Adaptive Bi-Level Planning of Photovoltaic Hosting Capacity for Hydro-Dominant Distribution Grids Considering Hydraulic Safety Constraints

Abstract

Hydro-dominant distribution grids with high penetrations of distributed photovoltaic (PV) generation exhibit a clear operational asymmetry. PV output changes rapidly at the minute scale, whereas hydropower regulation is constrained by reservoir water balance, turbine ramping capability, and hydraulic safety limits. During high-inflow periods, mandatory hydropower generation further reduces the downward regulation margin and restricts midday PV accommodation. To address this issue, this paper develops an asymmetry-aware adaptive bi-level planning framework for photovoltaic hosting capacity (PVHC) assessment. A db4 discrete wavelet transform is used to decompose PV output into low-frequency energy trends and high-frequency fluctuation components. The upper layer performs hourly economic dispatch while maintaining reservoir water balance, and the lower layer conducts minute-level constrained tracking under ramping and vibration-zone avoidance constraints. A bisection-type capacity-search procedure is then used to identify the PVHC boundary by jointly checking curtailment, ramping, frequency proxy, voltage, line-loading, point-of-common-coupling exchange, and vibration-zone residence constraints. Case studies based on a 15 min PV dataset from a 30 MW station, hydropower operation records, and a modified 15-node feeder in Southwest China show that hydrological asymmetry materially affects PV accommodation. The obtained PVHC ranges from 53.17 MW under the most restrictive high-proxy condition to 65.33 MW under low-proxy operation. Compared with the no-coordination case, representative-month PVHC increases from 49.80 MW to 65.33 MW, while the simulated residence time within the predefined vibration-prone zone decreases from 447 min to 0 min. These results indicate that PVHC evaluation in hydro-dominant feeders should jointly consider electrical constraints, hydrological asymmetry, and hydraulic safety limits.

1. Introduction

Against the backdrop of increasingly urgent low-carbon energy transitions, the development and utilization of renewable energy have become central to power-system planning worldwide [1,2]. Owing to its modular deployment and mature conversion technology, photovoltaic (PV) generation has expanded rapidly in recent years [3]. However, high PV penetration introduces intermittency, steep ramping, voltage rise, and reverse-power-flow risks, which complicate secure operation after large-scale grid integration [4,5,6,7,8]. Therefore, accurately assessing the capacity of a distribution grid to accommodate PV generation, namely photovoltaic hosting capacity (PVHC), has become an important planning problem [9,10].

Among various strategies to smooth PV output fluctuations and improve PVHC, energy storage systems and hybrid storage configurations are widely used because they can provide fast active-power compensation [11,12,13]. Nevertheless, electrochemical storage remains constrained by investment cost, aging, state-of-charge management, and cell-level balancing requirements [14,15]. These limitations motivate the use of existing hydropower flexibility where hydrological and mechanical operating constraints can be represented explicitly.

Hydropower provides a mature and flexible regulating resource with quasi-energy-storage characteristics, because reservoir storage supports long-duration energy balancing and turbine-governor action can supply shorter-term power regulation [16,17,18]. Relying on a hydro-PV hybrid system can therefore mitigate PV volatility while exploiting hydropower flexibility to enlarge feasible PV accommodation scenarios [19,20,21].

Existing studies on hydro-PV hybrid systems primarily address optimization problems at a single time scale, such as day-ahead economic dispatch or fast frequency-control studies [22,23,24]. Nevertheless, PV output volatility presents distinct multi-time-scale features, involving both energy imbalance over long durations and power shocks at short time scales [25]. A single-level control strategy may therefore underestimate the usable flexibility of hydropower resources or overlook hydraulic safety when estimating PVHC.

Hierarchical scheduling has recently been used to separate steady-state economic operation from fast feedback regulation in active distribution networks and renewable microgrids [26,27]. This idea is relevant to hydro-dominant feeders because the low-frequency component of net load is more naturally assigned to reservoir energy management, whereas the high-frequency component is better treated as a power-tracking task. However, directly transferring this architecture to hydropower-dominated systems remains nontrivial, because turbine nonlinearity, hydraulic transients, and operation near vibration-prone zones may increase mechanical risk [28,29,30].

