Capability Curve Modeling for Hydro-Power Generators in Optimal Power Flow Problems

Abstract

With the growing emphasis on sustainability in the power sector, it becomes imperative to ensure that every component of the power system operates optimally and efficiently. More sustainable power system operation can be achieved by relying on accurate models that ensure resources, especially those from renewable sources like hydro-power, are utilized to their fullest potential. In many optimization problems based on optimal power flow formulations, the steady-state operation characteristics of hydro-power plants are modeled in an approximate manner, which could potentially lead to solutions that do not fully exploit their capabilities or even to solutions that jeopardize their stable operation. This work proposes a formulation for the complete capability curve of the plant, including the exact modeling of its generator stability limits. The ability of the proposed formulation to reproduce the plant operation boundaries is appropriately demonstrated through a test case. Furthermore, two approximate formulations commonly used in the literature are solved, highlighting their limitations. It is concluded that the complete representation of the capability curve can improve the quality of solutions provided by OPF-based problems.

1. Introduction

Optimal power flow (OPF) stands as a cornerstone in the operation and planning of power systems [1,2,3]. It involves determining the best operating conditions for the system, ensuring the minimization of production costs while adhering to constraints like the power balance, voltage limits, and equipment capabilities. The significance of OPF has grown exponentially with the modernization of power grids, especially as they integrate more renewable energy, distributed generation, and fluctuating consumer demands. Through its capability to derive the best operational strategies, OPF enhances overall system efficiency, boosts reliability, and ensures the electricity provided is in line with both technical standards and economic considerations. As power systems continue to progress, the pivotal role of OPF in synchronizing generation, transmission, and demand becomes increasingly pronounced, emphasizing its crucial role in the future of sustainable and efficient power systems.

Improving the mathematical models of power system components within the OPF framework is paramount for the accurate and efficient operation of contemporary power grids. As the power network becomes more intricate with the integration of diverse energy sources and advanced technologies, the models used to represent system components need to be refined to capture their interactions. In this context, the modeling of power plants’ production limits is a key aspect to ensure appropriate operation of the power system. These limits, traditionally specified as minimum and maximum power outputs [4], define the boundaries within which a power plant can operate without violating technical constraints. Expanding this simple representation of limits to include the generator capability curve provides a more comprehensive and accurate means of modeling power plant operations. The capability curve represents the feasible operating regions of the power plant, offering a more precise representation of the real and reactive power production constraints than the “rectangle” limitations derived from the minimum and maximum power outputs. By incorporating this curve into the OPF, system operators can align decisions more closely with the actual operational constraints and capabilities of power plants, thereby ensuring enhanced system reliability and efficiency.

In the technical literature, there are a few studies that incorporate the capability curve of generators in the formulation of the OPF, e.g., [5,6,7,8]. In these references, the representation of the capability curve is approximate in the sense that it does not include the limitations imposed by stability criteria. This study focuses on modeling the capability curve of a hydroelectric plant, explicitly including the steady-state stability limit of its synchronous generator. The accuracy of the proposed formulation is tested by solving a maximum loading condition problem on a one-machine isolated power system. In this optimization problem, a homotopy modeling technique is used to drive the hydro-power plant to its operating limits. The results obtained with the proposed model are compared with those provided by two approximate models that are frequently applied in the literature.

2. Capability Curve Background

The capability curve of a synchronous generator provides a graphical representation of the operational limits of the electric machine. It represents the boundaries within which the generator can operate safely without overheating or violating stability constraints. If turbine operation constraints are incorporated, the resulting curve offers a comprehensive view of the operational capacity of the power plant. Figure 1 illustrates a representative capability curve with all values expressed per unit relative to the rated power of the machine. It is basically a plot that shows the active power on the x-axis and the reactive power on the y-axis. The curve defines the maximum and minimum limits for generating or absorbing reactive power at various levels of active power.

Above the x-axis lies the lagging operating zone, corresponding to the overexcited operation of the generator, where it supplies reactive power to the system. Conversely, below the x-axis is the leading operating zone, associated with its underexcited operation, where the machine absorbs reactive power from the system.

