}{2}$$

(5)

$$m_{02}={\rho }_{02}\cdot c_{02}\cdot \sin {\alpha }_{02}\cdot A_{out,stator}$$

(6)

Supersonic flow is common at the stator outlet in ORC, which leads to a critical flow rate. In such case, the stator outlet flow angle will deviate from the installation angle and should be corrected:

$$m_{02}=m_{cr}={\rho }_{cr,02}\cdot c_{cr,02}\cdot A_{\min ,stator}$$

(7)

$$\sin {\alpha }_{02,correct}=\frac{m_{cr}}{{\rho }_{02}\cdot c_{02}\cdot A_{out,stator}}$$

(8)

A small clearance is set between the stator and rotor to avoid collision, and the tangential momentum is supposed to be conserved [32]:

$$c_{02,u}\cdot r_{02}=c_{03,u}\cdot r_{03}$$

(9)

$$m_{03}={\rho }_{03}\cdot c_{03,\mathrm{r}}\cdot A_{in,rotor}$$

(10)

In the rotors, the flow process can be expressed by:

$$h_{03}+\frac{w_{03}^2}{2}-\frac{u_{03}^2}{2}=h_{04}+\frac{w_{04}^2}{2}-\frac{u_{04}^2}{2}$$

(11)

$$m_{04}={\rho }_{04}\cdot c_{04,r}\cdot A_{out,rotor}$$

(12)

Losses models which include stator loss (L_stator), incidence loss (L_i), passage loss (L_p), disc friction loss (L_df), clearance loss (L_c) and exit loss (L_e) are introduced in this work [33,34]:

$$L_{stator}=h_{02}-h_{02,s}=\frac{c_{02,s}^2(1-K_{stator}^2)}{2}$$

(13)

$$L_i=\frac{w_{03}^{2}si{n}^{2}n}{2}$$

(14)

$$L_p=K_p\left(\frac{w_{03}^{2}co{s}^{2}n+w_{04}^2}{2}\right)$$

(15)

$$L_{df}=\frac{0.02125{\rho }_{03}{u}_{03}^2{r}_{03}^2}{m{({\rho }_{03}{u}_{03}{r}_{03}/\mu )}^{0.2}}$$

(16)

$$L_{\mathrm{c}}=\frac{u_{03}^3{Z}_{rotor}}{8\pi }(K_z{\delta }_z{M}_z+K_r{\delta }_r{M}_r+K_{zr}\sqrt{{\delta }_z{\delta }_r{M}_z{M}_r})$$

(17)

$$L_e=\frac{c_{04}^2}{2}$$

(18)

Then the turbine efficiency is given as:

$${\eta }_{Turbine}=\frac{h_{01}-h_{04}^{st}}{h_{01}-h_{04,s}}$$

(19)

Based on the above equations, a turbine submodel is established to calculate the turbine performance on different operating conditions. By inputting geometric dimensions, rotational speed, backpressure and inlet thermodynamic parameters of the turbine, the efficiency and mass flow rate can be predicted through an iteration process. Here the pressures at stator outlet and rotor inlet are previously assumed, then the thermodynamic parameters and mass flow rate of each section of the turbine can be obtained. The iteration continues by adjusting the pressures until the mass flow rate of each section becomes equivalent, then the turbine performance can be acquired.

To verify the validity of the turbine submodel, the prediction result has been compared with the data from Hu et al. [19]. As displayed in

Table 3. Turbine submodel validation.