The remaining research gap is therefore not only whether hydropower can smooth PV fluctuations, but how PVHC can be assessed when temporal asymmetry, hydrological asymmetry, and hydraulic safety constraints are represented in the same planning framework. To bridge this gap, this paper proposes an adaptive bi-level planning framework for hydro-dominant distribution grids.

The hydro-PV system studied here is asymmetric in time scale because reservoir energy dynamics evolve slowly, whereas PV fluctuations appear at minute-level resolution; it is also asymmetric in resource availability because high-inflow months force hydropower units toward higher base output and reduce downward regulation margin. The proposed controller is therefore interpreted as a symmetry-restoration mechanism that reallocates low-frequency energy and high-frequency power disturbances while respecting nonconvex hydraulic safety intervals.

The contributions of this paper are as follows:

• A spectral-decoupled bi-level dispatch framework is proposed to coordinate long-term reservoir energy scheduling and short-term photovoltaic fluctuation smoothing.
• A vibration-zone avoidance constraint is embedded into the lower-layer control model to reduce operation within predefined disjoint unsafe power regions.
• An iterative PVHC assessment method is developed by integrating hydropower ramping limits, curtailment constraints, frequency-proxy limits, and distribution-network security constraints.

2. Materials and Methods

To quantify PVHC under multi-time-scale and hydrological asymmetries, this study proposes an asymmetry-aware dispatch-and-tracking framework composed of an hourly energy-management layer, a minute-level power-regulation layer, and a PVHC boundary-search module. As illustrated in Figure 1, the framework decouples the hydro-PV dispatch problem into an upper energy-management layer and a lower power-regulation layer. The upper layer performs hourly predictive dispatch and coordinates reservoir water balance with economic operation, while the lower layer operates at 1 min resolution as a constrained tracking controller for high-frequency PV deviations within the remaining hydropower ramping margin. Embedded in a PVHC boundary-search algorithm, the framework identifies the maximum allowable PV integration level subject to hydraulic coupling, predefined vibration-zone avoidance, and feeder-security constraints. Simulations and graphical post-processing were conducted using MATLAB R2025a (The MathWorks, Inc., Natick, MA, USA).

2.1. PV Uncertainty Decomposition Model

Due to random disturbances such as cloud shading and atmospheric flow, photovoltaic generation exhibits significant non-stationary characteristics. As illustrated in Figure 2a, this behavior is characterized by the superposition of diurnal energy trends (low-frequency components) and stochastic power fluctuations (high-frequency components). Traditional signal decomposition methods often fail to effectively separate these two components. For instance, empirical mode decomposition is prone to mode mixing, whereas variational mode decomposition depends on preset parameters and lacks strict orthogonality.

To address these complexities, this study constructs an uncertainty decomposition model using the discrete wavelet transform based on multi-resolution analysis. Daubechies db4 wavelets are selected because their compact support is suitable for capturing sharp PV transients, and the measured 15 min PV profile from the 30 MW station is interpolated to the 1 min lower-layer control interval before decomposition. The original PV sequence $P_{\mathrm{PV}}\left(t\right)$ is decomposed into the level-J approximation and the reconstructed detail component as

$$P_{\mathrm{PV}}\left(t\right)=P_{\mathrm{PV}}^{\mathrm{Lf}}\left(t\right)+P_{\mathrm{PV}}^{\mathrm{Hf}}\left(t\right)=\sum _k{c}_{J,k}{\varphi }_{J,k}\left(t\right)+{\displaystyle \sum _{j=1}^J}\sum _k{d}_{j,k}{\psi }_{j,k}\left(t\right).$$

(1)

In Equation (1), ${\varphi }_{J,k}$ and ${\psi }_{j,k}$ denote the scaling and wavelet functions, respectively. The term $P_{\mathrm{PV}}^{\mathrm{Lf}}\left(t\right)$ captures the deterministic energy trend used by the upper-layer dispatch, whereas $P_{\mathrm{PV}}^{\mathrm{Hf}}\left(t\right)$ represents minute-scale stochastic deviations that must be handled by fast hydro tracking. The spectral properties and statistical distribution are illustrated in Figure 2b,c.