The grey area of the capability curve (between limits) represents the safe operating points of the generator. Operating outside this area could result in serious damage to the turbine-generator unit equipment. A description of the limits follows:

• Stator current limit. The outer boundary of the capability curve is primarily defined by the stator current limit. The stator winding can only carry a certain amount of current before overheating. Both active power and reactive power contribute to the stator current.
• Maximum excitation limit. The upper portion of the capability curve is constrained by the maximum allowable excitation (or field) current. This direct current produces the magnetic field in the rotor. There is a maximum limit to how much current the field winding can carry without overheating. When the field current is at its maximum, the generator is at its maximum lagging reactive power production [9].
• Minimum excitation limit. The lower segment of the capability curve is limited by the minimum permissible excitation current. Typically, this current is zero, indicative of a no-excitation scenario. Such a condition could lead to a reactive power flow into the generator well beyond its rated capacity. Furthermore, some exciter configurations have inherent operational constraints, preventing them from supplying the field winding with a current below a specific positive threshold, thereby establishing a minimum excitation current above zero [10].
• Maximum active power limit. The rightmost part of the curve represents the rated active power output of the turbine-generator unit, limited by the maximum capacity of its turbine.
• Minimum active power limit. Some turbines also have a minimum active power limit below which their operation is not advisable. Francis and Kaplan hydro turbines operate inefficiently when they run below certain percentages of their peak capacity [11]. As breaching these limits does not result in actual damage to the turbine-generator unit, they have not been included in the capability curve of Figure 1.
• Steady-state stability limit. The lower portion of the capability curve is also constrained by steady-state stability considerations. Beyond the theoretical stability limit, the generator becomes unstable and risks falling out of synchrony with the grid. Given that this situation can be very dangerous for the machine’s integrity, in practice, a security margin is typically imposed [12,13], leading to the practical steady-state stability limit shown in the figure.

The capability curve for round-rotor synchronous generators has an additional limit due to stator end core heating. This heating originates from the flux that radiates from the end of the stator core when the generator operates at a low field current. Though this heating effect is also observed in salient-pole synchronous generators, it is not commonly a limiting factor [9,10].

Finally, the capability curve of a synchronous generator is intrinsically linked to the terminal voltage. Variations in this voltage can shift the operational limits [9]. Regarding this fact, the capability curve presented in Figure 1 corresponds to a generator terminal voltage of 1 pu.

3. Steady-State Stability Limits

According to d-q theory [12], the active and reactive powers produced by a salient-pole synchronous generator can be formulated as follows:

$$\begin{array}{cc}\hfill p^{\mathrm{G}}& =\frac{e^{\mathrm{f}}v}{X_{\mathrm{d}}}sin\delta +v^2\left(\frac{1}{X_{\mathrm{q}}}-\frac{1}{X_{\mathrm{d}}}\right)sin\delta cos\delta \hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill q^{\mathrm{G}}& =\frac{e^{\mathrm{f}}v}{X_{\mathrm{d}}}cos\delta -v^2\left(\frac{{cos}^2\delta }{X_{\mathrm{d}}}+\frac{{sin}^2\delta }{X_{\mathrm{q}}}\right)\hfill \end{array}$$

(2)

where $p^{\mathrm{G}}$ and $q^{\mathrm{G}}$ are the active and reactive output powers, $e^{\mathrm{f}}$ is the magnitude of the internal voltage (induced electromagnetic force), v is the magnitude of the terminal voltage, $\delta$ is the angle between the internal and terminal voltages, and $X_{\mathrm{d}}$ and $X_{\mathrm{q}}$ are the direct- and quadrature-axis synchronous reactances, respectively.

For given values of the internal and terminal voltage magnitudes, the theoretical steady-state stability limit of the generator corresponds to the value of $\delta$ that maximizes the active power production of the generator. Therefore, such a limit can be obtained by solving

$$\begin{array}{c}\hfill \frac{\partial p}{\partial \delta }=0\end{array}$$

(3)

which, according to (1), results in the following equation:

$$\begin{array}{c}\hfill \frac{e^{\mathrm{f}}v}{X_{\mathrm{d}}}cos\delta +v^2\left(\frac{1}{X_{\mathrm{q}}}-\frac{1}{X_{\mathrm{d}}}\right)cos2\delta =0\end{array}$$

(4)