Parameter	Reference Data	Prediction Result	Deviation
Turbine efficiency (%)	82.3	82.7	0.5%
Mass flow rate (kg/s)	5.85	5.74	1.9%
Turbine outlet temperature (°C)	47.3	46.9	0.9%
Power (kW)	68.5	67.5	1.5%

, the submodel can predict the turbine performance within an acceptable error range. Subsequently, the turbines of subcritical and supercritical ORCs are specially designed based on the optimal thermodynamic parameters for the design conditions. A radial turbine design program was developed based on the method presented by Hu et al. [19]. In this program, the inlet pressure and temperature, mass flow rate and outlet pressure of the turbine listed in Table 2 are taken as input variables, and α₀₂, β₀₂, β₀₄, N, U/C₀ and r₀₄/r₀₃ are selected as decision variables to regulate the geometric dimensions and turbine performance. In addition, structure restrictions are considered to guarantee the rationality of design. The efficiency of designed turbines should be close to 80%, which is the setting value in the thermodynamic parameter optimization on design condition. Satisfactory designs of turbines are obtained by adjusting the decision variables, and the design results are displayed in

Table 4. Radial inflow turbines design results.

Parameter	Subcritical Turbine	Supercritical Turbine
r01 (mm)	159.8	119.3
r02 (mm)	135.8	101.4
r03 (mm)	123.5	92.2
r04,tip (mm)	85.8	52.7
r04,hub (mm)	60.2	25.8
bstator (mm)	5.9	4.1
bin,rotor (mm)	5.9	4.1
α02 (°)	19	17
β04 (°)	29	36
Zstator (mm)	14	14
Zrotor (mm)	11	11
N (rpm)	17000	19000
PTurbine (kW)	402.9	498
ηTurbine (%)	80.1	79.9

.

3.2. Heat Exchangers

Plate heat exchanger (PHE) has been increasingly used in various industries owing to its high heat transfer coefficient, simple maintenance, compact structure and less fouling. Queensland Geothermal Energy Centre of Excellence (QGECE) has been considering PHE as a favorite candidate for use in geothermal power systems [35]. A welded construction PHE without any gaskets which can work under temperatures beyond 400 °C and pressures up to 100 bar has been proposed for high temperature and pressure conditions [36], which means that welded construction PHEs can be used as vapor generators in supercritical ORCs.

In this study, PHEs are selected as evaporator (vapor generator) and condenser in both subcritical and supercritical ORCs. Chevron-type corrugation plates are adopted and the basic configuration of the PHE is displayed in Figure 5.

The heat transfer process in each heat changer is divided into many small sections, and log mean temperature difference method is employed to calculate the heat transfer rate in each section:

$$Q=UA\mathrm{\Delta }{T}_m$$

(20)

The pressure drop in heat exchanger is given by:

$$\mathrm{\Delta }P=\frac{2f{G}^2}{\rho }\cdot \frac{Pl}{D}$$

(21)

For single-phase flow in evaporator and condenser, the Nusselt number and friction factor are given by Equations (22) and (23) [37,38].

$$Nu=0.724{\left(\frac{6\theta }{\mathsf{\pi }}\right)}^{0.646}R{e}^{0.583}P{r}^{1/3}$$

(22)

$$f=2.99/R{e}^{0.183}$$

(23)

A correlation obtained by experimental analysis is employed to calculate the heat transfer of supercritical fluid in vapor generator [35]:

$$Nu=0.187R{e}^{0.71}P{r}^{0.35}{\left(\frac{{\tilde{c}}_p}{c_{p,b}}\right)}^{0.5}{\left(\frac{{\rho }_w}{{\rho }_b}\right)}^{0.3}$$

(24)

For the evaporation process in evaporator, the following correlations are adopted to determine the pressure drop and heat transfer, and θ_max equals to 70° [39].

$$f=15.698C{\left(\frac{G^{2}D}{{\rho }_m\sigma }\right)}^{-0.475}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{-0.571}B{d}^{0.255}$$

(25)

If Bd < 4

$$Nu=982{\left(\frac{\theta }{{\theta }_{\max }}\right)}^{1.101}{\left(\frac{G^{2}D}{{\rho }_m\sigma }\right)}^{0.315}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{-0.224}B{o}^{0.320}$$

(26)

If Bd ≥ 4

$$Nu=18.495{\left(\frac{\theta }{{\theta }_{\max }}\right)}^{0.248}{\left(\frac{xGD}{{\mu }_g}\right)}^{0.135}{\left(\frac{GD}{{\mu }_l}\right)}^{0.351}{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{0.223}B{o}^{0.198}B{d}^{0.235}$$

