Hybrid Analytical--EMT Method for HVDC Protection System Component-Level Design

Abstract: Protection system design for multi-terminal HVDC grids is challenging due to the complexity of the system and the often conflicting design requirements. Effective specification of protection component parameters (e.g., DC circuit breakers and series DC inductors) during component-level design is crucial due to interdependencies among components, the need for detailed modeling, and the complex interactions between the protection system and converter control systems. Both analytical and simulation-based approaches have been proposed as solutions for component-level design. However, analytical methods may not accurately represent system behavior given that approximation is necessary, and simulation-based approaches often require extensive computational effort and time. Therefore, this paper presents an efficient systematic design method, combining both approaches. First, a fundamental analytical solution is derived to consider the protection system requirements. Then, a hybrid analytical--EMT methodology is proposed to accelerate convergence toward the required design parameters, after which detailed models are applied to ensure accuracy in design and validation. The approach is applicable to component-level design for both fully and partially selective protection strategies in HVDC grids.

• The paper introduces a hybrid analytical–EMT design method that uses closed-form inductance estimates, targeted simulations, and critical-case selection to co-design DC inductors, DCCBs, and converter overcurrent limits.
• The method produced final inductances of roughly 50–450 mH across case studies, with requirements strongly influenced by DCCB operation time, interruption capability, converter overcurrent factor, grid configuration, and fault location.
• The paper reduced computational effort to at most six EMT iterations for single-inductor designs and averages of nine and 88 iterations across two case studies, compared with hundreds or thousands for simulation-only approaches.

Motivation and problem statement

Component-level design of protection systems for multi-terminal HVDC grids requires the joint specification of DC circuit breaker (DCCB) parameters, series DC inductors, and converter overcurrent capabilities. These quantities are interdependent: a slower or lower-rated DCCB demands more inductance, while larger inductors raise installation cost and can degrade system stability. Industrial practice reflects this tension — the Zhangbei project uses 150–200 mH reactors, while other studies propose values exceeding 500 mH. The paper addresses the gap between purely analytical sizing methods, which rely on approximations that may misrepresent system behavior, and exhaustive EMT-simulation-based methods, which require hundreds to thousands of iterations.

The proposed method takes as inputs the outputs of system-level design — topology, protection strategy (fully, partially, or non-selective), zone boundaries, and converter DC fault ride-through (DC-FRT) requirements — and produces component-level specifications, with the DC inductor treated as the primary design variable since DCCB operation time and converter overcurrent capacity are constrained by available technology.

Analytical derivation of design constraints

The method derives two approximate inductance requirements per protection zone. For converter requirements, the bus voltage at the DCCB location is expressed via an inductor relation over the neutralization time $t_n = t_{relay} + t_{cb}$, and equated to a KVL expression from the converter terminal, where the converter is modeled as a constant voltage source behind an equivalent inductance $L_{eq} = L_{con} + L_{cab}$ with $L_{con} = (2/3)L_{arm}$. This yields:

$$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$

where $\Delta I_{in}$ (adjacent-zone infeed) and $\Delta I_{cab}$ (connection cable discharge) are not computed analytically but measured from EMT simulations — this is the key hybrid element. The critical converter current change is set by the worst-case operating point: rated rectifier current with maximum reactive power, giving $\Delta I_{con}^c = I_{con}^m - I_{con}^r$, with $I_{con}^m$ derived from the arm current limit and AC current controller limit. The DCCB requirement is approximated as:

$$L_{dc}^{cb} \approx \frac{(U_{dc}-\bar{U}_{m})\, t_n}{I_{cb}^m - I_{cb}^r}$$

Two structural consequences follow directly from these expressions. First, the absence of adjacent cables and connection cables (i.e., $\Delta I_{in} = \Delta I_{cab} = 0$) maximizes the required inductance, so this condition defines the conservative initial estimate. Second, the required inductance depends on grid configuration, not only on component ratings.

Critical case identification

To minimize simulation count, the method identifies three classes of critical conditions. Critical converters are those reaching their overcurrent limits fastest for out-of-zone faults; when converters share parameters, the one connected directly to the DCCB or via the shortest line is critical due to low wave propagation delay and small $L_{eq} = L_{con} + L_{cab}$0. Critical power flows are those maximizing loading of the critical converter in rectifier mode and of the DCCB under study. Critical faults are pole-to-ground faults with zero resistance; critically, the worst-case location depends on the neutralization time relative to a critical time $L_{eq} = L_{con} + L_{cab}$1 at which the cable voltage envelope $L_{eq} = L_{con} + L_{cab}$2: if $L_{eq} = L_{con} + L_{cab}$3, a terminal fault governs ($L_{eq} = L_{con} + L_{cab}$4); otherwise a non-terminal fault must be considered because traveling-wave reflections produce fast current rise at intermediate locations. In the case studies, with DCCB operation times of at least 2 ms, terminal faults were governing.