Solving (4) for $cos\delta$ yields

$$\begin{array}{c}\hfill cos\delta =\frac{1}{4}·\left(-\frac{e^{\mathrm{f}}}{v}·\frac{X_{\mathrm{q}}}{X_{\mathrm{d}}-X_{\mathrm{q}}}+\sqrt{{\left(\frac{e^{\mathrm{f}}}{v}·\frac{X_{\mathrm{q}}}{X_{\mathrm{d}}-X_{\mathrm{q}}}\right)}^2+8}\right)\end{array}$$

(5)

The values of $\delta$ computed by Equation (5), which hereinafter will be denoted by ${\delta }^{\mathrm{st}}$, together with the values of the internal and terminal voltage magnitudes, define the values of the active and reactive powers corresponding to the steady-state stability limit, i.e.,

$$\begin{array}{cc}\hfill p^{\mathrm{G},\mathrm{st}}& =\frac{e^{\mathrm{f}}v}{X_{\mathrm{d}}}sin{\delta }^{\mathrm{st}}+v^2\left(\frac{1}{X_{\mathrm{q}}}-\frac{1}{X_{\mathrm{d}}}\right)sin{\delta }^{\mathrm{st}}cos{\delta }^{\mathrm{st}}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill q^{\mathrm{G},\mathrm{st}}& =\frac{e^{\mathrm{f}}v}{X_{\mathrm{d}}}cos{\delta }^{\mathrm{st}}-v^2\left(\frac{{cos}^2{\delta }^{\mathrm{st}}}{X_{\mathrm{d}}}+\frac{{sin}^2{\delta }^{\mathrm{st}}}{X_{\mathrm{q}}}\right)\hfill \end{array}$$

(7)

An example of the values obtained from Equations (6) and (7) are the points on the curve that represent the theoretical stability limit in Figure 1. Due to the risk involved in operating the generator close to its steady-state stability limit, in practice, this limit is often reduced by a constant value corresponding to 10% of the generator rated power [12,13]. This reduction results in the practical stability limit shown in Figure 1.

4. Capability Curve Model for OPF Problems

Based on what was described in the previous sections, the proposed formulation for the operational limits related to the capability curve of a hydro-power plant is as follows:

• Stator current limit

  $$\begin{array}{cc}\hfill & {\left(p^{\mathrm{G}}\right)}^2+{\left(q^{\mathrm{G}}\right)}^2\le {\left(v·I^{\mathrm{G},\max }\right)}^2\hfill \end{array}$$

  (8)

  where $I^{\mathrm{G},\max }$ is the rated power of the generator.
• Maximum excitation limit.

  $$\begin{array}{cc}\hfill & e^{\mathrm{f}}\le E^{\mathrm{f},\max }\hfill \end{array}$$

  (9)

  where $E^{\mathrm{f},\max }$ is the internal voltage that corresponds to the rated field current of the generator.
• Minimum excitation limit.

  $$\begin{array}{cc}\hfill & e^{\mathrm{f}}\ge E^{\mathrm{f},\min }\hfill \end{array}$$

  (10)

  where $E^{\mathrm{f},\min }$ is either zero or a small percentage of $E^{\mathrm{f},\max }$.
• Maximum active power limit.

  $$\begin{array}{cc}\hfill & p^{\mathrm{G}}\le P^{\mathrm{G},\max }\hfill \end{array}$$

  (11)

  where $P^{\mathrm{G},\max }$ is the rated power of the hydro turbine.
• Minimum active power limit.

  $$\begin{array}{cc}\hfill & p^{\mathrm{G}}\ge P^{\mathrm{G},\min }\hfill \end{array}$$

  (12)

  where $P^{\mathrm{G},\min }$ is either zero or a percentage of $P^{\mathrm{G},\max }$.
• Steady-state stability limit.

  $$\begin{array}{cc}\hfill & cos{\delta }^{\mathrm{st}}=\frac{1}{4}\left(-\frac{e^{\mathrm{f}}}{v}\frac{X_{\mathrm{q}}}{X_{\mathrm{d}}-X_{\mathrm{q}}}+\sqrt{{\left(\frac{e^{\mathrm{f}}}{v}·\frac{X_{\mathrm{q}}}{X_{\mathrm{d}}-X_{\mathrm{q}}}\right)}^2+8}\right)\hfill \end{array}$$