(27)

where:

$$Bd=\frac{\left({\rho }_l-{\rho }_g\right)g{D}^2}{\sigma }$$

(28)

$$Bo=\frac{q}{G\gamma }$$

(29)

$$C=2.125{\left(\frac{\theta }{{\theta }_{\max }}\right)}^{9.993}+0.955$$

(30)

For condensation process in condenser, the Nusselt number and friction factor are respectively expressed as [40]:

$$Nu=4.118R{e}_{eq}^{0.4}P{r}_l^{1/3}$$

(31)

$$f=94.75{\left(\frac{p_m}{P_{cr}}\right)}^{0.8}B{o}^{0.5}R{e}^{-0.4}R{e_{eq}}^{-0.4}$$

(32)

in which P_cr denotes critical pressure, and equivalent Reynolds number is given as:

$$R{e}_{eq}=\frac{G_{eq}D}{{\mu }_l}$$

(33)

where:

$$G_{eq}=G\left[\left(1-x\right)+x{\left(\frac{{\rho }_l}{{\rho }_g}\right)}^{0.5}\right]$$

(34)

If the geometric dimensions of a PHE are specified, the required heat transfer area can be calculated by above corresponding correlations, mass flow rates and thermodynamic parameters on both sides of the PHE. Under off-design conditions, the thermodynamic parameters at the ends of PHE and mass flow rates are not all known, and the required heat transfer area can be acquired by assuming the unknown quantities. The assumed quantities should be adjusted until the obtained area is consistent with the actual heat transfer area of PHE [20].

Afterwards the specific designs of evaporator (vapor generator) and condenser are performed in accordance to the optimal thermodynamic parameters on the design condition. All the PHEs are arranged as counterflow single-pass flow [10,19], and the pressure drops are limited to 2%. The design results of PHEs are displayed in

Table 5. Heat exchangers design results.

Parameter	Evaporator (Subcritical ORC)	Condenser (Subcritical ORC)	Vapor Generator (Supercritical ORC)	Condenser (Supercritical ORC)
Plate width (m)	0.46	0.87	0.75	0.67
Plate length (m)	0.7	1.3	1.5	1.38
Plate thickness (mm)	0.6	0.6	0.6	0.6
Channel distance (mm)	3.0	3.1	2.5	3.1
Chevron angle (°)	60	60	60	60
Heat transfer area (m2)	48.0	168.5	120.4	182.2
Pressure drop (kPa)	16.9	8.4	21.1	7.9

.

3.3. Pump

The methodology of the pump adopted in [16,19] is employed in this work. The non-dimensional performance curve of the refrigerant pump is derived from a centrifugal pump in a real power plant, as presented in Figure 6. The nominal rotational speed is assumed as 1500 rpm, and the efficiency comes to the maximum value of 70% at the design flow rate and head. For off-design conditions, the rotational speed can be adjusted, and the off-design performance can be acquired base on the non-dimensional performance curve and affinity laws.

The cooling water pump makes cooling water flow circulate between condenser and cooling tower, and the power consumed by it is quite small. For simplicity, the head and efficiency of cooling water pump are taken as constant values displayed in Table 1 for off-design conditions.

4. Results and Discussion

4.1. Off-Design Performance Optimization

In a geothermal ORC system, the heat source and heat sink conditions may frequently vary due to the variations of geological conditions, ambient temperature and actual operation, then a specific control strategy should be adopted to keep the system operating stably by adjusting the operating parameters. In this work, for the off-design operation, the subcooling is fixed at 2 K for both ORCs, and the superheating is fixed at 5 K for subcritical ORC. A control strategy which maximizes the net power output is employed, and off-design performance analysis is carried out with variable geothermal water mass flow rate (m_gw), geothermal water inlet temperature (T_gw) and cooling water inlet temperature (T_cw).