Design algorithm

The core algorithm proceeds in three parts. Part A fixes system-level requirements and identifies critical cases. Part B iterates between the analytical expressions and EMT simulations: starting from the conservative estimate ($L_{eq} = L_{con} + L_{cab}$5), measured infeed and discharge currents update $L_{eq} = L_{con} + L_{cab}$6 until successive iterations agree within a tolerance (e.g., 5%). If the DCCB constraint dominates from the outset, no iteration is needed. Part C refines the value downward using a current-margin KPI — the normalized difference between each component's maximum current limit and the achieved current — terminating when the minimum of the converter and DCCB margins falls within a target band (5% in the case studies). Inductor reduction uses a dynamic rate bounded by $L_{eq} = L_{con} + L_{cab}$7 to avoid overshooting constraints. A generalized algorithm extends this to $L_{eq} = L_{con} + L_{cab}$8 zones by designing each inductor against both adjacent zones, taking the maximum, updating the system, and adding a final re-iteration pass to handle inter-inductor dependencies.

A notable practical feature is that any converter model — including black-box vendor models — can be used in the EMT stage, provided the scalar parameters $L_{eq} = L_{con} + L_{cab}$9, $L_{con} = (2/3)L_{arm}$0, and $L_{con} = (2/3)L_{arm}$1 are known.

Case study results

Two bipolar systems modeled in PSCAD with frequency-dependent 525 kV cables, Type-5 average HB-MMC models, CIGRE-representative controls, and Type-5 DCCB models were studied: a five-terminal partially selective system (three zones, two inductors) and a four-terminal fully selective meshed system (ten DCCBs/inductors, nine zones). Converter overcurrent factor $L_{con} = (2/3)L_{arm}$2 pu, DCCB interruption capability $L_{con} = (2/3)L_{arm}$3 kA, and operation time $L_{con} = (2/3)L_{arm}$4 ms defined twelve scenarios for Case Study 1.

Key quantitative findings include:

Scenario	$L_{con} = (2/3)L_{arm}$5	$L_{con} = (2/3)L_{arm}$6	$L_{con} = (2/3)L_{arm}$7	$L_{con} = (2/3)L_{arm}$8 (mH)	Limiting component
1	1.5	12 kA	2 ms	165	Converter
3	1.5	12 kA	5 ms	414	Converter
4	2	12 kA	2 ms	109	DCCB
10	2	24 kA	2 ms	80	DCCB

Across all scenarios, final inductances ranged from roughly 50 to 450 mH, confirming strong sensitivity to both DCCB technology and converter overcurrent rating. Several results deserve emphasis. First, for $L_{con} = (2/3)L_{arm}$9, Zone 1 was always limiting because its critical converter connects directly to the DCCB, whereas Zone 2's critical converter sits 250 km away, so cable propagation delay and discharge reduce its inductor demand. Second, for $$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$0, the limiting zone varied across input parameter sets — demonstrating that assuming a fixed limiting zone without full multi-zone analysis is unjustified. Third, in Case Study 2, inductors adjacent to two cables (e.g., $$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$1) required smaller values than those with one adjacent cable ($$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$2), quantifying the infeed-current effect embedded in the analytical expression. Fourth, a comparison of $$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$3 versus $$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$4 — identical except for converter control mode — indicated that DC-voltage droop control demands a larger inductor than PQ control, though the authors explicitly caution that different control implementations could reverse this observation.

Validation runs confirmed that designed inductors keep arm currents and DCCB currents below their limits for terminal faults, which produced the highest currents, and that non-terminal faults did not violate constraints for the considered breaker times.

Computational efficiency

The efficiency claim is the paper's strongest numerical result: the core algorithm converged in at most six EMT iterations per single-inductor design, with averages of nine iterations (Case Study 1) and 88 iterations (Case Study 2) per scenario, completing in minutes to about an hour. By contrast, simulation-only approaches cited in the literature require hundreds to thousands of iterations and can take days. The reduction stems from two mechanisms: analytical initialization narrows the search space before simulation begins, and critical-case identification eliminates evaluation of non-governing fault locations and operating points. An implication is that the method scales plausibly toward large multi-terminal grids where exhaustive EMT-based optimization becomes computationally prohibitive.

Limitations and open questions

Several limitations are stated or implicit. The analytical expressions rest on simplifications — constant converter internal voltage, lumped connection-cable inductance, constant voltage across the DC inductor — which bias the estimate conservatively; accuracy therefore depends on the Part C refinement rather than on the closed-form expressions themselves. The method requires knowledge of specific converter parameters ($$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$5, $$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$6, $$L_{dc}^{con} \approx \frac{(U_{dc}-\bar{U}_{m}) t_n - L_{eq}\,\Delta I_{con}}{\Delta I_{con} + \Delta I_{in} + \Delta I_{cab}}$$7), which may not be fully disclosed by all vendors. The converter-control-mode finding (droop control requiring larger inductance) is acknowledged as implementation-dependent and left as an open question requiring further investigation. The claim that inter-inductor dependency effects are negligible when each inductor has at most one neighbor is supported only for Case Study 1's topology and may not generalize to denser meshes. Finally, the case studies use Type-5 average converter models; while the authors argue vendor models can be substituted, no industrial-grade validation with proprietary black-box models is presented.

Conclusion

This paper contributes a systematic, computationally efficient procedure for co-designing DC inductors, DCCB parameters, and converter overcurrent allowances in multi-terminal HVDC protection systems, applicable to both fully and partially selective strategies. Its central mechanism — analytical initialization refined by targeted EMT simulation with critical-case pre-selection — reduces simulation counts by roughly an order of magnitude relative to simulation-only alternatives while retaining model fidelity through direct use of detailed EMT models. The demonstrated sensitivity of required inductance (roughly 50–450 mH) to DCCB speed, interruption rating, and converter overcurrent factor underscores that these components cannot be specified independently.