2.2. Energy-Power Decoupled Hydropower Model

The hydropower plant is represented through an energy-power decoupled abstraction: the reservoir provides slow energy buffering, whereas the turbine-generator unit supplies short-term active-power regulation. This treatment follows the same modeling logic used in hybrid energy-storage coordination, but it keeps the hydraulic constraints that are essential for a hydro-dominant feeder.

2.2.1. Reservoir Energy Balance

As a large-capacity energy buffering unit, the reservoir serves as the core of the upper-layer constraint framework. By regulating the storage and release of water flow over an extended time window, the reservoir achieves the spatiotemporal transfer of hydraulic resources and maintains full-cycle energy balance, as illustrated in Figure 3.

The reservoir continuity equation is expressed by Equations (2) and (3), replacing the former ambiguous reference to “Expression (2–3)”.

$$\begin{array}{cc}\hfill V_{k+1}& =V_k+\left(Q_{i,k}-Q_{g,k}-Q_{s,k}\right)\Delta T,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill V_{\min }& \le V_k\le V_{\max },0\le Q_{g,k}\le Q_g^{\max },0\le Q_{s,k}\le Q_s^{\max }.\hfill \end{array}$$

(3)

Here, $V_k$ denotes the reservoir storage at hour k, while $Q_{i,k}$, $Q_{g,k}$, and $Q_{s,k}$ represent natural inflow, generation discharge, and spillage, respectively. These equations enforce mass conservation and provide the long-term hydraulic coupling mechanism required for PVHC assessment.

2.2.2. Hydro-Turbine Power Regulation

The hydro-turbine generator serves as the power-regulation element of the system. The hydropower output $P_H\left(t\right)$ is determined by the net head $H_n\left(t\right)$ and generation flow $Q_g\left(t\right)$, as shown in Figure 4.

Because reservoir storage is large relative to minute-level dispatch adjustments, short-horizon variations in the hydraulic head are assumed to be small in the dispatch layer. Accordingly, the hydropower production relationship is described by

$$P_H\left(t\right)={\eta }_{w}g{H}_n\left(t\right)Q_g\left(t\right)\approx k{Q}_g\left(t\right),$$

(4)

where k denotes the local hydropower production coefficient around the operating point. This approximation is used only for dispatch-level PVHC assessment. The corresponding linearization error is summarized in Figure 5.

The power support capability of the hydro-turbine is bounded by capacity limits and ramping limits:

$$\begin{array}{cc}\hfill P_{H,\min }& \le P_H\left(\tau \right)\le P_{H,\max },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill -R_d& \le {\displaystyle \frac{P_H\left(\tau \right)-P_H(\tau -1)}{\Delta t}}\le R_u.\hfill \end{array}$$

(6)

In Equations (5) and (6), $\Delta t$ denotes the fast lower-layer time step, which is 1 min in the case study. The simulated hydro unit is rated at 50 MW, the steady-state dispatchable range is 5–50 MW before the hydrological forced-output adjustment, and $R_u=R_d=2.0$ MW/min. The predefined vibration-prone interval is 18–24 MW, so the feasible safe operating zones are [5, 18] MW and [24, 50] MW under the representative setting (Figure 6). The case-study parameters are summarized in

Table 1. Main hydropower, feeder, and simulation parameters used in the case study.