To predict the off-design performance of a specified operation point, the values of m_gw, T_gw and T_cw should be given, the rotational speed of refrigerant pump (N_RP), turbine guide vane angle (α₀₂) and cooling water mass flow rate (m_cw) are regarded as control variables to adjust the system performance. The ORC system in this work is for power generation and supposed to connect to the grid, therefore the rotational speed of turbine remains constant. To conduct the submodel performance prediction, some operation parameters need to be previously assumed. For subcritical ORC, the assumed variables include turbine inlet pressure (P₁), condensation pressure (P₂). The turbine inlet temperature (T₁) can be obtained by the fixed superheating and P₁. While for supercritical ORC, one more variable, namely T₁ should be assumed. For both subcritical and supercritical ORCs, according to the values of P₁, P₂, α₀₂ and T₁, the working fluid mass flow rate (m_wf), thermodynamic state at the outlet, as well as the off-design performance of the turbine can be determined by the turbine submodel. Then the off-design performances of condenser, refrigerant pump and evaporator (vapor generator) can be successively calculated by the corresponding submodels. It is noteworthy that the assumed operation parameters need to be adjusted until the calculated results agree with the actual component properties. In this work, optimizations based on genetic algorithm are respectively performed for subcritical and supercritical ORCs. The net power output is chosen as optimization objective, and the control variables mentioned above are regarded as decision variables. The detailed flow chart of off-design performance optimization is shown in Figure 7.

4.2. Various Geothermal Water Mass Flow Rates

In this section, subcritical and supercritical ORCs with various m_gw are optimized, meanwhile T_gw and T_cw are fixed at the designed values. As displayed in Figure 8, the variations of the optimal control variables with m_gw are similar for subcritical and supercritical ORCs. As m_gw grows, α₀₂ and N_RP both gradually increase. The growth of α₀₂ is obvious, while the increase in N_RP is slight. In addition, m_cw first gradually increases and then becomes stable with m_gw for both ORCs, but the overall change is fairly small.

To further look into the off-design performances of ORCs with m_gw, the optimal operation parameters are also analyzed. For both ORCs, P₁, namely the pressure of heater (evaporator or vapor generator), is roughly constant, and P₂ slowly rises with m_gw, as shown in Figure 9a,b. For supercritical ORC, T₁ nearly remains constant. The m_wf of the whole system is equivalent to the turbine mass flow rate. In this paper, supersonic flow always appears in the turbine nozzle due to the high expansion ratio, then the mass flow rate reaches the critical value. Thus, m_wf only depends on P₁, T₁ and α₀₂.

As displayed in Figure 9c, for both ORCs, m_wf grows with m_gw, which is mainly due to the increase in α₀₂. The enthalpy rise in the evaporator (vapor generator) is stable due to the roughly constant T₁ and P₁, and m_wf increases to absorb more heat from geothermal water. Meanwhile, the heat transfer rate in the condenser also grows, but m_cw changes very little. The slow increase in P₂ means the condensation temperature increases, then the condenser pinch temperature difference tends to be larger, which could satisfy the higher heat transfer rate in the condenser. As to the refrigerant pump, for both ORCs, with the increase in m_gw, the pump head is almost constant while the volumetric flow rate gradually grows due to the rising m_wf. Thus, a slow increase in N_RP simultaneously satisfies the pump head and volumetric flow rate (see Figure 8b).

The lower the geothermal water reinjection temperature (T_re), the more fully the geothermal energy is utilized. However, T_re should not be lower than 70 °C to avoid the silica oversaturation. As displayed in Figure 9d, for subcritical ORC, as m_gw grows, the utilization of geothermal energy drops, which can be deduced by the growing T_re. This may be due to the limited heat transfer area of the evaporator, while for supercritical ORC, when m_gw is smaller than the design value, T_re comes to the minimum allowed value of 70 °C, and m_wf degrades rapidly as m_gw falls (see Figure 9c). This can be explained by the fact that if m_wf does not decrease fast, the working fluid would absorb more heat from geothermal water, leading to the consequence that T_re is lower than 70 °C When m_gw is greater than designed value, T_re grows rapidly with m_gw, which implies the heat amount absorbed by the working fluid becomes relatively small, resulting in a slow increase in m_wf.