  (13)

  $$\begin{array}{cc}\hfill & {sin}^2{\delta }^{\mathrm{st}}+{cos}^2{\delta }^{\mathrm{st}}=1\hfill \end{array}$$

  (14)

  $$\begin{array}{cc}\hfill & p^{\mathrm{G},\mathrm{st}}=\frac{e^{\mathrm{f}}·v}{X_{\mathrm{d}}}·sin{\delta }^{\mathrm{st}}+v^2\left(\frac{1}{X_{\mathrm{q}}}-\frac{1}{X_{\mathrm{d}}}\right)sin{\delta }^{\mathrm{st}}cos{\delta }^{\mathrm{st}}\hfill \end{array}$$

  (15)

  $$\begin{array}{cc}\hfill & p^{\mathrm{G}}\le p^{\mathrm{G},\mathrm{st}}-0.1\hfill \end{array}$$

  (16)

The previous formulation of the capability curve constraints assumes that the parameters and limits are expressed in per unit values with respect to the generator ratings. Therefore, the value $0.1$ in Equation (16) represents the stability margin equal to 10% of the rated power of the generator, as established in Section 3. Observe that all these values would have to be converted to a common power base when the model is incorporated in an OPF problem involving a whole power system.

5. Test Case

The proposed formulation of the capacity curve constraints was tested by solving an OPF problem of the type known as a maximum loading condition problem [14]. Such problems focus on maximizing a homotopy parameter, which usually represents the load margin for a specific operating point of a power system. However, analogous models have been applied in different contexts [15]. The ad hoc problem formulation used here is as follows:

$$\begin{array}{cc}\hfill & \mathrm{Maximize}z=\lambda \hfill \\ \hfill & \mathrm{s}.\mathrm{t}.\hfill \\ \hfill & \mathrm{Power}\mathrm{balance}\mathrm{constraints}(\mathrm{standard}\mathrm{power}\mathrm{flow}\mathrm{equations})\hfill \\ \hfill & \mathrm{Constraints}\left(1\right),\left(2\right),\mathrm{and}\left(8\right)–\left(16\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & p^{\mathrm{D}}=\lambda ·cos\phi \hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & q^{\mathrm{D}}=\lambda ·sin\phi \hfill \end{array}$$

(18)

where variable $\lambda$ is the homotopy parameter, and $p^{\mathrm{D}}$ and $q^{\mathrm{D}}$ are variables representing the active and reactive power demands, respectively. The parameter $\phi$ acts as the power factor of the demand and guides the demand increase modeled by Equations (17) and (18). The power balance constraints are represented by the well-known power flow equations (see, e.g., [14]).

The test system consists of a hydro-power plant supplying a single load through a transmission line, as depicted in Figure 2. The parameters of the synchronous machine and the limits of the power plant capability curve are provided in

Table 1. Turbine-generator unit data, [p.u.].

${\mathit{X}}_{\mathbf{d}}$	${\mathit{X}}_{\mathbf{q}}$	${\mathit{P}}^{\mathbf{G},\mathbf{max}}$	${\mathit{P}}^{\mathbf{G},\mathbf{min}}$	${\mathit{I}}^{\mathbf{G},\mathbf{max}}$	${\mathit{E}}^{\mathbf{G},\mathbf{max}}$	${\mathit{E}}^{\mathbf{G},\mathbf{min}}$
$1.1$	$0.7$	$0.9$	$0.0$	$1.0$	$1.76$	$0.088$

. For simplicity, the system data are expressed per unit relative to a common power and voltage base, which aligns with the rated values of the generator. The transmission line is modeled by a reactance of $0.15$ pu. The generator controls the voltage magnitude at bus 1, which is set to $1.0$ pu.