Parameter	Value	Purpose or Interpretation
Rated capacity of the measured PV station	30 MW nominal	Used to build normalized PV profiles; PVHC is a scalable planning boundary, not the present installed capacity.
Rated capacity of the simulated hydropower unit	50 MW	Rated active-power capacity of the representative hydropower unit.
Basic dispatchable hydropower range	5–50 MW	Steady-state output range before applying the forced minimum-output adjustment and excluding the vibration-prone interval.
Forced minimum hydro output	9.2–17.6 MW	Monthly value mapped from the desensitized inflow proxy.
Generation-flow limit	70 m3/s	Maximum turbine discharge used in the production model.
Head range	60–100 m	Desensitized operating-head range used for dispatch-level calculation.
Reservoir storage range	500–3000 Mm3	Upper-layer energy-storage state constraint.
Turbine efficiency	0.88	Steady-state production coefficient in Equation (4).
Ramp-up/ramp-down rate	2.0 MW/min	Lower-layer turbine response limit.
Vibration-prone interval	18–24 MW	Predefined prohibited partial-load region for the simulated unit.
Upper/lower time steps	1 h/1 min	Hourly reservoir dispatch linked to minute-level power tracking; $A_T=60$.
Frequency-proxy model	$f_0=50$ Hz, $T_f=3600$ s, $D=1.5$, base = 100 MVA	Dispatch-level screening proxy; not an electromechanical transient model.
PCC limits	+35 MW export/−45 MW import	External-grid exchange corridor.

.

The measured PV output data were obtained from an existing 30 MW PV station. For PVHC assessment, the measured profile is first normalized by the station rating and then scaled to each candidate PV capacity in the boundary search. Thus, the reported PVHC is a scalable planning boundary rather than the existing installed capacity.

2.3. Proposed Two-Layer Coupled Control Strategy

To address the spatiotemporal contradiction encountered during PV accommodation, this paper proposes a two-layer cooperative dispatch-and-tracking strategy. The upper layer uses a 24 h rolling horizon with a 1 h scheduling step to maintain reservoir water balance and issue the hourly reference $P_H^r\left(k\right)$. The lower layer then holds the current hourly reference over the corresponding 60 one-minute intervals and corrects it using the high-frequency PV component $P_{\mathrm{PV}}^{\mathrm{Hf}}$. In this way, the two layers are coupled through a sample-and-hold reference trajectory and receding-horizon feedback rather than operating as independent optimizers. The control architecture is illustrated in Figure 7.

2.3.1. MPC-Based Upper-Layer Economic Dispatch

The upper-layer energy-management system adopts a rolling-horizon mechanism based on model predictive control. The prediction horizon $N_p$ is set to 24 h, and the scheduling time step $\Delta T$ is set to 1 h. At each decision instance, the system solves for the optimal control sequence based on the latest state feedback and applies the first dispatch instruction.

The objective function introduces the hydroelectric unit wear cost $C_w$ and the energy-curtailment cost $C_p$:

$$\min J={\displaystyle \sum _{k=t}^{t+N_p-1}}\left[C_w\left(k\right)+C_p\left(k\right)\right].$$

(7)

The regulation-wear term is

$$C_w\left(k\right)=c_w\left|P_H^r\left(k\right)-P_H^r(k-1)\right|,$$

(8)

where $c_w=0.5$ in the normalized simulation objective. For the point-of-common-coupling (PCC) convention, positive $P_{\mathrm{PCC}}$ denotes export to the external grid. The exchange corridor is constrained by +35 MW export and −45 MW import limits.

The curtailment and spillage cost is defined as

$$C_p\left(k\right)={\lambda }_s{Q}_s\left(k\right)+{\lambda }_c{P}_c\left(k\right),$$

(9)

where $P_c\left(k\right)$ denotes curtailed PV power and $Q_s\left(k\right)$ denotes spillage flow. In the case study, ${\lambda }_s={\lambda }_c=1$ after per-unit normalization. Because the final PVHC boundary is determined by explicit security constraints, these weights mainly affect the selection of smooth feasible trajectories rather than the stopping thresholds.