For both ORCs, as m_gw grows, both the turbine power output and the power consumed by refrigerant pump rise owing to the increase in m_wf, and the power consumed by cooling water pump changes very little because of the small variation of m_cw. Since the increased turbine power output more than compensates for the increase in power consumed by the pumps, the net power output (P_net) rises with m_gw, as shown in Figure 10. Although P_net of supercritical ORC is always greater than that of subcritical ORC, the difference between them becomes smaller when m_gw deviates from the design value, and this is mostly due to the variation of turbine power output. For supercritical ORC, when m_gw is smaller than the design value, the m_wf degrades rapidly to satisfy the constraint of T_re, resulting in a quick decline in turbine power output, while when m_gw is greater than the design value, the slow growth of m_wf leads to a slow increase in turbine power output.

4.3. Various Geothermal Water Inlet Temperatures

In this part, subcritical and supercritical ORCs with various T_gw are optimized, meanwhile m_gw and T_cw are fixed at the designed values. The variations of optimal control variables are showed in Figure 11. For supercritical ORC, α₀₂ basically remains the same value, while for subcritical ORC, a slightly decline can be observed. It is noteworthy that abnormal small α₀₂ emerges when T_gw reaches the lowest value of 130 °C for both ORCs. Besides, for both ORCs, N_RP gradually increases with T_gw, and m_cw remains basically stable.

The variations of optimal operation parameters with T_gw are shown in Figure 12. For both ORCs, P₁ gradually increases and P₂ slightly rises with T_gw, as shown in Figure 12a,b. For supercritical ORC, as T_gw grows, T₁ also gradually increases to generate a better temperature match between working fluid and geothermal water.

As presented in Figure 12c, for both ORCs, m_wf gradually increases with T_gw. For supercritical ORC, α₀₂ is nearly constant, the increase of m_wf is mainly determined by higher P₁ and T₁. For subcritical ORC, although α₀₂ basically decreases slowly, which means that m_wf has a decline trend, higher P₁ and T₁ can make up for this trend, consequently m_wf grows. As to the refrigerant pump, for both ORCs, N_RP adds with T_gw to satisfy the increases in volumetric flow rate and pump head, which are mainly caused by the increasing m_wf and difference between P₁ and P₂, respectively.

As displayed in Figure 12d, for both ORCs, T_re generally increases as T_gw rises. Likewise, when T_gw comes to the minimum value of 130 °C, T_re becomes abnormal. For subcritical ORC, when T_gw changes from 135 °C to 130 °C, the decline rate of T_re slows down significantly, this may be due to the unusual small α₀₂ (see Figure 11a), which leads to a small m_wf, then the restriction on the lowest T_re (70 °C) can be satisfied. For supercritical ORC, abnormal high T_re emerges when T_gw reaches 130 °C. This is because when T_gw reaches 130 °C, P₁ tends to be lower than the critical pressure of 4.06 Mpa for R134a (see the dashed line in Figure 12a). However, P₁ must be higher than the critical pressure to meet the supercritical condition, and high P₁ contributes to large m_wf, which leads to abnormal small α₀₂ (see Figure 11a) to balance m_wf, eventually the geothermal energy fails to be fully utilized, and abnormal high T_re emerges.

For both ORCs, P_net gradually increases with T_gw, and P_net of supercritical ORC is greater than that of subcritical ORC, as displayed in Figure 13. As T_gw rises, working fluid per unit mass flow rate can produce more power due to the higher T₁ and P₁, and m_wf also grows, thus turbine power output increases. The power consumed by refrigerant pump also grows owing to the increases of m_wf and difference between P₁ and P₂. The power consumed by cooling water pump changes very little due to the small variation of m_cw. Eventually, P_net increases since the increased turbine power output can make up for the increase of power consumed by the pumps.