The objective of this test case is to ensure that the proposed formulation accurately captures the constraints placed on the generator production by its capability curve. In simple terms, a maximum loading condition problem seeks to maximize the load of the system, that is, to elevate as much as possible the load supplied by the system. The solution is obtained when increasing the load inevitably leads to the violation of some of the technical limits represented by the constraints of the problem. The test system is set up so that the only factor that could limit the increase in load are the limits imposed by the power plant production limits. In order to explore the whole operating range of the plant, the maximum loading condition problem is solved for the entire spectrum of demand power factor values, i.e., $\phi \in [-90°,90°]$. Therefore, stepping through this range in increments of 1 degree, the optimization problem was solved for every respective value of $\phi$. The solver applied was CONOPT [16] under GAMS [17]. The operating points of the plant corresponding to each solution obtained with the maximum loading condition problem are depicted in Figure 3 as green solid circles. Dotted lines are the limits described in Section 2.

It can be observed how the solutions are distributed throughout the perimeter of the grey area. Therefore, by comparison with Figure 1, it can be concluded that the proposed formulation is capable of modeling all the limitations included in the complete capability curve of the power plant.

To graphically illustrate the implications of the approximate modeling of the capability curve, the same simulation has been carried out on two alternative formulations of the maximum loading condition problem described above. The first alternative consists of eliminating the constraints (1), (2), (8)–(10), and (13)–(16), from the original problem, and including constraints that directly limit the maximum and minimum reactive power production, as follows:

$$\begin{array}{cc}\hfill & q^{\mathrm{G}}\le Q^{\mathrm{G},\max }\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & q^{\mathrm{G}}\ge Q^{\mathrm{G},\min }\hfill \end{array}$$

(20)

This is the way to handle the generator output limitations in traditional OPF formulations. The modified version of the maximum loading condition problem is solved for $Q^{\mathrm{G},\max }=0.436$ pu and $Q^{\mathrm{G},\min }=-0.436$ pu. In this case, the optimization problem increases the load until an active or a reactive power limit of the plant is found. The solutions obtained for each value of $\phi$ correspond to the blue crosses in Figure 3. It can be seen that this approximation undervalues the capabilities of the hydroelectric plant in both the overexcited operating zone and the underexcited operating zone, since the entire gray area between the outside of the “rectangle” limits and the limits of the capacity curve is overlooked by this modeling approach.

The second alternative corresponds to the approximate modeling of the capability curve where the stability limits of the generator are not included. For this, the problem maintains the original formulation, but removing constraints (13)–(16). The resulting maximum loading condition problem is solved again for each value of $\phi$. The plant operating points corresponding to the solutions obtained are indicated as red crosses in Figure 3. It can be seen that when the power system requires the power plant to absorb high levels of reactive power (underexcited operation), the optimization problem yields solutions corresponding to operating points outside the secure (gray) operating area, thereby violating the stability margin of the generator. This poses a risk to both the integrity of the electric machine and the operation of the power system. Note that under these operating conditions, a disturbance could cause the protections of the generator to decouple the power plant from the system.

From the previous discussions, the advantages of using the proposed model in decision-making tools based on OPF formulations can be discerned. For example, from the perspective of a power system operator, the proposed model provides a more accurate understanding of the actual reactive power reserves available in the managed system and enables better quantification of the effect of potential contingencies on the network voltage levels. This allows for better-informed decision making, which can result in lower operating costs and reduced risk in terms of system stability. In the context of electricity markets, hydroelectric plant owners have at their disposal an improved OPF model to design their strategies for participating in ancillary services markets related to reactive power and voltage control, thus enabling higher profits.

6. Conclusions

In this study, a model has been proposed to explicitly represent the steady-state stability limit of a salient-pole synchronous generator in OPF problems. This model allows for an improved representation of the capacity curve of hydro-power plants, enhancing the quality of the solutions provided by these optimization problems. The proposed model was tested using a maximum loading condition problem, verifying that the suggested formulation is capable of reproducing the desired operating limitations. The same optimization problem was solved by substituting the proposed formulation with two approximate formulations commonly used in the technical literature, namely, (i) constant (maximum and minimum) production limits, and (ii) a capability curve without considering the stability limits of the generator. From the obtained results, it is concluded that the formulation based on constant limits underestimates the production capacity of the hydro-power plant, not allowing its full utilization. On the other hand, the formulation that ignores the stability limits allows solutions corresponding to operating points that violate the generator security margins. Both results highlight the advantages of the proposed formulation. Immediate future work will focus on developing a model to represent the capability curve of pumped hydroelectric plants and analysis of its use in decision-making tools based on OPF formulations.