Because the feeder can exchange power with the external utility grid, the upper-layer balance is written through the PCC exchange variable:

$$P_{\mathrm{PCC}}\left(k\right)=P_H^r\left(k\right)+P_{\mathrm{PV}}^{\mathrm{Lf}}\left(k\right)-P_c\left(k\right)-P_l\left(k\right).$$

(10)

The final output of the upper layer is the reference trajectory $P_H^r\left(k\right)$, which guides the hydropower unit in smoothing PV power fluctuations while maintaining reservoir feasibility.

2.3.2. Lower-Layer Control: Real-Time Fluctuation Smoothing and Turbine Regulation

The lower-layer control operates on a 1 min time scale to address the stochastic high-frequency component of PV power. Its target hydro output is constructed as

$$P_H^t\left(\tau \right)=P_H^r\left(\tau \right)-\xi \left(\tau \right),$$

(11)

where $\xi \left(\tau \right)$ is the high-frequency PV prediction deviation. Considering hydraulic inertia and mechanical transmission constraints, the actual output approximates the target within the feasible region:

$$\min {\left(P_H^r\left(\tau \right)-P_H^t\left(\tau \right)\right)}^2.$$

(12)

To avoid operation in the vibration-prone interval ${\mathsf{\Omega }}_v$, the model imposes

$$P_H^r\left(\tau \right)\notin {\mathsf{\Omega }}_v.$$

(13)

This nonconvex feasible set is represented using binary variables $z_m\in \{0,1\}$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{m=1}^M}{z}_m& =1,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill z_m{P}_{\min ,m}& \le P_{H,m}\left(\tau \right)\le z_m{P}_{\max ,m},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill P_H^r\left(\tau \right)& ={\displaystyle \sum _{m=1}^M}{P}_{H,m}\left(\tau \right).\hfill \end{array}$$

(16)

Here, M is the number of safe operating zones and $[P_{\min ,m},P_{\max ,m}]$ is the power range of the mth safe zone.

2.4. PVHC Boundary-Search Algorithm

Building upon the bi-level dispatch-and-tracking model, this section evaluates PVHC as a constrained boundary-identification problem. PVHC is defined as the maximum scalable PV active-power capacity that the feeder can accommodate without violating hydraulic safety, curtailment, ramping, frequency proxy, voltage, line-loading, and PCC-exchange constraints. The measured 30 MW PV station is therefore used as a normalized profile source, and the PV capacity is scaled during the search.

The original step-increment description is replaced by a bisection-type boundary search, which is consistent with the exported simulation results and avoids unnecessary iterations near the feasibility boundary. The lower and upper PV capacity bounds are initialized as $P_{\mathrm{low}}=0$ MW and $P_{\mathrm{high}}=150$ MW, respectively, and the search tolerance is set to $\epsilon =0.25$ MW. In each iteration, the candidate capacity is $P_{\mathrm{mid}}=(P_{\mathrm{low}}+P_{\mathrm{high}})/2$. If all security and economic criteria are satisfied, $P_{\mathrm{low}}$ is updated to $P_{\mathrm{mid}}$; otherwise, $P_{\mathrm{high}}$ is updated to $P_{\mathrm{mid}}$. The process stops when $P_{\mathrm{high}}-P_{\mathrm{low}}\le \epsilon$, and $P_{\mathrm{low}}$ is reported as the PVHC.

Frequency-proxy security is evaluated by a dispatch-level screening proxy rather than a detailed electromechanical transient model. The fast active-power imbalance is obtained by removing a 30 min moving average from PCC exchange power, normalizing it on a 100 MVA base, and propagating it through a first-order response with $f_0=50$ Hz, $T_f=3600$ s, and $D=1.5$. The adopted threshold is $|\Delta f_{\mathrm{proxy}}|\le 0.20$ Hz. The boundary-search process is summarized in Figure 8.