4.4. Various Cooling Water Inlet Temperatures

In this section, subcritical and supercritical ORCs with various T_cw are optimized, meanwhile m_gw and T_gw are fixed at the designed values. As shown in Figure 14a, for both ORCs, α₀₂ first rises and then drops with T_cw, but the maximum α₀₂ corresponds to different T_cw which is 25 °C for supercritical ORC and 15 °C for subcritical ORC. As presented in Figure 14b, N_RP is nearly constant for subcritical ORC, while N_RP first increases and then decreases with T_cw for supercritical ORC, and the maximum N_RP also corresponds to T_cw of 25 °C Nonetheless, the variations of α₀₂ and N_RP are overall small for both ORCs. In addition, m_cw is approximately steady for both ORCs.

As presented in Figure 15a, as T_cw increases, for supercritical ORC, P₁ keeps roughly constant, while for subcritical ORC, P₁ slowly rises except the abnormal high P₁ which corresponds to T_cw of 10 °C. For both systems, P₂ gradually increases, as shown in Figure 15b.

Figure 15c illustrates the variation of m_wf with T_cw. For supercritical ORC, owing to the slight variations of T₁ and P₁, the variation of m_wf is similar with that of α₀₂. For subcritical ORC, m_wf is almost constant, this is mainly because the variations of P₁ and α₀₂ are just opposite (see Figure 14a and Figure 15a), and their effects on m_wf offset each other. As to the refrigerant pump, for both ORCs, the pump head is roughly constant as T_cw increases, N_RP mainly varies to satisfied the variation of volumetric flow rate, which is consistent with the change in m_wf. Thus, the variation of N_RP is just similar with that of m_wf for both ORCs. As displayed in Figure 15d, for both ORCs, T_re generally decreases as T_cw drops. This indicates that ORC with lower T_cw tends to utilize geothermal energy more fully, which is mainly determined by the lower P₂. However, the decline rate of T_re is uneven because of the lowest T_re restriction of 70 °C. For supercritical ORC, when T_cw is lower than 25 °C, T_re is close to 70 °C, and the decline rate of T_re obviously slows down. In this case, the lowest T_re restriction is activated, α₀₂ reduces as T_cw declines (see Figure 14a); meanwhile P₁ and T₁ are roughly constant, resulting in the decrease of m_wf, so that T_re is not lower than 70 °C. As to subcritical ORC, when T_cw is lower than 15 °C, the decline rate of T_re slows down, and the lowest T_re restriction is activated. Thus, P₁ and α₀₂ become abnormal when T_cw reaches 10 °C (see Figure 14a and Figure 15a). For both ORCs, as T_cw drops, the turbine power output grows owing to the decrease in P₂, the power consumed by refrigerant pump is almost constant due to the small changes of m_wf and difference between P₁ and P₂, and the power consumed by cooling water pump changes very little_. Thus, P_net grows as T_cw drops, as shown in Figure 16. On the whole, P_net of subcritical ORC is always smaller than that of supercritical ORC, however, the difference between them presents a decrease trend as T_cw drops, which is owing to the reduction in m_wf of supercritical ORC.

4.5. Various Geothermal Water Inlet Temperatures and Cooling Water Inlet Temperatures

In practical operation of geothermal ORC power system, m_gw can be actively regulated to some extent, but T_gw influenced by geological condition and T_cw determined by ambient temperature depend on objective conditions. These two variables are independent of each other and can simultaneously change. Thus, the off-design performances of both ORCs are further discussed with various T_gw and T_cw.

Figure 17a shows that the variations of α₀₂ are quite different for subcritical and supercritical ORC. For subcritical ORC, α₀₂ first basically remains constant then drops with T_gw, while for supercritical ORC, α₀₂ first increases then becomes roughly stable with T_gw. As displayed in Figure 17b, N_RP gradually grows with T_gw for both ORCs. As T_cw varies, N_RP is roughly constant for subcritical ORC, while for supercritical ORC, when T_cw is lower than 20 °C, N_RP has a declining trend as T_cw drops.