3. Results

Empirical validation is conducted on a modified 15-node radial distribution feeder, rather than on an abstract test network, so that the PVHC boundary can be assessed under hydropower-dominated operating conditions. The network topology, retained from the submitted manuscript, is shown in Figure 9. It represents a desensitized small-hydropower-dominated distribution system located in Yunnan, Southwest China. Because of the high resistance-to-reactance ratio and restricted upstream transmission corridor, voltage security, line thermal limits, PCC exchange, and hydraulic safety jointly shape the feasible PVHC boundary.

3.1. Seasonal Interplay of Hydro and PV Resources

The hydrological condition is represented by a desensitized inflow proxy derived from nonnegative monthly hydropower operation records after smoothing and robust scaling. Figure 10 has been re-exported with larger axis labels, thicker lines, and a vector source file so that the month labels and dual-axis units remain readable in the final MDPI layout. The results indicate a clear asymmetric resource effect: low-proxy months retain relatively high PVHC, with the representative January value reaching 65.33 MW, whereas the high-proxy August–September period reduces feasible hosting capacity to 55.81 MW and 53.17 MW, respectively.

The hydrological asymmetry indicator is defined as $A_H=P_{H,f}/P_r$, where $P_{H,f}$ is the monthly forced minimum hydropower output and $P_r=50$ MW. As $A_H$ increases, forced hydropower output occupies a larger share of unit capacity and reduces downward regulation margin for midday PV accommodation. Based on

Table 2. Seasonal hydrological asymmetry indicator and proposed PVHC.

Month	Hydrological Proxy (m3/s)	${\mathit{A}}_{\mathit{H}}$	Proposed PVHC (MW)
Jan	12.23	0.19	65.33
Feb	12.00	0.18	64.75
Mar	13.42	0.20	65.19
Apr	13.19	0.20	63.87
May	14.14	0.21	60.35
Jun	16.30	0.24	56.69
Jul	23.09	0.32	61.67
Aug	26.00	0.35	55.81
Sep	25.56	0.35	53.17
Oct	20.13	0.28	61.38
Nov	17.23	0.25	64.31
Dec	16.12	0.23	64.45

, the Pearson correlation coefficient between $A_H$ and PVHC is approximately −0.753, indicating a negative relationship between hydrological asymmetry and PV accommodation capability.

3.2. Efficacy of Multi-Scale Spectral Decomposition

The fundamental mechanism of the proposed control architecture relies on the effective spectral decoupling of low-frequency energy flows from high-frequency power fluctuations. Figure 11 verifies this signal decomposition mechanism. The raw photovoltaic power profile exhibits a diurnal envelope with stochastic volatility induced by cloud movements and atmospheric turbulence. Direct injection of such unprocessed stochastic signals into the electrical grid may increase nodal-voltage and power-balance risks.

In contrast, Figure 11c characterizes the high-frequency component comprising zero-mean stochastic fluctuations and rapid ramping events. With amplitudes distributed between approximately −10 MW and +15 MW, this signal requires agile regulation capability. Feeding this component to the lower-layer tracking controller reduces the need to solve the full global optimization problem at every fast control interval.

3.3. Dynamic Power Balance and Grid Safety Verification

Figure 12 presents detailed dispatch results for the representative low-inflow month selected from the simulation outputs. During the midday interval, PV generation surges and hydropower output is guided toward the safe lower operating region, creating accommodation headroom while avoiding excessive residence in the vibration-prone interval. The PCC exchange remains within the adopted export/import corridor after PV curtailment and hydro tracking are applied. The operating security checks are summarized in

Table 3. Operating security indicators of the proposed method in the representative month.

Indicator	Value	Limit	Status
Max PCC export	35.00 MW	35 MW	Satisfied
Max PCC import	17.00 MW	45 MW	Satisfied
Max voltage	1.039 p.u.	1.05 p.u.	Satisfied
Min voltage	0.981 p.u.	0.95 p.u.	Satisfied
Max line loading	83.33%	100%	Satisfied
PV curtailment	4.93%	5%	Satisfied
Max frequency-proxy deviation	0.143 Hz	0.20 Hz	Satisfied
Ramping violation	0.00%	1%	Satisfied
Vibration-zone residence	0 min	2% of steps	Satisfied
Runtime	0.033 s	Reported	Reported

.

All indicators remain within the adopted PVHC stopping thresholds. Because PVHC is obtained by bisection boundary identification, the reported value corresponds to the last feasible capacity before any constraint violation occurs. The search accuracy is determined by $\epsilon =0.25$ MW. The frequency-proxy sensitivity is quantified in

Table 4. Sensitivity of representative-month PVHC to the adopted frequency-proxy threshold.

Frequency-Proxy Limit (Hz)	PVHC (MW)	Max $\mathbf{|}\mathbf{\Delta }{\mathit{f}}_{\mathbf{proxy}}\mathbf{|}$ (Hz)	Curtailment (%)	Max Import (MW)
0.10	50.83	0.100	0.00	17.0
0.15	65.33	0.143	4.93	17.0
0.20	65.33	0.143	4.93	17.0
0.25	65.33	0.143	4.93	17.0
0.30	65.33	0.143	4.93	17.0

.

The sensitivity test indicates that a very strict ±0.10 Hz proxy threshold reduces PVHC to 50.83 MW. For thresholds from ±0.15 Hz to ±0.30 Hz, PVHC remains 65.33 MW because curtailment, PCC exchange, and hydraulic constraints become the active limiting conditions. Thus, the reported result is not determined solely by an arbitrary ±0.20 Hz choice. The same trend is visualized in Figure 13.

3.4. Vibration-Zone Avoidance and Mechanical Safety

A key constraint in hydropower operation is the avoidance of nonconvex vibration-prone zones, where hydraulic instability may increase mechanical stress on the turbine runner. Figure 14 verifies the vibration-zone avoidance strategy from both statistical and time-domain perspectives. The predefined prohibited interval of 18–24 MW is treated as a dispatch-level engineering constraint derived from desensitized operation and maintenance restrictions for a representative Francis-type small hydropower unit, rather than as a universal hydraulic-instability boundary.

Figure 14a shows that the hydropower active-power distribution is concentrated in two safe operating regions around the predefined vibration-prone interval. Figure 14b provides a temporal examination of the avoidance mechanism; the controller keeps the accepted operating points outside the 18–24 MW interval and prevents sustained residence within the prohibited zone. The connecting line in the plot is a visual interpolation between minute-level points and should not be interpreted as a physically validated continuous crossing trajectory inside the prohibited region. In the proposed case, the calculated residence time inside the interval is 0 min.

To quantify the gain of the proposed formulation,

Table 5. Comparative PVHC and security indices for the representative month.

(a) Capacity and Feeder-Security Indicators
Case	PVHC (MW)	Curt. (%)	Voltage Range (p.u.)	Line Loading (%)
No coordination	49.80	4.93	0.996–1.036	83.33
Single-layer MPC	65.77	4.92	0.995–1.039	83.33
Bi-level tracking w/o guidance	54.93	0.40	0.989–1.039	83.33
Proposed	65.33	4.93	0.981–1.039	83.33
(b) Dynamic and Hydraulic-Security Indicators
Case	Max $\mathbf{|}\mathbf{\Delta }{\mathit{f}}_{\mathbf{proxy}}\mathbf{|}$ (Hz)	PCC Export/Import (MW)	Ramp Viol. (%)	Vib. (min)
No coordination	0.072	35.0/0.3	0.00	447
Single-layer MPC	0.088	35.0/0.2	0.00	154
Bi-level tracking w/o guidance	0.200	35.0/10.1	0.00	180
Proposed	0.143	35.0/17.0	0.00	0

compares four representative PVHC assessment/control cases: no coordinated control, single-layer MPC, bi-level tracking without vibration-zone guidance, and the proposed bi-level method with vibration-zone avoidance. These baselines represent common static or single-layer PVHC screening approaches and an ablation of the proposed hydraulic-safety module.