Figure 18a,b illustrate that P₁ gradually increases and P₂ slowly rises with T_gw, while P₁ does not change significantly and P₂ gradually increases with T_cw, for both ORCs. As shown in Figure 18c, m_wf gradually grows with T_gw for both ORCs. As T_cw varies, m_wf is roughly constant for subcritical ORC, while for supercritical ORC, the variations of m_wf are similar with that of N_RP (see Figure 17b). When T_cw is low, the decline trend of m_wf ensures that T_re is not lower than 70 °C.

Figure 18d illustrates that for both ORCS, lower T_gw and T_cw leads to lower T_re, and supercritical ORC is more likely to activate the constraint of T_re. This is mainly due to the varying temperature of the working fluid during the transition from liquid to vapor in the vapor generator, which leads to a better temperature match between working fluid and geothermal water. Notably, for supercritical ORC, when T_gw comes to 130 °C, abnormal high T_re appears, which may be owing to the supercritical condition restriction. As presented in Figure 19, for both ORCs, higher T_gw and lower T_cw bring about the increase in P_net. The P_net of supercritical ORC is always greater than that of subcritical ORC. However, for supercritical ORC, the growth rate of P_net slows down with the decrease of T_cw (below 20 °C), leading to smaller difference of P_net between supercritical and subcritical ORCs. This is mainly owing to the reduction of m_wf (see Figure 18c), which is caused by the constraint of T_re.

5. Conclusions

In this study, the parameter designs of subcritical and supercritical ORCs applied in geothermal power systems were firstly conducted under the design conditions. Then the off-design performance prediction model of ORC was established based on specially designed turbine and heat exchangers. Lastly, a specific optimal control strategy was proposed to maximum the net power output, and the off-design performances of subcritical and supercritical ORCs were comprehensively analyzed and compared under the optimal control strategy. The main conclusions of this work can be summarized as follows:

For both ORCs, under the optimal control strategy, greater geothermal water mass flow rate, higher geothermal water inlet temperature, lower cooling water inlet temperature bring about larger net power output. As geothermal water mass flow rate grows, the turbine guide vane angle gradually increases, and the refrigerant pump rotational speed slowly increases; the turbine inlet pressure changes very little, and the condensing pressure slowly rises. Owing to the increase in turbine guide vane angle, the working fluid mass flow rate rises to absorb more heat from geothermal water, leading to the growth in net power output. As the geothermal water inlet temperature increases, the turbine guide vane angle is relatively stable while the refrigerant pump rotational speed grows; the turbine inlet pressure gradually increases, the condensing pressure slowly rises, and the working fluid mass flow rate grows. The rise of net power output is mainly owing to the increases of turbine inlet pressure and working fluid mass flow rate. With cooling water inlet temperature drops, the turbine guide vane angle and the refrigerant pump rotational speed change a little; the turbine inlet pressure and the working fluid mass flow rate are roughly constant, while the condensing pressure significantly declines. The increase in net power output is mainly determined by the reduced condensing pressure.

Generally, under off-design operating conditions, the optimal control strategies of subcritical and supercritical ORCs are roughly approximate, and the variations of net power output of them are similar. The net power output of supercritical ORC is always greater than that of subcritical ORC, and the difference between them tends to drop when the supercritical ORC activates the restriction of geothermal water reinjection temperature. It is worth noting that smaller the geothermal water mass flow rate, the lower geothermal water inlet temperature and lower cooling water inlet temperature are inclined to activate the restriction of reinjection temperature, especially for supercritical ORC, which is due to the better temperature match in the vapor generator. When the restriction of reinjection temperature is activated, the turbine guide vane angle tends to decrease, resulting in a reduction in the working fluid mass flow rate to ensure that the reinjection temperature is not less than 70 °C, which further leads to a decreasing tendency of the net power output.

The proposed optimal control strategy can provide references and guidelines for the off-design operation regulation of subcritical and supercritical ORC systems. On the basis of this research, the next steps of our work are to establish a comprehensive dynamic model of ORC systems, study the dynamic characteristics under off-design conditions, and put forward a reasonable control logic and specific controller model.