The proposed method raises representative-month PVHC by 15.53 MW compared with no coordinated control while keeping voltage, line loading, frequency proxy, PCC exchange, ramping, and curtailment within their limits. Although the single-layer MPC obtains a slightly higher PVHC, it leaves 154 min of residence within the predefined vibration-prone zone. The proposed method reduces this residence to zero with only a 0.44 MW, or approximately 0.67%, PVHC reduction relative to the single-layer case. The bi-level tracking case without vibration-zone guidance removes the avoidance mechanism from the tracking controller; its trajectory can therefore trigger other operational constraints earlier, especially the frequency-proxy security constraint, which explains why its PVHC is lower than that of the proposed guided controller. The comparison is further illustrated in Figure 15.

4. Discussion

The proposed two-layer dispatch-and-tracking framework addresses the temporal asymmetry between slow reservoir energy evolution and short-term stochastic PV fluctuations by separating energy scheduling from power tracking. The time-scale indicator $A_T=60$ confirms that the two layers operate on strongly different temporal resolutions, while the hydrological indicator $A_H$ explains why high-inflow months reduce the downward regulation margin. In the symmetry framing, the controller uses this quantified asymmetry to restore long-horizon energy balance and short-horizon power balance subject to hydraulic safety limits.

Simulation results underscore the importance of incorporating hydraulic safety constraints, ramping limits, frequency-proxy security checks, PCC exchange limits, and distribution-network security checks into PVHC assessment. The frequency-threshold sensitivity in Table 4 shows that the adopted ±0.20 Hz proxy is not the sole determinant of the reported PVHC, while the comparison in Table 5 shows that the single-layer MPC obtains a slightly higher PVHC but leaves non-negligible vibration-zone residence. The proposed formulation therefore prioritizes hydraulic safety with negligible sacrifice in hosting capacity, improving the quality of the feasible operating trajectory rather than merely maximizing the installed PV boundary.

Several limitations should be stated explicitly. First, the hydro production function is linearized around the normal operating range, and detailed hydraulic transient dynamics are not fully represented. Second, the vibration-prone zone is modeled as a fixed forbidden interval, whereas its boundary may vary with water head, guide-vane opening, discharge, unit aging, and turbine characteristics. Third, the feeder-security model uses a radial LinDistFlow screening representation and is not equivalent to a full three-phase unbalanced AC power-flow study. Fourth, the frequency-proxy response is a dispatch-level screening model and should not be interpreted as an electromagnetic or electromechanical transient simulation. Fifth, the inflow variable, raw PV measurements, hydropower operation records, and branch-level feeder parameters are desensitized from field records; future work should use measured discharge, reservoir head, vibration, and complete branch-level data when confidentiality restrictions allow.

5. Conclusions

This paper proposes an asymmetry-aware adaptive bi-level planning framework for assessing PV hosting capacity in hydro-dominant distribution grids under hydraulic safety constraints. The framework separates low-frequency PV energy trends from short-term fluctuations, coordinates reservoir energy management in the upper layer, and performs minute-level hydro tracking with ramping and vibration-zone avoidance constraints in the lower layer.

The case study on a modified 15-node feeder indicates that hydrological asymmetry materially affects PV accommodation. In the tested simulation, the proposed method yields 65.33 MW PVHC under the representative low-proxy condition and 53.17–55.81 MW under the most restrictive high-proxy conditions. In the representative comparison month, PVHC increases from 49.80 MW under no coordinated control to 65.33 MW under the proposed method, while modeled vibration-zone residence decreases from 447 min to 0 min.

The engineering implication is that hydropower flexibility can be treated as a dispatch-level resource for PV accommodation only when hydraulic safety and feeder security are modeled explicitly. Future work should validate the framework with larger real feeders, complete AC power-flow data, detailed hydraulic-transient models, hardware-in-the-loop tests, and broader benchmark strategies.